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

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

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Partially - Commutative Context - Free Languages Wojciech Czerwi ski S awomir Lasota G MES 2010 Outline Outline Context Free Languages + di ff erent notions of commutativity ( PCCFL ) Outline Context Free Languages + di ff erent

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Partially-Commutative Context-Free Languages

Wojciech Czerwiński Sławomir Lasota

G∀MES 2010

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Outline

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Outline

  • Context Free Languages + different notions of

commutativity (PCCFL)

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Outline

  • Context Free Languages + different notions of

commutativity (PCCFL)

  • Examples
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Outline

  • Context Free Languages + different notions of

commutativity (PCCFL)

  • Examples
  • Our results (expressivity, pumping lemmas)
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Outline

  • Context Free Languages + different notions of

commutativity (PCCFL)

  • Examples
  • Our results (expressivity, pumping lemmas)
  • Further research
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X XBC Greibach normal form a

Context-Free Languages

Left derivation

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X XBC Greibach normal form a

Context-Free Languages

Left derivation

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X XBC a

Context-Free Languages

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X XBC a B C b c X BC a ε ε

Context-Free Languages

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X XBC a B C b c X BC a ε ε L(X) = ⋃n≥1 an (bc)n

Context-Free Languages

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X XBC a B C b c X BC a ε ε

Context-Free Languages

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X XBC a B C b c X BC a ε ε

Context-Free Languages Commutative

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X XBC a B C b c L(X) = #a = #b = #c a „preceds” b and c X BC a ε ε

Context-Free Languages Commutative

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Mixing sequentiality with commutativity

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Mixing sequentiality with commutativity

Three different notions

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Mixing sequentiality with commutativity

  • X (Y

;Z) || W L(X) = a((L(Y)‧L(Z) || L(W)) PA languages (CFL with interleaving) aThree different notions

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Mixing sequentiality with commutativity

  • X (Y

;Z) || W L(X) = a((L(Y)‧L(Z) || L(W)) PA languages (CFL with interleaving)

  • X YZ + relation on nonterminals

Partially Commutative CFL (PCCFL) a a

Three different notions

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Mixing sequentiality with commutativity

  • X (Y

;Z) || W L(X) = a((L(Y)‧L(Z) || L(W)) PA languages (CFL with interleaving)

  • X YZ + relation on nonterminals

Partially Commutative CFL (PCCFL)

  • X YZ + plus relation on terminals

Mazurkiewicz traces a a a

Three different notions

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PA languages

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PA languages

S P;C a

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PA languages

S P;C a S s ε

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PA languages

S P;C a P S || D b S s ε

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PA languages

S P;C a P S || D b C c ε S s ε

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PA languages

S P;C a P S || D b C c ε D d ε S s ε

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S; a P C

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S b|| S D ; a P C

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S b|| S D ; a P C ε s

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S b|| S D ; a P C d ε ε s

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PA languages

S P;C a P S || D b C c ε D d ε S s ε S b|| S D ; a P C c ε d ε ε s

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PA languages

S P;C a P S || D b C c ε D d ε S s ε absdc, abdsc ∊ L(S) S b|| S D ; a P C c ε d ε ε s

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PCCFL

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PCCFL

  • V - nonterminal symbols (V = {X,B,C})
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PCCFL

  • V - nonterminal symbols (V = {X,B,C})
  • set of transitions (X XBC, ...)

a

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PCCFL

  • V - nonterminal symbols (V = {X,B,C})
  • set of transitions (X XBC, ...)
  • symmetric independence relation I⊆ V×V

(I = {(B,C),(C,B)}) a

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PCCFL

  • V - nonterminal symbols (V = {X,B,C})
  • set of transitions (X XBC, ...)
  • symmetric independence relation I⊆ V×V

(I = {(B,C),(C,B)})

  • context-free grammar ⇒ I = ∅

a

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PCCFL

  • V - nonterminal symbols (V = {X,B,C})
  • set of transitions (X XBC, ...)
  • symmetric independence relation I⊆ V×V

(I = {(B,C),(C,B)})

  • context-free grammar ⇒ I = ∅
  • commutative context-free grammar ⇒ I = V×V

a

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PCCFL example

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PCCFL example

P WBCT a

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PCCFL example

P WBCT a W WBC a

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PCCFL example

P WBCT a W WBC a W U s

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PCCFL example

P WBCT a W WBC a W U s B b ε

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PCCFL example

P WBCT a W WBC a W U s B b ε C c ε

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PCCFL example

P WBCT a W WBC a W U s B b ε C c ε T t ε

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PCCFL example

P WBCT a W WBC a W U s B b ε C c ε T t ε U u ε

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3 s U(BC)3T

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3 s U(BC)3T ~I B3TUC3

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3 TUC3 b3 s U(BC)3T ~I B3TUC3

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3 TUC3 b3 C3 tu s U(BC)3T ~I B3TUC3

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε P W(BC)3T a3 TUC3 b3 C3 tu c3 ε s U(BC)3T ~I B3TUC3

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PCCFL example

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} a3sb3tuc3∊L(P) B b ε C c ε T t ε U u ε P W(BC)3T a3 TUC3 b3 C3 tu c3 ε s U(BC)3T ~I B3TUC3

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transitive PCCFL

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transitive PCCFL

  • dependence relation D = V×V∖I
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transitive PCCFL

  • dependence relation D = V×V∖I
  • transitive dependence relation D
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transitive PCCFL

  • dependence relation D = V×V∖I
  • transitive dependence relation D
  • equivalence class of D - thread
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Example of transitive PCCFL

S ε s

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Example of transitive PCCFL

S ε s S SA a

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Example of transitive PCCFL

S ε s S SA a S SB b

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’}

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} S

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} S SA a

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} S SA a b SBA

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} S SA a b SBA a SABA

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Example of transitive PCCFL

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} S SA a b SBA a SABA s ABA

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Example of transitive PCCFL

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

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Our results

PA tPCCFL PCCFL

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Our results

  • pumping lemma for transitive PCCFL

PA tPCCFL PCCFL

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Our results

  • pumping lemma for transitive PCCFL
  • transitive PCCFL is a strict subclass of PCCFL

PA tPCCFL PCCFL L1

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Our results

  • pumping lemma for transitive PCCFL
  • transitive PCCFL is a strict subclass of PCCFL
  • transitive PCCFL and PA languages are

incomparable PA tPCCFL PCCFL L1 L2 L3

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Pumping lemma for transitive PCCFL

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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=xyz, so that there exists words t and u such that tu≠ε and for each m≥0, xtmyumz ∊L

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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=xyz, so that there exists words t and u such that tu≠ε and for each m≥0, xtmyumz ∊L Properties

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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=xyz, so that there exists words t and u such that tu≠ε and for each m≥0, xtmyumz ∊L

  • there are two places for new words

Properties

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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=xyz, so that there exists words t and u such that tu≠ε and for each m≥0, xtmyumz ∊L

  • there are two places for new words
  • t and u are arbitrary words

Properties

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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=xyz, so that there exists words t and u such that tu≠ε and for each m≥0, xtmyumz ∊L

  • there are two places for new words
  • t and u are arbitrary words
  • the same lemma holds for PA languages

Properties

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Pumping lemmas

Regular Languages Context-Free Languages Commutative Context-Free Languages transitive Partiay-Commutative Context-Free Languages and PA Languages

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Pumping lemmas

Regular Languages Context-Free Languages Commutative Context-Free Languages transitive Partiay-Commutative Context-Free Languages and PA Languages

words from word w arbitrary words

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Pumping lemmas

Regular Languages Context-Free Languages Commutative Context-Free Languages transitive Partiay-Commutative Context-Free Languages and PA Languages

two places

  • ne place

words from word w arbitrary words

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Example languages

PA tPCCFL PCCFL

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Example languages

PA tPCCFL PCCFL L1

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Example languages

PA tPCCFL PCCFL L1 L2

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Example languages

PA tPCCFL PCCFL L1 L2 L3

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L1 (tPCCFL ⊊ PCCFL)

L1 PA tPCCFL PCCFL

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L1 (tPCCFL ⊊ PCCFL)

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε L1 PA tPCCFL PCCFL

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L1 (tPCCFL ⊊ PCCFL)

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε L1 PA tPCCFL PCCFL =L(P)

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L1 (tPCCFL ⊊ PCCFL)

P WBCT a W WBC a W U s I = {(B,C),(T,C),(B,U),(T,U),...} B b ε C c ε T t ε U u ε L1 Pumping lemma for tPCCFL (also for PA) PA tPCCFL PCCFL =L(P)

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L2 (tPCCFL ⊈ PA)

L2 PA tPCCFL PCCFL

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L2 (tPCCFL ⊈ PA)

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε L2 Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} PA tPCCFL PCCFL

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L2 (tPCCFL ⊈ PA)

S ε s S SA a S SB b A c A ’ B c B’ A ’ a ε B’ b ε L2 Equivalence classes of D - threads = {S, A, B}, {A ’}, {B’} PA tPCCFL PCCFL =L(S)

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L3 (PA ⊈ tPCCFL)

L3 PA tPCCFL PCCFL

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L3 (PA ⊈ tPCCFL)

S P;C a P S || D b A a ε B b ε C c ε D d ε S s ε L3 PA tPCCFL PCCFL

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L3 (PA ⊈ tPCCFL)

S P;C a P S || D b A a ε B b ε C c ε D d ε S s ε L3 PA tPCCFL PCCFL =L(S)

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Further research

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Further research

  • PA ⊈ PCCFL (not only tPCCFL)? Maybe L3
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Further research

  • PA ⊈ PCCFL (not only tPCCFL)? Maybe L3
  • corresponding automata model (stateless

automata with multiply stacks)

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Further research

  • PA ⊈ PCCFL (not only tPCCFL)? Maybe L3
  • corresponding automata model (stateless

automata with multiply stacks)

  • applications in modeling (multiply threads

architecture)

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Thank you!