1000 automatically generated exercises with n-gram grammars

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}ab,{\rtimes}bb,aab,aba,abb,baa,bbb,bb{\ltimes},b{\ltimes}{\ltimes}\}\)

1. aab
2. bb
3. abb
4. bbb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,abd,caa,cba,dad,dcc,ddc\}\)

1. cba
2. dabcdbcd
3. dcdbacac
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaac,abaa,abba,abbb,cabb,cbab\}\)

1. \(\varepsilon\)
2. b
3. bcccccca
4. bcaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,acba,adac,beca,cbdb,cecb,dacd,ebcd\}\)

1. d
2. deddbed
3. baeeedd
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}d,ba,cc,cd,db,dc,a{\ltimes}\}\)

1. cabd
2. dba
3. edced
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}bd,{\rtimes}{\rtimes}da,{\rtimes}da{\ltimes},{\rtimes}bdb,{\rtimes}bcc,accc,bcca,cacc,ccac,cccd,ccdb,cdbc,bdb{\ltimes},dbc{\ltimes},db{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},da{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. bdb
2. dd
3. da
4. cca

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bab,ccb,dac,dba,dbd\}\)

1. acab
2. bbbdcbba
3. d
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,da,de,ed,ee\}\)

1. \(\varepsilon\)
2. cbab
3. e
4. bebbec

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}bbb,{\rtimes}b{\ltimes}{\ltimes},aabb,abbb,baab,bbaa,bbba,bbbb,bbb{\ltimes},bb{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. aabb
3. b
4. aab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{adc,bdb,bdd,cad\}\)

1. baabdd
2. bca
3. bbaddcb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}bcc,bcbc,bccb,bcdc,bddb,cbcd,cbdd,ccbd,dbcb,ddbc,cdc{\ltimes},dc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. dbaddcdcaaa
3. cad
4. d

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ab,{\rtimes}bb,aaa,aab,aba,baa,bab,bba,ab{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. bbabbab
2. aabaabbb
3. \(\varepsilon\)
4. baba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aba,abb,bab,bba\}\)

1. baabb
2. baa
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ae,cc,cd,dd,ec\}\)

1. daaaeb
2. dee
3. e
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aac,abc,acb,baa,bab,bca,cbb\}\)

1. b
2. \(\varepsilon\)
3. bc
4. aac

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}b{\ltimes},{\rtimes}ab,{\rtimes}ba,aba,abb,acb,bab,bac,bba,bbb,bbc,cbb,bc{\ltimes},ba{\ltimes},c{\ltimes}{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. cccc
2. baabbacab
3. b
4. cab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{acbd,bdbc,cadd,cbab,cbcc,ccca,cdcc,ddca\}\)

1. cdab
2. c
3. \(\varepsilon\)
4. bcc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}c,{\rtimes}c{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}cc,aba,acb,bab,bac,bba,bbb,bca,cbb,cbc,ccb,ca{\ltimes},a{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. bacbccca
3. c
4. abcbcbccbb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abb,bad,bbd,bda,dca,dcb,dcd,dda\}\)

1. dcccda
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{adaa,baaa,bbdc,caab,ddad\}\)

1. \(\varepsilon\)
2. bbddada
3. cacccbad
4. abba

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{adab,cccb,cccc,dccc,dcdd\}\)

1. \(\varepsilon\)
2. bbadbcd
3. b
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{bebb,cbce,cddb,ebaa\}\)

1. bbbede
2. e
3. \(\varepsilon\)
4. dea

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}dc,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}cc,{\rtimes}{\rtimes}ad,{\rtimes}cca,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}dc{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}ada,aada,abdd,adab,caad,ccaa,dabd,bdd{\ltimes},ada{\ltimes},dc{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},da{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. cbccbad
2. \(\varepsilon\)
3. ccc
4. ba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abc,acc,bad,cac,ccd,daa,dbb,dda\}\)

1. dcbddbdd
2. bdccab
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}bb,aca,ada,bac,bba,bbb,cad,dad,ad{\ltimes},d{\ltimes}{\ltimes}\}\)

1. cbabbaba
2. bbbacad
3. \(\varepsilon\)
4. bbacad

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{bacd,bbdd,cbba,ccbb,cccb,dabc,dbbc\}\)

1. cd
2. dca
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,caa,cab,cba\}\)

1. cbccaaa
2. c
3. cc
4. bcbcb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}d{\ltimes},{\rtimes}{\rtimes}ca,{\rtimes}cab,{\rtimes}d{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},abcd,bcdc,cabc,cdcc,dcca,cca{\ltimes},ca{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. abcb
2. aaacd
3. acdbcddb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bc,cb,da,dc,eb\}\)

1. acc
2. bd
3. dcbec
4. aecbdce

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}aaa,{\rtimes}acc,{\rtimes}bc{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},accb,bcbb,cbcb,ccbc,aaa{\ltimes},cbb{\ltimes},aa{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. bbac
2. aababc
3. cbbbc
4. ccc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bd,cc,da,db,dc,dd\}\)

1. bdddcdcb
2. \(\varepsilon\)
3. acacbaab
4. ddb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}db,{\rtimes}{\rtimes}bd,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}db{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}bdb,acbd,bacb,bdba,bddb,cbdd,dbac,ddbb,dbb{\ltimes},db{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. abb
2. ddc
3. aacddddc
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{baaa,baba,bbaa,bbab,bbbb\}\)

1. bbbabaabaaab
2. bbaa
3. bbaababbb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,abba,bbce,ddea,dded,ecde,edbd,eeee\}\)

1. acbb
2. b
3. e
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}aa,{\rtimes}aae,{\rtimes}aca,aaec,aeac,aece,cedd,daea,ddae,eced,edda,aca{\ltimes},eac{\ltimes},ac{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. dd
3. aca
4. cbaabbccd

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{caae,cbdb,ccdb,cdce,dceb\}\)

1. b
2. d
3. \(\varepsilon\)
4. e

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,aba,abb,baa,bab,bba,bbb\}\)

1. aaabbba
2. bbbbaab
3. \(\varepsilon\)
4. baababa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aaba,abaa,baaa,bbba,bbbb\}\)

1. \(\varepsilon\)
2. babba
3. aa
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aabb,abbb,baab,bbaa,bbab,bbbb\}\)

1. ab
2. baba
3. bbbab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abaa,abab,abba,babb,bbab,bbbb\}\)

1. a
2. abbaaaaa
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ad,bb,bc,ca,cb,cc\}\)

1. cb
2. bc
3. cabbabbba
4. bcbc

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}be,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}d{\ltimes}{\ltimes},{\rtimes}bec,{\rtimes}ac{\ltimes},becd,cdce,ecdc,dce{\ltimes},ce{\ltimes}{\ltimes},ac{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. d
2. ab
3. abbbe
4. bb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}acb,aaab,acaa,acba,baca,caaa,cbac,aab{\ltimes},ab{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. cabcaa
2. a
3. cbbcccaa
4. bbaca

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes},{\rtimes}ca,aab,aba,abc,bab,bca,caa,cab,ba{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. bbcacbb
3. b
4. bcbbbbc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a,{\rtimes}a{\ltimes},{\rtimes}ca,abb,bbc,bca,cab,ab{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. a
2. cab
3. ccab
4. cbbca

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abaa,bedd,cbbd,cbca,dbcb,eeec\}\)

1. ddadbda
2. ecc
3. eaebdea
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,abb,bab,bba,bbb\}\)

1. \(\varepsilon\)
2. ababa
3. a
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}ab,abc,aca,bac,bcb,cba,ca{\ltimes},a{\ltimes}{\ltimes}\}\)

1. cabaca
2. bcbcb
3. \(\varepsilon\)
4. aabcabb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\ltimes},{\rtimes}c{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ee,aad,ada,dad,eaa,eea,ad{\ltimes},d{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. dbdad
2. edad
3. daccac
4. adbe

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bec,cab,cad,cdb,ceb,daa,ebb\}\)

1. b
2. ed
3. e
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aec,bae,bcc,cee,eae,eba\}\)

1. b
2. abee
3. cbcbddca
4. dbacc

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}eb,{\rtimes}{\rtimes}ce,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}cec,{\rtimes}eb{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},bdce,cbdc,ccbd,cecc,ceee,dcee,eccb,eee{\ltimes},eb{\ltimes}{\ltimes},ee{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. eb
2. \(\varepsilon\)
3. aaeaedabd
4. bee

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,bc,cc,cd,db,b{\ltimes}\}\)

1. \(\varepsilon\)
2. a
3. dacb
4. dadcaa

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}b,{\rtimes}cb,{\rtimes}db,{\rtimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes},aba,bbd,bdb,bee,dbb,dbe,eab,eea,cb{\ltimes},ba{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. cb
2. \(\varepsilon\)
3. dbddaee
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}e,bd,ce,db,dc,ed,e{\ltimes},b{\ltimes}\}\)

1. b
2. eaa
3. \(\varepsilon\)
4. adaac

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aca,bac,bca,cba,ccc\}\)

1. ca
2. abcccaa
3. acbc
4. bcba

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ac,ba,bc,cb,cc\}\)

1. \(\varepsilon\)
2. cccbcbac
3. abcbcabbbb
4. cbbbacacc

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}cc,{\rtimes}{\rtimes}bc,{\rtimes}cca,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}bcb,adcd,cadc,ccad,dcdc,cdc{\ltimes},bcb{\ltimes},dc{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. d
2. dcadb
3. bcd
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}bcb,{\rtimes}ca{\ltimes},{\rtimes}aca,{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes}{\ltimes},acad,adad,adcc,cada,dadc,dcc{\ltimes},bcb{\ltimes},cb{\ltimes}{\ltimes},cc{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. aabacbdb
3. b
4. aca

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ab,{\rtimes}b{\ltimes},{\rtimes}ba,aab,abc,baa,bca,bcc,caa,ccc,ab{\ltimes},cc{\ltimes},b{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. abbbcca
2. bbab
3. acaacc
4. ccbbcca

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaac,babc,caaa,cbab,cbcc\}\)

1. caacac
2. cab
3. \(\varepsilon\)
4. abcab

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}cb,{\rtimes}ca,abb,aca,cab,cac,bb{\ltimes},cb{\ltimes},b{\ltimes}{\ltimes}\}\)

1. cb
2. cabb
3. abccb
4. bac

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}ab,{\rtimes}aba,{\rtimes}aaa,{\rtimes}ab{\ltimes},aaba,aabb,abaa,abba,baab,bbaa,baa{\ltimes},aaa{\ltimes},ab{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. abaa
2. \(\varepsilon\)
3. aaa
4. ab

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ab,ac,ba,bb,bc,ca,cb,cc,a{\ltimes},b{\ltimes}\}\)

1. ab
2. a
3. bbabbbbcca
4. acb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,cc,db,dc\}\)

1. ddc
2. ddaa
3. dabdaa
4. aaacbcc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},{\rtimes}c,aa,ab,bb,bc,ca,cb,a{\ltimes},c{\ltimes}\}\)

1. c
2. \(\varepsilon\)
3. a
4. cbcbb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}aba,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaba,aabb,abaa,abab,baab,baba,aba{\ltimes},abb{\ltimes},ba{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. babbbbaba
3. bbbb
4. aba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ba,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}b{\ltimes}{\ltimes},{\rtimes}baa,aaab,aabb,abbb,baaa,bbaa,bbba,baa{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. aaabbbbbb
3. \(\varepsilon\)
4. babbbaaaa

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}b{\ltimes},{\rtimes}bb,bac,bbc,bcb,cba,ac{\ltimes},b{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. bcca
2. cb
3. c
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bac,bcb,cab,cac,cca,ccb\}\)

1. acaaaaa
2. baccab
3. \(\varepsilon\)
4. ab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,abe,bcc,cab,dca,edb,eea\}\)

1. dbbe
2. cebbbccebb
3. cbdbbee
4. e

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abe,ada,adc,aed,cad,cea,deb\}\)

1. \(\varepsilon\)
2. b
3. ebbca
4. daabeea

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,abab,abbb,baab,bbba,bbbb\}\)

1. ba
2. \(\varepsilon\)
3. b
4. bb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{acb,bab,bbb,bbc,cac,cba,cbc\}\)

1. bccbbc
2. \(\varepsilon\)
3. b
4. cccbaaca

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}ea,{\rtimes}ead,{\rtimes}{\ltimes}{\ltimes}{\ltimes},adbe,becc,ccea,cead,dbec,eadb,ecce,adb{\ltimes},db{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. eabcc
2. e
3. adbabdaa
4. ebaadcedae

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,ac,ca,cc,c{\ltimes}\}\)

1. a
2. bbcb
3. caabaccc
4. c

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,aba,baa,bba,bbb\}\)

1. bba
2. \(\varepsilon\)
3. b
4. abbbab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aabb,abab,abbb,baab,bbbb\}\)

1. \(\varepsilon\)
2. abbbab
3. bbaaaaa
4. aaaaaaaa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,bc,cc,cd,da,db,dc,dd\}\)

1. bbc
2. abad
3. abbacab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abb,abc,acc,bbc,cab,cbb,cbc,ccc\}\)

1. cccc
2. \(\varepsilon\)
3. aaca
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aad,abc,bbd,bda,bdc,dad,dcb,dcc\}\)

1. bacc
2. cd
3. aa
4. adcc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}e,ad,ba,bb,bc,cb,de,eb,ec,b{\ltimes}\}\)

1. eddcddce
2. eddcdab
3. abbcaeeaba
4. c

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}d,ab,bc,cd,da,dd,d{\ltimes}\}\)

1. \(\varepsilon\)
2. dc
3. cb
4. bbaada

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ab,ac,ba,bc,ca,cb,cc,c{\ltimes}\}\)

1. cabc
2. aa
3. bbb
4. aaabbaaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{cabe,cdaa,cdca,cddc,cecc,dddd\}\)

1. ade
2. dc
3. aeda
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}d,{\rtimes}a{\ltimes},{\rtimes}dd,{\rtimes}cc,aad,aca,adb,bac,cad,daa,dba,dda,ad{\ltimes},cc{\ltimes},d{\ltimes}{\ltimes},c{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. a
2. dbdbc
3. dcacbdbcdc
4. cc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}c{\ltimes},{\rtimes}ba,abc,aee,bae,dab,dda,edd,eed,bc{\ltimes},c{\ltimes}{\ltimes}\}\)

1. dbcba
2. c
3. cdae
4. dded

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}bcb,{\rtimes}cbc,{\rtimes}{\ltimes}{\ltimes}{\ltimes},babb,bbab,bbba,bbbb,bcbb,cbbb,cbc{\ltimes},abb{\ltimes},bc{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. cacb
2. baccccca
3. aac
4. abcaabcb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aca,bac,bba,bbd,bcd,dcb,dcc\}\)

1. \(\varepsilon\)
2. b
3. cbddddd
4. dda

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aca,bbc,beb,ceb,cee,dde,ded\}\)

1. e
2. \(\varepsilon\)
3. b
4. ec

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}b{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}ba,{\rtimes}aaa,{\rtimes}baa,{\rtimes}ba{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes}{\ltimes},aaaa,aaab,aabb,abba,baa{\ltimes},bba{\ltimes},ba{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. bbbaa
2. \(\varepsilon\)
3. baaa
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abca,bbbc,bbcc,bcca,caaa,cbbb,ccbc,ccca\}\)

1. b
2. a
3. \(\varepsilon\)
4. abbbaabbbc

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,abb,baa,bab,bbb\}\)

1. bb
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes},{\rtimes}bb,{\rtimes}aa,aaa,aab,aba,baa,bb{\ltimes},aa{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. abb
2. abababa
3. aaabb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaba,abbb,baba,babb,bbba\}\)

1. baa
2. abaaaa
3. \(\varepsilon\)
4. abbbbaaab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,acce,cbae,ceee,dbba,ddcc,ecdd,edbc\}\)

1. eeaecb
2. aadaa
3. \(\varepsilon\)
4. e

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aba,baa,bab,bba\}\)

1. b
2. \(\varepsilon\)
3. bbabaabaa
4. aabbabaa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bb,cb,cc\}\)

1. c
2. cabaaaa
3. \(\varepsilon\)
4. accbaaa

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}ba,{\rtimes}aa,aab,abb,baa,bba,bbb,aa{\ltimes},ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. abbbab
2. aa
3. babaaa
4. ba

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,abba,abbb,baba,babb,bbab,bbba\}\)

1. b
2. \(\varepsilon\)
3. a
4. baabbab

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bd,cc,dd\}\)

1. ab
2. bcbddaba
3. acadb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abe,bbb,ccb,ddb,dde,eca,eee\}\)

1. \(\varepsilon\)
2. e
3. b
4. eb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes},{\rtimes}aa,aaa,aab,aba,abb,baa,bab,bba,aa{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. b
3. baba
4. aa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{adac,adcd,cbed,eaee,ebdc\}\)

1. b
2. e
3. cdbada
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}ba,{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}baa,aabb,abba,baab,baba,bbab,aba{\ltimes},ba{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. babaab
2. bbbba
3. ab
4. abaabb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,aa,ab,ac,ba,ca,cc,c{\ltimes}\}\)

1. cbc
2. \(\varepsilon\)
3. abcaac
4. c

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{cdec,dbea,dcba,eebb\}\)

1. cd
2. caedbdeeb
3. \(\varepsilon\)
4. aaedbaeec

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,abb,caa,cba,cbb,cca,ccb\}\)

1. b
2. aac
3. \(\varepsilon\)
4. ba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}be,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}bb{\ltimes},{\rtimes}bed,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},acdb,bede,deac,eacd,edea,cdb{\ltimes},db{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. eceab
2. dacacaab
3. daaeebd
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bcd,bdb,cba,cda,cdd,ced,dbb,eae\}\)

1. ddaddcbda
2. cdbaaaed
3. ced
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,baa,bab,bdb,ccc,daa,dcc\}\)

1. b
2. a
3. \(\varepsilon\)
4. cabbacbdbd

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,{\rtimes}cd,abd,bab,bdc,cce,cde,cec,dcc,ecd,de{\ltimes},cd{\ltimes},e{\ltimes}{\ltimes},d{\ltimes}{\ltimes}\}\)

1. deebebab
2. d
3. dcddcdcee
4. abcdecdbc

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abc,bcb,beb,cae,cbd,dcd,ebd\}\)

1. ecebabce
2. dbaa
3. eb
4. ccac

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aeb,bce,bea,dbb\}\)

1. \(\varepsilon\)
2. dddceed
3. e
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abbb,baaa,babb,bbbb\}\)

1. b
2. \(\varepsilon\)
3. abbbbabbb
4. a

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}ab,{\rtimes}b{\ltimes},aab,abb,baa,bba,bb{\ltimes},ab{\ltimes},b{\ltimes}{\ltimes}\}\)

1. aabbab
2. b
3. abbbaa
4. aaaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}bb,{\rtimes}bba,{\rtimes}bb{\ltimes},{\rtimes}bbb,babb,bbab,bbba,bbbb,bba{\ltimes},abb{\ltimes},ba{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. bb
3. aaaa
4. bba

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,aaa,aab,aba,baa,bab,ab{\ltimes},b{\ltimes}{\ltimes}\}\)

1. abaaabb
2. bbbabbba
3. \(\varepsilon\)
4. ababbbb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bcc,cad,ccd,cdc\}\)

1. \(\varepsilon\)
2. d
3. dd
4. bacbadb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaca,accb,bacb,cccb\}\)

1. acaccbcca
2. bcba
3. \(\varepsilon\)
4. bca

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ae,de,ea,ec\}\)

1. badbbacc
2. bba
3. cdce
4. aaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abda,adcb,bccc,ccca,daab\}\)

1. a
2. dcbb
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{adae,ccdc,cebd,dedc\}\)

1. adedc
2. \(\varepsilon\)
3. b
4. e

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}aab,aabb,abba,baaa,bbaa,aaa{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. bbbaa
2. ab
3. \(\varepsilon\)
4. abbaba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}ba,{\rtimes}aa{\ltimes},{\rtimes}aaa,{\rtimes}baa,aaab,aaba,abaa,baab,baa{\ltimes},aba{\ltimes},aa{\ltimes}{\ltimes},ba{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. aa
2. \(\varepsilon\)
3. ccc
4. aacc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ab,ac,ad,ba,ca,de,ea,c{\ltimes}\}\)

1. a
2. ac
3. abac
4. acac

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abb,acb,bbc,ccb,cce,edc\}\)

1. eede
2. e
3. abaed
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ac,ad,bc,ca,cb,cc,cd,da,db,c{\ltimes}\}\)

1. aacd
2. bcadadbd
3. ccbbb
4. ac

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}d,{\rtimes}d{\ltimes},{\rtimes}ac,aab,aac,abd,aca,acb,bcd,caa,cbc,cda,daa,bd{\ltimes},d{\ltimes}{\ltimes}\}\)

1. d
2. acaabd
3. caac
4. cacacbba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{acd,bac,bda,cbb,daa,dab,dcc,ddb\}\)

1. a
2. \(\varepsilon\)
3. adcac
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a,{\rtimes}c{\ltimes},{\rtimes}ac,abb,aca,bbb,bbc,bcb,cab,cbb,bb{\ltimes},b{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. bbb
2. \(\varepsilon\)
3. ca
4. abcc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ba,bb,bd,db,a{\ltimes}\}\)

1. a
2. aa
3. dba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,bacc,bccb,caaa,caba,cabb,cbcc\}\)

1. \(\varepsilon\)
2. cabc
3. accb
4. bbbaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,abba,baaa,baab,baba\}\)

1. \(\varepsilon\)
2. a
3. b
4. abbab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{acdc,addb,bddd,cbbb,dbdc,ddca\}\)

1. b
2. \(\varepsilon\)
3. da
4. a

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,bab,bba,bbb\}\)

1. abbabaaba
2. \(\varepsilon\)
3. baaabaab
4. bbbb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}c,bc,bd,cb,cc,cd,db,dc,dd,c{\ltimes},d{\ltimes}\}\)

1. dabbbca
2. c
3. \(\varepsilon\)
4. abdbc

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aba,abb,bba,bbb\}\)

1. abbbaabaa
2. baaba
3. bbab
4. bbb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{babd,bcac,bcdb,cbbd,cdab,ddad,dddd\}\)

1. adccca
2. \(\varepsilon\)
3. bdadcdaad
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}d,{\rtimes}ca,{\rtimes}de,{\rtimes}d{\ltimes},aba,aec,bae,cab,ced,dec,eca,ece,edc,dc{\ltimes},ca{\ltimes},d{\ltimes}{\ltimes},c{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. eadb
3. eadbaebeaa
4. dedb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}d,ac,bd,da,dd,de,eb,ee,c{\ltimes}\}\)

1. \(\varepsilon\)
2. accebcbbc
3. eecdbbbebe
4. beedbebece

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,cc,cd,dc,dd,c{\ltimes}\}\)

1. ace
2. deeda
3. c
4. cc

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}ba,{\rtimes}aab,{\rtimes}baa,aabb,abbb,baab,babb,bbab,bbba,aab{\ltimes},abb{\ltimes},ab{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. aabbbbb
2. \(\varepsilon\)
3. babb
4. b

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ea,{\rtimes}ead,adbb,bbee,beee,dbbe,eadb,eddb,eedd,eeed,eeee,ddb{\ltimes},db{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. ebeeacd
2. cbceced
3. eba
4. eadbbeeeddb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ac,ad,ba,cd,da,dd,a{\ltimes},b{\ltimes}\}\)

1. \(\varepsilon\)
2. ababdabc
3. b
4. ddb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,abb,baa,bba\}\)

1. \(\varepsilon\)
2. abaababa
3. a
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ba,bd,ca,dc\}\)

1. cddbdcc
2. ddd
3. \(\varepsilon\)
4. bba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,abb,bbc,cca,ccc\}\)

1. cbacab
2. \(\varepsilon\)
3. aacacab
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,ab,ba,bb,a{\ltimes},b{\ltimes}\}\)

1. b
2. ababbaaaa
3. aaba
4. bbabba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}acb,{\rtimes}{\ltimes}{\ltimes}{\ltimes},acbd,bdca,cbdc,acb{\ltimes},dca{\ltimes},cb{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. cccdd
3. b
4. bacaab

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}bc,{\rtimes}bb,aac,aba,aca,baa,bab,bbb,bbc,bcc,cba,ccb,bc{\ltimes},ca{\ltimes},c{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. bc
2. bbbc
3. bbc
4. cbabaaaab

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}ca,{\rtimes}bca,{\rtimes}cab,aaaa,aaab,aabc,abcd,bcaa,bcda,caaa,cdaa,cab{\ltimes},daa{\ltimes},ab{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. bcaaabcdaa
2. dcddcca
3. cab
4. cdddddcbdad

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. abaaaa
2. ba
3. bbbbbbbba
4. bbbbabbabb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}c,aa,ac,ad,ca,cd,da,d{\ltimes}\}\)

1. cd
2. ca
3. \(\varepsilon\)
4. cad

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaac,aabc,abbc,acaa,bbcb,bcbc,cabc,cbbc\}\)

1. abc
2. acaccaaa
3. bbbacc
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}bba,{\rtimes}a{\ltimes}{\ltimes},aaab,abba,baaa,babb,bbaa,bbab,aab{\ltimes},ab{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. abbbbaa
3. baaba
4. babbbb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ba,bc,cb\}\)

1. cbacc
2. b
3. \(\varepsilon\)
4. cba

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bc,ca,cb,a{\ltimes},b{\ltimes}\}\)

1. bcabccacbabb
2. c
3. bbccabacaacb
4. bbabaa

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,aca,bbd,ccd\}\)

1. adbd
2. abaa
3. d
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}c,aa,ac,bb,bc,cb,cc,c{\ltimes}\}\)

1. babd
2. c
3. cc
4. ac

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ba,bd,cd,db,dc,dd\}\)

1. \(\varepsilon\)
2. accaddc
3. bcac
4. ddacdbada

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\ltimes}{\ltimes},{\rtimes}bc,{\rtimes}aa,abc,bab,bba,bcb,cbb,bc{\ltimes},aa{\ltimes},a{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. bc
2. bab
3. \(\varepsilon\)
4. aa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ba,bb,cb,cc\}\)

1. \(\varepsilon\)
2. ccb
3. b
4. a

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}a,{\rtimes}{\ltimes}{\ltimes},{\rtimes}ac,acd,ace,cdd,cec,dac,dda,ecd,cd{\ltimes},d{\ltimes}{\ltimes}\}\)

1. bbdccceab
2. acecd
3. \(\varepsilon\)
4. acd

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ad,ca,cc,db\}\)

1. ea
2. \(\varepsilon\)
3. dbaea
4. abb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bbc,bcb,bda,bdb,cbd,ccc,dad,dcc\}\)

1. bcac
2. ddaacaa
3. c
4. aca

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ba,bb,b{\ltimes}\}\)

1. abbbab
2. ab
3. abbbbabbb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abcb,beba,cbdb,cddc,ceed,ceee,dcba,eadb\}\)

1. e
2. b
3. dedc
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{acab,babb,bcbb,bcca,ccca\}\)

1. a
2. b
3. bcb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abc,acb,dab,dad,ddb\}\)

1. b
2. ccdaccdca
3. \(\varepsilon\)
4. aadcacba

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}c,ad,bd,cb,da,a{\ltimes},b{\ltimes}\}\)

1. dcbbad
2. bccc
3. \(\varepsilon\)
4. dcab

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,{\rtimes}d,aa,ab,bc,bd,cd,da,db,a{\ltimes},c{\ltimes}\}\)

1. ca
2. dbdc
3. \(\varepsilon\)
4. ddadcdbca

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ab,ba,a{\ltimes}\}\)

1. aaabbaba
2. a
3. bba
4. baaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,abaa,baab,bbaa\}\)

1. bbbbb
2. aba
3. bbaa
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}ab,{\rtimes}aa,aaa,aab,aba,baa,ab{\ltimes},aa{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. abaabaaa
2. ab
3. bbbbaabaab
4. aa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{cb,ce,da,ea\}\)

1. cbeeccadca
2. e
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}e{\ltimes},{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}ba,{\rtimes}cac,{\rtimes}e{\ltimes}{\ltimes},{\rtimes}ba{\ltimes},acbb,bacb,bbac,bbdc,bdcb,cacb,cbba,cbbd,dcb{\ltimes},ba{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. ba
2. cacbbdcb
3. e
4. cbbabaacdc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,b{\ltimes}\}\)

1. baaaba
2. bb
3. baaab
4. abbaabaaa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{bd,cb,cc,cd,da,dc\}\)

1. cbcdad
2. \(\varepsilon\)
3. b
4. dcbccdabd

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,abb,baa,bba,bbb\}\)

1. \(\varepsilon\)
2. b
3. aabba
4. aaa

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ac,ca,cc,c{\ltimes}\}\)

1. aac
2. acc
3. cbbc
4. ac

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}da,acc,acd,bac,cba,ccb,dac,cd{\ltimes},d{\ltimes}{\ltimes}\}\)

1. acededeb
2. bbd
3. ddbecae
4. cca

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}c,{\rtimes}{\ltimes}{\ltimes},{\rtimes}ca,abc,bcc,cab,cba,ccb,ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ac
2. accca
3. \(\varepsilon\)
4. acbbbc

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{acd,bce,bda,cab,cad,ddb,deb\}\)

1. b
2. e
3. \(\varepsilon\)
4. dee

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}ac{\ltimes},{\rtimes}cbb,{\rtimes}{\ltimes}{\ltimes}{\ltimes},abca,babc,bbcb,bcab,bcba,cabc,cbab,cbbc,abc{\ltimes},ac{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. ac
2. \(\varepsilon\)
3. cbbcba
4. ccbbcbabc

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{bcab,dabd,dacd,dbda,dcdc\}\)

1. baadacbc
2. abdbccddcd
3. badddbdc
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ea,{\rtimes}eac,accc,ccca,cccc,eacc,cca{\ltimes},ca{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. eaccca
2. cdbcce
3. a
4. bdc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ba,bb,db,dc,ec,ed\}\)

1. dbced
2. daebcaeea
3. abdaccece
4. baebedaaac

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aaab,aabb,babb,bbaa,bbab,bbba\}\)

1. aabbaa
2. \(\varepsilon\)
3. babaab
4. bbbb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}a,{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ac,aab,acb,baa,bba,cbb,ab{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. bba
2. c
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ba,bb,b{\ltimes}\}\)

1. \(\varepsilon\)
2. bba
3. aaabaaba
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abaa,abbc,bbca,bcaa,cbac,cbbc,cccc\}\)

1. b
2. ababcbca
3. cbbcac
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}ba,aaa,aab,aba,abb,baa,bab,bba,ab{\ltimes},b{\ltimes}{\ltimes}\}\)

1. baaba
2. bab
3. bbaaab
4. baab

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}de,ace,ada,ced,dac,dda,dea,ead,edd,da{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ebaeceda
2. daaaaaccac
3. \(\varepsilon\)
4. deada

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aadc,adde,bacc,bdea,cded,daea,ddcc\}\)

1. bac
2. bdddb
3. e
4. abcc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}da,aba,abb,bbe,bed,dab,eda,ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. cdccada
2. d
3. ab
4. daba

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ab,{\rtimes}b{\ltimes},abb,bbb,bb{\ltimes},b{\ltimes}{\ltimes}\}\)

1. aaa
2. \(\varepsilon\)
3. aab
4. babaab

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}eb,{\rtimes}{\rtimes}ed,{\rtimes}ebd,{\rtimes}eda,aace,aced,cedb,daac,dbda,edaa,edbd,ebd{\ltimes},bda{\ltimes},bd{\ltimes}{\ltimes},da{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. ceaee
2. aacccca
3. aedbb
4. beeede

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ad,bc,da\}\)

1. \(\varepsilon\)
2. b
3. cddaa
4. a

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}cc,{\rtimes}{\rtimes}cd,{\rtimes}cda,{\rtimes}d{\ltimes}{\ltimes},{\rtimes}cc{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},abbe,cdab,dabb,bbe{\ltimes},be{\ltimes}{\ltimes},cc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes}\}\)

1. d
2. \(\varepsilon\)
3. b
4. da

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bd,be,db,ea,ec,ed,ee\}\)

1. e
2. ecbaece
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aaba,abba,bbba\}\)

1. \(\varepsilon\)
2. abababa
3. b
4. baaabb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aabd,acbd,baad,bbab,cbba,cddb,cddc,ddbb\}\)

1. abbdacb
2. cd
3. bacdbaa
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}bc,aac,baa,bbc,bcb,bcd,cba,cbb,cdc,dcb,ac{\ltimes},c{\ltimes}{\ltimes}\}\)

1. abdba
2. abdccca
3. cd
4. aa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ac,ba,bb\}\)

1. cbcabbba
2. baccb
3. bbc
4. caccbbc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}ab,{\rtimes}bb,aab,aba,baa,bb{\ltimes},ba{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. aba
2. bb
3. c
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaed,accb,baec,dcdd,deeb,eacc,ecac,eeac\}\)

1. c
2. e
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaba,acba,bbba,cbaa\}\)

1. aa
2. aabaac
3. cabba
4. bb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abdb,baaa,bbbd,bcbb,cdbc,cddc,dbdb,dcad\}\)

1. \(\varepsilon\)
2. a
3. b
4. dbbb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ab,ac,ba,bc,ca,cb,cc,a{\ltimes}\}\)

1. aaa
2. ab
3. a
4. aa

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,{\rtimes}c,ac,ca,cc,a{\ltimes},b{\ltimes}\}\)

1. accb
2. bcbc
3. \(\varepsilon\)
4. caaac

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}abc,abca,abcb,bcab,bcbb,cabc,cbb{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. cc
2. cabcba
3. a
4. abcbb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bd,ca,cd\}\)

1. \(\varepsilon\)
2. b
3. a
4. daacbacccc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,ad,bd,dd\}\)

1. \(\varepsilon\)
2. ababbbbccc
3. a
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}e,aa,ab,ad,ae,be,de,ea,e{\ltimes}\}\)

1. \(\varepsilon\)
2. eabcccb
3. baddcabece
4. adbbcaad

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,abb,bcb,cbb,cbc\}\)

1. acabca
2. cacacccaac
3. accbaacb
4. cccbb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}bb,{\rtimes}ad,acb,bac,bba,bbd,bde,cbb,ad{\ltimes},de{\ltimes},e{\ltimes}{\ltimes},d{\ltimes}{\ltimes}\}\)

1. caaeac
2. bbde
3. ad
4. abb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ae,ba,ce,da\}\)

1. e
2. acbaebe
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}b{\ltimes},{\rtimes}bb,{\rtimes}aa,aab,aba,abb,baa,bab,bba,bbb,aa{\ltimes},ba{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. b
2. aa
3. bba
4. aaa

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,aba,bba,bbb\}\)

1. a
2. \(\varepsilon\)
3. b
4. abbb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{adb,cac,dbd,ece,edb\}\)

1. aacaaeda
2. eccaed
3. cddce
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{acd,bde,ccb,dea\}\)

1. abbbcbecbabb
2. bedbcbadae
3. \(\varepsilon\)
4. abcaeeeb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aabb,abaa,abbb,babb,bbab,bbbb\}\)

1. aa
2. \(\varepsilon\)
3. a
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}d,aa,bb,be,cb,cc,ce,dc,ea,ec,a{\ltimes},d{\ltimes}\}\)

1. dcea
2. d
3. dccea
4. adcad

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aaba,abbb,baaa,babb,bbaa,bbbb\}\)

1. ababa
2. bbaaab
3. \(\varepsilon\)
4. bbaaa

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}d,{\rtimes}{\ltimes}{\ltimes},{\rtimes}d{\ltimes},{\rtimes}aa,{\rtimes}bd,bbc,bcc,bdb,cbd,ccb,dbb,aa{\ltimes},bd{\ltimes},d{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ab
3. badbad
4. bd

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}d,ab,ac,ba,bb,be,ca,da,ea,a{\ltimes}\}\)

1. beeee
2. aca
3. a
4. deebaeab

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,aa,ac,bc,ca,cb,c{\ltimes}\}\)

1. cbcc
2. a
3. bc
4. ccacbca

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,abba,baaa,baab,babb\}\)

1. bbbba
2. babbab
3. b
4. aababb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,aab,aba,abb,baa,bab,bb{\ltimes},b{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. aa
3. ba
4. aab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aac,aca,acb,ccc\}\)

1. c
2. \(\varepsilon\)
3. b
4. ccb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aabb,abaa,baba,babb,bbab,bbbb\}\)

1. aabbba
2. \(\varepsilon\)
3. baa
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ba,bb,bc,ca,cb\}\)

1. a
2. cabb
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{baa,bab,bbd,bdc,cda,dbe,ddc,edc\}\)

1. ba
2. a
3. adeaecadcaa
4. ceead

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{add,cab,cbc,dad,dbb,dbd,dcd\}\)

1. bdacbc
2. b
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}aba,{\rtimes}a{\ltimes}{\ltimes},aabb,abaa,baab,abb{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. abbc
3. caab
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{da,de,eb,ee\}\)

1. \(\varepsilon\)
2. e
3. dcacdd
4. cbdedebc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ad,bb,cc,cd,da,dd\}\)

1. \(\varepsilon\)
2. bcccd
3. b
4. dd

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,b{\ltimes}\}\)

1. bbab
2. aabb
3. abbaa
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,ae,ba,ca\}\)

1. b
2. ad
3. \(\varepsilon\)
4. ecac

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bc,ca,cc\}\)

1. \(\varepsilon\)
2. b
3. acbbcc
4. a

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a,{\rtimes}cb,{\rtimes}a{\ltimes},acb,bac,bca,cac,cba,cbc,cb{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ccacaa
2. baabbccc
3. bccac
4. cb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ad,ca,cc,da,db,dd\}\)

1. \(\varepsilon\)
2. bdbdbadcb
3. dbca
4. db

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}bb,aab,abd,baa,bba,bdd,daa,dda,ab{\ltimes},b{\ltimes}{\ltimes}\}\)

1. deccae
2. \(\varepsilon\)
3. bbaab
4. bbdaedbe

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}a{\ltimes},{\rtimes}bd,aba,bab,bdb,dba,ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. a
2. bdba
3. d
4. dabca

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ac,ba,bb,bc,ca,cb\}\)

1. bbcaa
2. \(\varepsilon\)
3. bcbb
4. a

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bd,cb,cc,dd\}\)

1. da
2. d
3. ccda
4. a

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,bd,ca,cd,db,dd\}\)

1. dbaccabb
2. ca
3. cddbca
4. aaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aacb,abac,accb,baaa,bbac,bbbb,cacc,cbab\}\)

1. b
2. abcc
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,{\rtimes}bb,aab,aba,abb,baa,bab,bba,ab{\ltimes},bb{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. aab
2. \(\varepsilon\)
3. aaabbbbabbb
4. bbabbaa

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ae,bc,ca,cd,de,eb,ec,ee,c{\ltimes}\}\)

1. baccac
2. aec
3. ce
4. aeec

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abc,aca,bab,bcb,caa,cca\}\)

1. b
2. caabbc
3. \(\varepsilon\)
4. cba

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ce,db,dc,ec\}\)

1. caacbdbedab
2. \(\varepsilon\)
3. cbdacadd
4. a

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,ba,ca,cb,cd,db,dc,dd\}\)

1. dcbc
2. \(\varepsilon\)
3. bbcbaaab
4. baabbb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}c,ab,bc,ca,cb,cc,a{\ltimes}\}\)

1. cabcabba
2. \(\varepsilon\)
3. ca
4. ccacca

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}d,aa,ab,ad,ba,bd,da,db,b{\ltimes},d{\ltimes}\}\)

1. b
2. \(\varepsilon\)
3. d
4. dcbb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,bb,bc,bd,cb,cc,dc,b{\ltimes}\}\)

1. \(\varepsilon\)
2. b
3. bdbabac
4. dabd

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aadb,bacc,bcac,bdbd,dcca\}\)

1. \(\varepsilon\)
2. dccccc
3. d
4. bbabaaad

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\ltimes},{\rtimes}c{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ec,ade,bad,cba,ecb,de{\ltimes},e{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. e
2. bec
3. aaab
4. dbcdcc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ab,ba,b{\ltimes}\}\)

1. \(\varepsilon\)
2. abbbb
3. aa
4. bb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,bc,cb,cc\}\)

1. \(\varepsilon\)
2. b
3. ccbccccc
4. bcbcbacaaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}{\rtimes}ba,{\rtimes}bac,{\rtimes}acc,{\rtimes}ca{\ltimes},{\rtimes}b{\ltimes}{\ltimes},accc,ccca,cccc,bac{\ltimes},cca{\ltimes},ac{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. cac
2. b
3. cbc
4. ca

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ba,bb,cb,cc\}\)

1. bbb
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{baaa,baab,baba,babb,bbaa,bbba,bbbb\}\)

1. aa
2. \(\varepsilon\)
3. a
4. ab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abcc,adad,bdaa,dacb,dbbc\}\)

1. dcddcb
2. bdb
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,abba,baaa,baba,babb,bbaa\}\)

1. baabab
2. bb
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ab,ba,bb,b{\ltimes}\}\)

1. \(\varepsilon\)
2. bbabba
3. bba
4. aab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,caa,dcc,dda\}\)

1. \(\varepsilon\)
2. b
3. cdbca
4. ab

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}b,{\rtimes}c{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}bc,{\rtimes}bd,aab,abc,acc,bca,bda,caa,cca,dac,ca{\ltimes},bc{\ltimes},c{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. dde
2. aebbdbbcdb
3. \(\varepsilon\)
4. cebbaebd

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}cd,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}cdb,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aacd,acda,baac,cdaa,cdba,daac,dbaa,acd{\ltimes},cd{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. badddbbbdba
3. dcba
4. cddaaabbc

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aba,abb,bba\}\)

1. \(\varepsilon\)
2. b
3. aabaa
4. a

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,acb,bba,cca\}\)

1. a
2. \(\varepsilon\)
3. aaaa
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ab,ba,a{\ltimes}\}\)

1. aa
2. b
3. bbaaaaab
4. a

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bb,bc,bd,da,db,dc\}\)

1. b
2. \(\varepsilon\)
3. bcaa
4. abdbcca

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}b,aa,ac,bc,ca,cb,cc,a{\ltimes},b{\ltimes}\}\)

1. ac
2. babcabb
3. cacabbc
4. bccaba

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,abaa,abba,baaa,babb,bbaa,bbba\}\)

1. b
2. \(\varepsilon\)
3. aaaaa
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{baac,bbba,bcab,bcbc,bccc,cccc\}\)

1. abca
2. \(\varepsilon\)
3. aababbbb
4. bcbbacba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bba,bdb,dde,edd\}\)

1. e
2. cb
3. cbaeebccac
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}ba,{\rtimes}bab,{\rtimes}bb{\ltimes},{\rtimes}baa,abbb,babb,bbab,bbba,bbbb,baa{\ltimes},bab{\ltimes},ab{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. bab
2. baa
3. bb
4. a

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ae,{\rtimes}ea,ade,deb,ead,eba,ae{\ltimes},ba{\ltimes},e{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ea
2. c
3. \(\varepsilon\)
4. abdde

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}bb,{\rtimes}ad,ada,aec,cac,cda,dae,eca,ecd,bb{\ltimes},ac{\ltimes},b{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. aaebcba
2. bb
3. \(\varepsilon\)
4. adaecac

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ee,{\rtimes}eeb,{\rtimes}ee{\ltimes},abba,bcdc,cdcd,cdda,dabb,dcdd,ddab,ebcd,eebc,bba{\ltimes},ee{\ltimes}{\ltimes},ba{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes}\}\)

1. ee
2. bccddedcbcee
3. bedbcb
4. eaddab

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,b{\ltimes}\}\)

1. bab
2. \(\varepsilon\)
3. abbb
4. baaba

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ca,cb,cc,cd,da,dd\}\)

1. e
2. aaedddc
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ac,bc,ca,cc,c{\ltimes}\}\)

1. aab
2. bc
3. ba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ea,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}cc,{\rtimes}cca,{\rtimes}aa{\ltimes},{\rtimes}d{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}eac,acac,acba,baca,cbac,eacb,cca{\ltimes},cac{\ltimes},ac{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. aabd
2. bbdabed
3. d
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. \(\varepsilon\)
2. abaaaaabab
3. baaaaaabb
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aaba,abaa,bbaa,bbba\}\)

1. bbbbaa
2. \(\varepsilon\)
3. baababb
4. baa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}ae,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}c{\ltimes}{\ltimes},{\rtimes}ab{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}aee,aeed,dded,edde,eedd,ded{\ltimes},ab{\ltimes}{\ltimes},ed{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. babedee
2. cbcadbb
3. \(\varepsilon\)
4. cddbaa

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,ae,ba,cd,db,dd,ec,d{\ltimes}\}\)

1. dabcbeca
2. eaaae
3. cbccdd
4. cd

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\ltimes},{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ab,aac,abb,acb,baa,bba,bca,cbc,ab{\ltimes},ca{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. a
2. \(\varepsilon\)
3. abca
4. abccbaac

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,ab,ac,bc,bd,ca,cd,da,db,dd,a{\ltimes}\}\)

1. bca
2. dcabdcbadd
3. bcdabc
4. baabbabaacc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. baa
2. ba
3. bba
4. ababbb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}ab,{\rtimes}abb,{\rtimes}ca{\ltimes},abba,babc,bbab,abc{\ltimes},bc{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes}\}\)

1. aaabb
2. ccc
3. ca
4. ac

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}cc,{\rtimes}bb{\ltimes},{\rtimes}acb,{\rtimes}ccc,{\rtimes}{\ltimes}{\ltimes}{\ltimes},abba,bbac,bbcc,bcca,cabb,cbbc,ccab,ccbb,cccb,acb{\ltimes},bac{\ltimes},ac{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. \(\varepsilon\)
3. bbbbcbbcaca
4. bb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,{\rtimes}cc,aad,adc,caa,cbd,cca,dcb,ba{\ltimes},bd{\ltimes},d{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. bccdbbab
2. bbac
3. c
4. addb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}bb,{\rtimes}bba,{\rtimes}ab{\ltimes},abab,baba,bbab,bab{\ltimes},ab{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. aba
2. \(\varepsilon\)
3. ab
4. aa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}cd,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}cdb,abda,babd,bbab,bdad,cdbb,dadb,dbba,adb{\ltimes},db{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. dcbcdcbbc
2. a
3. ddccacac
4. dbdda

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,bb,bc,cb,cd,da,db,dd\}\)

1. \(\varepsilon\)
2. aabc
3. bbbb
4. dbb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ba,bb,bc,ca,cb\}\)

1. a
2. caabaaaabcba
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aba,ada,cab,cda,dca,ddb\}\)

1. d
2. \(\varepsilon\)
3. ddd
4. cac

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,a{\ltimes},b{\ltimes}\}\)

1. abbbabbb
2. b
3. abbbbb
4. aaaaaabaaab

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ae,{\rtimes}{\rtimes}ec,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}{\rtimes}ba,{\rtimes}bac,{\rtimes}ecb,{\rtimes}d{\ltimes}{\ltimes},{\rtimes}ae{\ltimes},acae,baca,caeb,ecb{\ltimes},aeb{\ltimes},ae{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},eb{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. ae
2. \(\varepsilon\)
3. b
4. d

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,aca,acb,bac,bdd,cac,cbd,dd{\ltimes},d{\ltimes}{\ltimes}\}\)

1. bbc
2. bcdccb
3. cec
4. addded

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,{\rtimes}aa,aab,aba,aca,bac,caa,ab{\ltimes},ba{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ba
2. \(\varepsilon\)
3. bb
4. ccbccacb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abba,baab,bbab,bbbb\}\)

1. a
2. aabb
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}db,{\rtimes}{\rtimes}ea,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}ead,{\rtimes}d{\ltimes}{\ltimes},{\rtimes}dbd,abde,bdda,bded,dabd,dbdd,ddab,dedc,edc{\ltimes},ead{\ltimes},dc{\ltimes}{\ltimes},ad{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. cd
2. eeecc
3. aebceeaae
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}bc,aac,acc,bcc,caa,cca,ca{\ltimes},a{\ltimes}{\ltimes}\}\)

1. acabcacb
2. a
3. bcca
4. bac

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}d,{\rtimes}db,{\rtimes}bd,bbd,bdd,dbb,ddb,db{\ltimes},bd{\ltimes},b{\ltimes}{\ltimes},d{\ltimes}{\ltimes}\}\)

1. aaadda
2. \(\varepsilon\)
3. db
4. bd

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ac,ba,bb,cc,cd,db,a{\ltimes},b{\ltimes}\}\)

1. bdccc
2. bb
3. bbdadbccc
4. abaabbbda

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ae,cc,da,dd,ed\}\)

1. cbdeda
2. bddeae
3. dddeb
4. ecceecb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}c,ab,ac,bd,ca,cc,da,a{\ltimes}\}\)

1. cca
2. ccaccbc
3. ca
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ba,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}ba{\ltimes},{\rtimes}bbb,aabb,abaa,abbb,baab,baba,bbab,bbba,bba{\ltimes},ba{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. \(\varepsilon\)
3. babaabbaa
4. aaaabab

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ac,ad,bb,cc,da,db,dc\}\)

1. b
2. dbdbd
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abba,abca,bacb,cacc\}\)

1. abbb
2. accbbbbbac
3. aac
4. accabbacc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ba,bb,ca\}\)

1. ac
2. a
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abdb,acca,adea,ddbc,deac,eacb,ebdc\}\)

1. \(\varepsilon\)
2. b
3. e
4. abeacb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}dc,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}ca{\ltimes},{\rtimes}c{\ltimes}{\ltimes},{\rtimes}dcc,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaaa,caaa,ccdc,cdca,dcaa,dccd,aaa{\ltimes},aa{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. ddcb
2. ccddacbbcb
3. aacbacd
4. abbbbdbc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}dd,aca,adb,bdd,caa,dac,dad,dbd,dda,aa{\ltimes},a{\ltimes}{\ltimes}\}\)

1. acba
2. \(\varepsilon\)
3. caabdcaabd
4. addcbbbbd

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,bab,bbb\}\)

1. aaba
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{baad,bdae,edaa,edee\}\)

1. e
2. cb
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abb,bbb,cab,cac,ccb\}\)

1. b
2. \(\varepsilon\)
3. a
4. cb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaac,bbaa,bbca,bbcc,caab,cdab,dbab\}\)

1. b
2. a
3. dcbcbcabb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}bb,{\rtimes}bbb,{\rtimes}cb{\ltimes},aaea,aeae,baae,bbaa,bbba,eaee,aee{\ltimes},ee{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. abeada
2. aaabdaae
3. dcdbea
4. ecda

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ac,bb,bc,cb\}\)

1. bba
2. cbb
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}bb,{\rtimes}aa{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}bbb,aabb,abbb,baab,bbaa,bbba,baa{\ltimes},aa{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. abab
2. abaaabbb
3. bbbbb
4. abbaabb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}a,{\rtimes}c{\ltimes},{\rtimes}ac,aca,adc,cad,dcb,cb{\ltimes},b{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ccadaa
3. c
4. cdbcb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aacb,aada,accd,addb,dacd,dbcc\}\)

1. \(\varepsilon\)
2. b
3. ddcaaccccbd
4. a

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ba,bb,b{\ltimes}\}\)

1. ab
2. aab
3. bbaab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}da,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}aaa,{\rtimes}dac,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaad,aada,acab,adac,daca,cab{\ltimes},dac{\ltimes},ac{\ltimes}{\ltimes},ab{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. dacab
2. \(\varepsilon\)
3. dac
4. cacd

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaba,aabb,abaa,abba,baba,bbaa\}\)

1. b
2. bababaa
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. babbaabaaa
2. ba
3. aabaaba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,ac,bb,bc,ca,c{\ltimes},b{\ltimes}\}\)

1. bb
2. b
3. ccbac
4. bbcbc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}dd,acb,adc,cac,cad,cba,cca,dca,dcc,ddc,ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ba
2. adcccbddbd
3. ddcacba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ab,ba,a{\ltimes}\}\)

1. ababa
2. babab
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,acd,bab,bbb\}\)

1. \(\varepsilon\)
2. ccdd
3. b
4. db

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}d,aa,ac,ae,ba,be,cb,ea,ec,ee,c{\ltimes},d{\ltimes}\}\)

1. cacdeebde
2. \(\varepsilon\)
3. d
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. ababaa
2. \(\varepsilon\)
3. abaaba
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}d,ab,bb,bc,ce,da,e{\ltimes}\}\)

1. dabce
2. ae
3. \(\varepsilon\)
4. ce

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,abaa,abab,abbb,baaa,babb\}\)

1. bba
2. aab
3. bb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,bab,bbb,caa,cbb,ccb\}\)

1. b
2. \(\varepsilon\)
3. a
4. acbcacc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ac,ba,bb,bc,ca,cb,cc\}\)

1. aaccaaac
2. aa
3. b
4. ccbbbcccaac

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,abc,baa,cac\}\)

1. b
2. \(\varepsilon\)
3. bbbccc
4. aabbab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abba,abca,acaa,bbaa,bbab,cabc,cccb\}\)

1. bba
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abcd,aebc,bade,baec,beeb,cadd,cdcb,decc\}\)

1. daddcba
2. b
3. ad
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}bb,{\rtimes}aa,aaa,aba,baa,bab,bba,bbb,aa{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. bababbb
3. bbabab
4. bbb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{adb,bcd,bda,bdb,cbb,cdb,dca,dcc\}\)

1. \(\varepsilon\)
2. ba
3. b
4. dbdd

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,abb,bca,bcb,bcc,caa,cca,ccb\}\)

1. aacc
2. bcbabcaabcca
3. ccc
4. acaccbbacaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,abbc,baab,cabb,caca,ccaa\}\)

1. b
2. \(\varepsilon\)
3. aabcacbbbccb
4. acccaaabcb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,{\rtimes}e,bd,de,eb,ed,ee,e{\ltimes},b{\ltimes}\}\)

1. bcbbba
2. \(\varepsilon\)
3. e
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ac,ca,cc,a{\ltimes}\}\)

1. aaacc
2. bababcb
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}db,{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}dbb,{\rtimes}ab{\ltimes},{\rtimes}b{\ltimes}{\ltimes},aabd,abda,babd,bbab,bdaa,daab,dbba,abd{\ltimes},ab{\ltimes}{\ltimes},bd{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. ddaa
3. ab
4. ebaecacbbb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aadd,bddd,dabb,dbbb,dcbb,ddda\}\)

1. dbbdabdb
2. aadbada
3. db
4. cbdbcd

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bc,db,dc,eb,ed,ee\}\)

1. b
2. bc
3. \(\varepsilon\)
4. e

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},aa,ab,ac,ba,bb,bc,ca,cb,cc,a{\ltimes},b{\ltimes}\}\)

1. abcaab
2. accaaaacaca
3. aaa
4. bcb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}b,{\rtimes}{\ltimes}{\ltimes},{\rtimes}be,{\rtimes}dc,aac,acc,caa,cba,cca,ccb,ccc,dcc,be{\ltimes},ba{\ltimes},e{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. be
2. eddbb
3. dccba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ab,ad,bd,da,dc,dd,c{\ltimes}\}\)

1. adc
2. aadc
3. abdc
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bd,{\rtimes}{\rtimes}ce,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}ceb,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}aa{\ltimes},{\rtimes}bdd,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aeec,bcea,cbdd,ceae,cebc,eaee,ebce,ecbd,eecb,bdd{\ltimes},aa{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. ebcdcddcba
2. \(\varepsilon\)
3. ba
4. aeeddeabddcb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aba,abb,bba,bbc,cbb,cca,ccc\}\)

1. bbbcbbaab
2. b
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}b,{\rtimes}b{\ltimes},{\rtimes}ea,aad,adc,ade,caa,dca,deb,ead,ebe,be{\ltimes},e{\ltimes}{\ltimes},b{\ltimes}{\ltimes}\}\)

1. ad
2. adba
3. caede
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,acc,baa,bcc,cbb,cbc,cca\}\)

1. b
2. abababab
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,aa,ab,ac,ba,bd,cb,da,dd,c{\ltimes}\}\)

1. cbac
2. cbcbdccccb
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aada,abac,abdb,bbac,cbbb,daca\}\)

1. cbdaaadda
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{acb,bac,bbb,bbc,cbb\}\)

1. \(\varepsilon\)
2. c
3. accabbc
4. cacca

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{acaa,addd,beec,cacc,cccd,dccd\}\)

1. ddedea
2. \(\varepsilon\)
3. e
4. bacdbcaa

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}e,{\rtimes}d,ab,ba,be,cc,ce,dc,eb,ec,e{\ltimes},c{\ltimes}\}\)

1. e
2. ababadbb
3. aeceeab
4. ec

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}dd,{\rtimes}{\rtimes}de,{\rtimes}{\rtimes}d{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}ddd,{\rtimes}d{\ltimes}{\ltimes},{\rtimes}deb,{\rtimes}bc{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},abba,babb,bade,bbad,bceb,ceba,debc,ebab,ebce,ddd{\ltimes},ade{\ltimes},de{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes}\}\)

1. eaaadeceddc
2. \(\varepsilon\)
3. baaede
4. abc

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,aeae,bded,dbee,ddec\}\)

1. dbdeea
2. d
3. edaaa
4. cb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abba,abbc,acab,ccbc\}\)

1. aaccbac
2. cacaaabc
3. aaabb
4. c

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},ab,ac,ad,ba,ce,ea,d{\ltimes}\}\)

1. ebcc
2. eb
3. \(\varepsilon\)
4. eeeb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}ba,{\rtimes}aaa,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}bbb,{\rtimes}ba{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaab,aaba,aabb,abaa,abba,baaa,bba{\ltimes},bbb{\ltimes},ba{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. aaba
3. bbaba
4. bbbabb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,ab,ba,bb,b{\ltimes}\}\)

1. aaababb
2. bb
3. baabbaa
4. aabbba

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}b{\ltimes},{\rtimes}ba,aab,aba,abb,baa,bab,bba,bb{\ltimes},b{\ltimes}{\ltimes}\}\)

1. bababa
2. ababa
3. baaaba
4. abbbbbbbb

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}ca,{\rtimes}be,aba,abb,bab,bbe,bea,beb,eab,ebc,bc{\ltimes},ca{\ltimes},a{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. bca
2. badb
3. bebc
4. ca

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}da,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}da{\ltimes},{\rtimes}dab,{\rtimes}bbe,{\rtimes}{\ltimes}{\ltimes}{\ltimes},acca,bbea,beac,cacd,ccac,eacc,acd{\ltimes},dab{\ltimes},cd{\ltimes}{\ltimes},ab{\ltimes}{\ltimes},da{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. e
2. abdded
3. cbdaddc
4. ecdbaaebc

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{bdcc,bdcd,cacd,cdab,cddc\}\)

1. dbbbac
2. \(\varepsilon\)
3. a
4. ba

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,bd,ca,dc,ed,ee\}\)

1. adddbe
2. ea
3. e
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ba,bb\}\)

1. aabbabb
2. bba
3. abbbbab
4. aba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}aad,{\rtimes}c{\ltimes}{\ltimes},aadd,addc,ddcd,dcd{\ltimes},cd{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. bb
3. aabdb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,abb,baa,bab,bba,bbb\}\)

1. bbbaababbb
2. aaaabbbbb
3. abaab
4. aaabbbaaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}db,{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}dbb,bbdb,bdbd,dbbd,dbdb,bdb{\ltimes},db{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. daddaa
2. \(\varepsilon\)
3. d
4. dcd

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}cc,{\rtimes}{\rtimes}ad,{\rtimes}cbd,{\rtimes}cc{\ltimes},{\rtimes}ada,acbd,bdac,cbda,cbdd,dacb,bdd{\ltimes},ada{\ltimes},cc{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},da{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. d
2. \(\varepsilon\)
3. cdc
4. ccaba

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,ab,ba,bc,bd,cb,cd,db,b{\ltimes}\}\)

1. bbbdca
2. cbadaacab
3. cb
4. cdddbdd

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}ca,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}ba,{\rtimes}bab,{\rtimes}cab,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aacb,abba,abca,babb,babc,bbab,bcaa,caac,acb{\ltimes},cab{\ltimes},ab{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. babcaacb
3. bbbccb
4. cab

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aabb,aadb,bbad,bbbb,dada,dbad\}\)

1. a
2. b
3. dddac
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}db,{\rtimes}ba,acb,bac,bda,cbd,dba,da{\ltimes},ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. ccdc
2. d
3. db
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}c,ac,bb,bc,bd,ca,cb,cc,cd,db,dc,c{\ltimes}\}\)

1. cc
2. bdbbabda
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}e,ac,ae,cd,da,ea,e{\ltimes}\}\)

1. e
2. ba
3. edc
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}e,{\rtimes}{\ltimes}{\ltimes},{\rtimes}d{\ltimes},{\rtimes}bc,{\rtimes}ed,acc,bcb,cbc,ccb,dac,dda,ddd,edd,cb{\ltimes},bc{\ltimes},d{\ltimes}{\ltimes},c{\ltimes}{\ltimes},b{\ltimes}{\ltimes}\}\)

1. cdbbadd
2. dcdeceab
3. acad
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,bab\}\)

1. b
2. abbaa
3. a
4. ba

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}b,ab,ac,ba,bd,cb,b{\ltimes},d{\ltimes}\}\)

1. bcddcbad
2. b
3. ab
4. ccc

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},ab,ba,bb,a{\ltimes},b{\ltimes}\}\)

1. \(\varepsilon\)
2. abbb
3. a
4. ab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bda,dce,dee,eaa,ebc\}\)

1. be
2. \(\varepsilon\)
3. bbabb
4. dcbdb

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ac,bb,bc,ca,cc\}\)

1. b
2. cbbb
3. a
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}aba,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},abab,abea,babe,bea{\ltimes},ea{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. abe
3. c
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,{\rtimes}{\ltimes},{\rtimes}d,aa,ad,ba,cd,db,dc,a{\ltimes},d{\ltimes}\}\)

1. \(\varepsilon\)
2. a
3. d
4. cbbb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,acca,baab,bacb,bcca,cbba,cbcb\}\)

1. ccbccc
2. b
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}a{\ltimes}{\ltimes},{\rtimes}bcc,aacc,accc,bccc,caac,ccaa,ccca,cccc,ccc{\ltimes},cc{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. aabc
3. bbaa
4. bccc

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abab,babc,bcab,caab,ccbc\}\)

1. abaca
2. \(\varepsilon\)
3. cbacca
4. aaabbcbc

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}e{\ltimes},{\rtimes}{\rtimes}da,{\rtimes}dae,{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}e{\ltimes}{\ltimes},aeaa,daea,eaad,aad{\ltimes},ad{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ddaebe
3. e
4. c

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,abb,baa,bab,bba,bbb\}\)

1. bbb
2. \(\varepsilon\)
3. bababab
4. baaabbbabb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,{\rtimes}c,bb,bc,cb,cc,b{\ltimes}\}\)

1. cb
2. bb
3. babbaa
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abdc,bbac,bccb,bccd,bcdc,ccbd,dbbc\}\)

1. bdcadab
2. c
3. ca
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}ec,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}bbb,{\rtimes}ecc,{\rtimes}{\ltimes}{\ltimes}{\ltimes},cccb,cccc,eccc,ccb{\ltimes},bbb{\ltimes},cb{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. eeee
2. ecba
3. ea
4. badc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,ae,bb,be,ca,dd,ed\}\)

1. b
2. \(\varepsilon\)
3. d
4. e

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,aca,bbb,bbc,bca,cbb\}\)

1. \(\varepsilon\)
2. aca
3. a
4. b

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}bd,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}bde,{\rtimes}bc{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},bdee,deeb,ebac,eeba,bac{\ltimes},ac{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. dceabe
2. \(\varepsilon\)
3. cabac
4. cececcd

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. aaba
2. aba
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ba,cc,cd,ce,dd,eb,ec\}\)

1. acdddeaa
2. ac
3. badbdedd
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ba,aaa,aab,aba,abb,baa,bab,bba,ab{\ltimes},b{\ltimes}{\ltimes}\}\)

1. bbabaa
2. aabbabbbba
3. aaaa
4. aba

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abda,bddd,dadd,dbaa,dbca,ddda\}\)

1. \(\varepsilon\)
2. b
3. ac
4. a

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{badd,beae,bebc,caca,ccad,dbce\}\)

1. cc
2. cccb
3. adb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,bd,cc,cd,dc,dd\}\)

1. \(\varepsilon\)
2. bababba
3. bbdacbc
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{accb,cabe,cada,ccec,dabb,dbdd\}\)

1. \(\varepsilon\)
2. b
3. cbade
4. addbdcbb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ba,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}b{\ltimes}{\ltimes},{\rtimes}baa,{\rtimes}ba{\ltimes},aaab,aabb,baaa,abb{\ltimes},ba{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. aabaa
3. \(\varepsilon\)
4. ab

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}d{\ltimes},{\rtimes}ce,acb,bac,bce,cbc,ceb,ced,dce,eba,edc,ed{\ltimes},d{\ltimes}{\ltimes}\}\)

1. dbdcd
2. aeeddbcaa
3. bcdebb
4. da

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaae,adab,baca,bbca,bccb,bdbc,ddcc,ddcd\}\)

1. \(\varepsilon\)
2. ccbedca
3. a
4. cceedd

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ba,bb,bc,ca,cc\}\)

1. ba
2. babb
3. ac
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,bb,bc,ca\}\)

1. cbc
2. b
3. \(\varepsilon\)
4. aaca

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}d,{\rtimes}c{\ltimes},{\rtimes}de,bec,ceb,dec,ebe,ece,ec{\ltimes},c{\ltimes}{\ltimes}\}\)

1. c
2. ecc
3. dccbdb
4. dec

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}c,bb,bc,cb,cc,c{\ltimes}\}\)

1. acac
2. cc
3. bcb
4. b

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aabb,abaa,abab,abba,baaa,babb,bbaa\}\)

1. b
2. a
3. \(\varepsilon\)
4. baab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{bba,bbc,bcc,bdb,bdc,cdd,dad\}\)

1. a
2. cbdacb
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}aba,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}abc,{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaaa,abab,abbc,acaa,babb,bbca,bcac,caaa,caca,abc{\ltimes},aaa{\ltimes},aa{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ccca
3. bbabcab
4. aba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}cc,{\rtimes}cca,abbb,bbbb,bbbc,bbcc,cabb,ccab,bcc{\ltimes},cc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes}\}\)

1. ccabbbcc
2. ba
3. cababaccc
4. cabccaaaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abba,accb,accc,bcbb,bccc\}\)

1. \(\varepsilon\)
2. caba
3. b
4. acabbabc

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}b{\ltimes},{\rtimes}ba,aaa,aab,aba,baa,bab,aa{\ltimes},b{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ba
3. bbabb
4. bbbabaabbbbb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{acdb,badc,bcaa,cbda,cbdb,ddbc\}\)

1. bbcd
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abe,acb,adc,cca,edb,eea,eed,eee\}\)

1. \(\varepsilon\)
2. e
3. b
4. aac

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}ea,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}ae,{\rtimes}{\rtimes}da,{\rtimes}ea{\ltimes},{\rtimes}aea,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}dae,aeae,aeec,eaee,ecdd,eecd,dae{\ltimes},cdd{\ltimes},ae{\ltimes}{\ltimes},ea{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes}\}\)

1. ea
2. cdbe
3. aaeabc
4. c

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aabb,abab,baab,baba\}\)

1. \(\varepsilon\)
2. ab
3. bbabbba
4. baabaaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}dd,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}c{\ltimes},{\rtimes}ddd,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}cbb,bbdd,bddc,cbbd,cdda,daaa,dcdd,ddaa,ddcd,ddd{\ltimes},aaa{\ltimes},aa{\ltimes}{\ltimes},dd{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. c
2. bcccddbbbab
3. aac
4. ddd

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}a{\ltimes},{\rtimes}a{\ltimes}{\ltimes},{\rtimes}abb,{\rtimes}bbb,aaaa,aaab,baaa,bbaa,bbba,aab{\ltimes},abb{\ltimes},ab{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. a
2. ab
3. aaaabaaa
4. abb

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aab,acb,acc,bab,bac,cab,cbb,ccb\}\)

1. ca
2. abc
3. bbbabbcba
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaab,aaba,aabb,abaa,abab,abba,baab,bbba\}\)

1. b
2. bbb
3. aaab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}bb,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}aaa,{\rtimes}bb{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},aaac,aacc,accb,adbd,badb,cbad,ccba,dbdd,bdd{\ltimes},dd{\ltimes}{\ltimes},bb{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. dddaddbacd
2. bdbcdbcd
3. bb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ba,bb,bc,ca,cb\}\)

1. bbaaba
2. a
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,b{\ltimes}\}\)

1. baaabaab
2. abb
3. \(\varepsilon\)
4. bababaaaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}ac,{\rtimes}acc,{\rtimes}aca,{\rtimes}cb{\ltimes},abcb,acca,baac,bcba,cabc,cbaa,ccab,aca{\ltimes},aac{\ltimes},ac{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},ca{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. ac
2. caccac
3. aacc
4. abbccc

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abca,bbbc,bcac,cacb,cbbb,ccbc\}\)

1. b
2. aab
3. \(\varepsilon\)
4. c

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}ac,aca,acb,bca,cac,cbc,ca{\ltimes},a{\ltimes}{\ltimes}\}\)

1. abcb
2. ac
3. bdca
4. ccdaabb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,aa,ac,ca,a{\ltimes}\}\)

1. cbbb
2. c
3. \(\varepsilon\)
4. bccb

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}aa,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},{\rtimes}aaa,aaab,aabb,abba,baaa,bbaa,abb{\ltimes},bb{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. aaabb
2. abbabbba
3. \(\varepsilon\)
4. bbaaabaaa

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}b,{\rtimes}c{\ltimes},{\rtimes}bb,bbc,bcb,cbb,bc{\ltimes},c{\ltimes}{\ltimes}\}\)

1. c
2. bbbcac
3. \(\varepsilon\)
4. bab

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,bb,b{\ltimes}\}\)

1. a
2. bbbbabb
3. \(\varepsilon\)
4. b

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}ad,{\rtimes}acc,{\rtimes}ad{\ltimes},accb,cbcd,ccbc,bcd{\ltimes},cd{\ltimes}{\ltimes},ad{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. ad
3. \(\varepsilon\)
4. a

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ab,ac,ba,bb,bc,ca,cb,cc\}\)

1. bcbbacb
2. accaaaacbc
3. b
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,aba,abb,baa,bab,bba\}\)

1. baaba
2. ba
3. bbbab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\ltimes},{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ab,{\rtimes}aa,aba,abb,bab,bba,aa{\ltimes},ba{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. a
3. b
4. baba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}e,{\rtimes}{\rtimes}e{\ltimes},{\rtimes}{\rtimes}ab,{\rtimes}{\rtimes}ce,{\rtimes}{\rtimes}ad,{\rtimes}e{\ltimes}{\ltimes},{\rtimes}cee,{\rtimes}ab{\ltimes},{\rtimes}adb,adbd,bddc,cade,dbdd,dcad,ddca,ade{\ltimes},cee{\ltimes},de{\ltimes}{\ltimes},ab{\ltimes}{\ltimes},ee{\ltimes}{\ltimes},e{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. eddade
2. abccbdece
3. bddaebbac
4. de

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,acb,adb,cdb,daa,dbd\}\)

1. dbb
2. b
3. \(\varepsilon\)
4. cc

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ac,bc,ca,cb,cc\}\)

1. \(\varepsilon\)
2. c
3. b
4. aa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}bcb,{\rtimes}acd,{\rtimes}cb{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},acda,adcd,cdad,dadc,bcb{\ltimes},dcd{\ltimes},cd{\ltimes}{\ltimes},cb{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. b
2. cb
3. \(\varepsilon\)
4. babca

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}a,{\rtimes}a{\ltimes},{\rtimes}{\ltimes}{\ltimes},{\rtimes}ad,aac,abd,acc,ada,bdd,ccd,daa,dab,dda,ddd,cd{\ltimes},ad{\ltimes},d{\ltimes}{\ltimes},a{\ltimes}{\ltimes}\}\)

1. \(\varepsilon\)
2. ad
3. a
4. acdb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}a,ad,ba,bb,cb,cc,dc,dd,de,ec,c{\ltimes}\}\)

1. abbb
2. adc
3. e
4. adec

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{bb,ca,cd,dd\}\)

1. dbbadbbcabd
2. b
3. dacbbaccda
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}b,aa,ab,ba,a{\ltimes}\}\)

1. aaaba
2. aa
3. ab
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aaab,abaa,abba,baab,bbbb\}\)

1. aaaaaa
2. bbababbab
3. bbba
4. aaa

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{abaa,accb,bcae,bebe,ecdb\}\)

1. b
2. dedbcacdceca
3. bcc
4. caeaabebb

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}{\ltimes},{\rtimes}c,{\rtimes}d,bb,bc,cb,cc,cd,db,dc,c{\ltimes},d{\ltimes}\}\)

1. c
2. \(\varepsilon\)
3. d
4. aabbddccaa

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,aab,abb,baa,bab,bba\}\)

1. b
2. bbbaa
3. \(\varepsilon\)
4. aa

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,ad,bb,bd,cb,cd\}\)

1. d
2. dccbd
3. bdadaadab
4. cbad

For each one of the strings below say whether it is generated by the following positive 3-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}b,{\rtimes}dd,{\rtimes}bc,aca,cad,dac,dda,bc{\ltimes},ad{\ltimes},d{\ltimes}{\ltimes},c{\ltimes}{\ltimes}\}\)

1. dbdadb
2. badc
3. dc
4. abae

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}db,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}{\rtimes}aa,{\rtimes}aa{\ltimes},{\rtimes}dbb,{\rtimes}{\ltimes}{\ltimes}{\ltimes},acdc,bbca,bcac,cacd,cdcd,dbbc,dcd{\ltimes},cd{\ltimes}{\ltimes},aa{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes}\}\)

1. aa
2. \(\varepsilon\)
3. cbdbabb
4. acbadcaa

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}cc,{\rtimes}{\rtimes}b{\ltimes},{\rtimes}ccb,{\rtimes}b{\ltimes}{\ltimes},abea,aecd,babe,beae,cbab,ccba,eaec,ecda,cda{\ltimes},da{\ltimes}{\ltimes},a{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. aedaaadae
2. bbb
3. b
4. ecaccbacccd

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaac,baab,cbba,cccb\}\)

1. babba
2. cac
3. abacab
4. ca

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{aaaa,aaab,abaa,abba,baaa,baab,babb\}\)

1. baa
2. aababaabaa
3. abaa
4. aababbb

For each one of the strings below say whether it is generated by the following negative 4-gram grammar:
\(G^-\): \(\{cbda,ccda,cdcb,ecea\}\)

1. acbabbce
2. de
3. ea
4. eaeaeda

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}a,{\rtimes}{\rtimes}{\rtimes}c,{\rtimes}{\rtimes}{\rtimes}{\ltimes},{\rtimes}{\rtimes}c{\ltimes},{\rtimes}{\rtimes}cb,{\rtimes}{\rtimes}ac,{\rtimes}{\rtimes}{\ltimes}{\ltimes},{\rtimes}acb,{\rtimes}c{\ltimes}{\ltimes},{\rtimes}cb{\ltimes},{\rtimes}{\ltimes}{\ltimes}{\ltimes},acbb,bbcb,bcbc,cbbc,cbcb,bcb{\ltimes},cb{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},b{\ltimes}{\ltimes}{\ltimes}\}\)

1. acaab
2. \(\varepsilon\)
3. cababa
4. acacac

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,acd,bab,bac,bca,cdb,dcd\}\)

1. \(\varepsilon\)
2. a
3. abcddba
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{ac,bc,cd,db\}\)

1. adadaccd
2. \(\varepsilon\)
3. dadccddcbdad
4. bccdbd

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aaa,abd,adc,bbb,cbd,cdd,dba,dca\}\)

1. \(\varepsilon\)
2. a
3. cbc
4. b

For each one of the strings below say whether it is generated by the following negative 2-gram grammar:
\(G^-\): \(\{aa,ab,ba,bb\}\)

1. abaab
2. a
3. baaa
4. aabab

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abd,acc,bac,cdd,dab,dbc\}\)

1. ddddbbb
2. dbacabbdcdb
3. bacb
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{ace,adb,cab,cdd\}\)

1. dbdabaeaccca
2. dacaea
3. dcea
4. \(\varepsilon\)

For each one of the strings below say whether it is generated by the following positive 2-gram grammar:
\(G^+\): \(\{{\rtimes}b,aa,ab,ba,bb,a{\ltimes}\}\)

1. aaa
2. \(\varepsilon\)
3. aba
4. ba

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{abd,baa,bac,bbc,cda\}\)

1. cdcaaccc
2. \(\varepsilon\)
3. ccadcda
4. cbdcbabba

For each one of the strings below say whether it is generated by the following positive 4-gram grammar:
\(G^+\): \(\{{\rtimes}{\rtimes}{\rtimes}b,{\rtimes}{\rtimes}{\rtimes}d,{\rtimes}{\rtimes}db,{\rtimes}{\rtimes}bc,{\rtimes}{\rtimes}bb,{\rtimes}bbc,{\rtimes}bc{\ltimes},{\rtimes}dbd,bbcd,bcdc,bcdd,bddb,cddb,dbcd,dbdd,ddbc,ddbd,cdc{\ltimes},dbd{\ltimes},dc{\ltimes}{\ltimes},bd{\ltimes}{\ltimes},bc{\ltimes}{\ltimes},c{\ltimes}{\ltimes}{\ltimes},d{\ltimes}{\ltimes}{\ltimes}\}\)

1. addbdbdddb
2. dbbcbcba
3. dabb
4. cadadda

For each one of the strings below say whether it is generated by the following negative 3-gram grammar:
\(G^-\): \(\{aac,bcc,bda,bdd,cba,dab,ddb\}\)

1. b
2. bdbddddca
3. \(\varepsilon\)
4. dc