Partially - Commutative Context - Free Languages Wojciech Czerwi ski - PowerPoint PPT Presentation

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c ~ BAA’ A ’BA B’AA ’

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b ~ BAA’ A ’BA B’AA ’ AA ’

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b c ~ BAA’ A ’BA B’AA ’ A ’A ’ AA ’

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b c ’ a ~ BAA’ A ’BA B’AA ’ A ’A AA ’ A ’

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b c ’ a a ~ BAA’ A ’BA B’AA ’ A ’A AA ’ ε A ’

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b c ’ a a ~ BAA’ A ’BA B’AA ’ A ’A AA ’ ε A ’ L = L ( S ) without letters c

Example of transitive PCCFL s a b S S SA S SB ε c c a b A ’ B’ A B A ’ ε B’ ε Equivalence classes of D - threads = { S, A, B } , { A ’ } , { B’ } a b a s c S SA ABA A ’BA SBA SABA c b c ’ a a ~ BAA’ A ’BA B’AA ’ A ’A AA ’ ε A ’ L = L ( S ) without letters c L = { wsv: w,v ∊ { a,b } *, #w ( a ) =#v ( a ) , #w ( b ) =#v ( b )}

Our results tPCCFL PA PCCFL

Our results • pumping lemma for transitive PCCFL tPCCFL PA PCCFL

Our results • pumping lemma for transitive PCCFL • transitive PCCFL is a strict subclass of PCCFL L 1 tPCCFL PA PCCFL

Our results • pumping lemma for transitive PCCFL • transitive PCCFL is a strict subclass of PCCFL • transitive PCCFL and PA languages are incomparable L 3 L 2 L 1 tPCCFL PA PCCFL

Pumping lemma for transitive PCCFL

Pumping lemma for transitive PCCFL For any transitive PCCFL language L there is some n ≥ 0 such that any w in L of length at least n may be split into words w =xy z , so that there exists words t and u such that tu ≠ ε and for each m ≥ 0, x t m yu m z ∊ L

Pumping lemma for transitive PCCFL For any transitive PCCFL language L there is some n ≥ 0 such that any w in L of length at least n may be split into words w =xy z , so that there exists words t and u such that tu ≠ ε and for each m ≥ 0, x t m yu m z ∊ L Properties

Pumping lemma for transitive PCCFL For any transitive PCCFL language L there is some n ≥ 0 such that any w in L of length at least n may be split into words w =xy z , so that there exists words t and u such that tu ≠ ε and for each m ≥ 0, x t m yu m z ∊ L Properties • there are two places for new words

Pumping lemma for transitive PCCFL For any transitive PCCFL language L there is some n ≥ 0 such that any w in L of length at least n may be split into words w =xy z , so that there exists words t and u such that tu ≠ ε and for each m ≥ 0, x t m yu m z ∊ L Properties • there are two places for new words • t and u are arbitrary words

Pumping lemma for transitive PCCFL For any transitive PCCFL language L there is some n ≥ 0 such that any w in L of length at least n may be split into words w =xy z , so that there exists words t and u such that tu ≠ ε and for each m ≥ 0, x t m yu m z ∊ L Properties • there are two places for new words • t and u are arbitrary words • the same lemma holds for PA languages

Pumping lemmas transitiv e Partia � y - Commutativ e Context - Free Languages and PA Languages Commutativ e Context - Fre e Context - Fre e Languages Languages Regular Languages

Pumping lemmas transitiv e arbitrary words Partia � y - Commutativ e Context - Free Languages and PA Languages Commutativ e Context - Fre e Context - Fre e Languages Languages Regular words from word w Languages

Pumping lemmas transitiv e arbitrary words two places Partia � y - Commutativ e Context - Free Languages and PA Languages Commutativ e Context - Fre e Context - Fre e Languages Languages Regular one place words from word w Languages

Example languages tPCCFL PA PCCFL

Example languages L 1 tPCCFL PA PCCFL

Example languages L 2 L 1 tPCCFL PA PCCFL

Example languages L 3 L 2 L 1 tPCCFL PA PCCFL

L 1 ( tPCCFL ⊊ PCCFL ) L 1 tPCCFL PA PCCFL