Petri automata for Kleene Allegories Paul Brunet & Damien Pous - PowerPoint PPT Presentation

Petri automata for Kleene Allegories Paul Brunet & Damien Pous Plume team – LIP, CNRS, ENS de Lyon, Inria, UCBL, Université de Lyon, UMR 5668 April 8, 2015 Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Regular expressions e , f ∈ Reg X � 0 | 1 | x ∈ X | e · f | e ∪ f | e ⋆ Interpretations Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Regular expressions e , f ∈ Reg X � 0 | 1 | x ∈ X | e · f | e ∪ f | e ⋆ Interpretations languages: Σ a finite set, σ : X → P (Σ ⋆ ), ∅ , { ǫ } , concatenation, union Rationnal languages correspond to � _ � : X → P ( X ⋆ ) x �→ { x } . Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Regular expressions e , f ∈ Reg X � 0 | 1 | x ∈ X | e · f | e ∪ f | e ⋆ Interpretations languages: Σ a finite set, σ : X → P (Σ ⋆ ), ∅ , { ǫ } , concatenation, union relations: S a set, σ : X → P ( S × S ), ∅ , Id S , composition, union → P ( X ⋆ ) Rationnal languages correspond to � _ � : X �→ { x } . x Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Regular expressions e , f ∈ Reg X � 0 | 1 | x ∈ X | e · f | e ∪ f | e ⋆ Interpretations languages: Σ a finite set, σ : X → P (Σ ⋆ ), ∅ , { ǫ } , concatenation, union relations: S a set, σ : X → P ( S × S ) , ∅ , Id S , composition, union → P ( X ⋆ ) Rationnal languages correspond to � _ � : X �→ { x } . x Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Model equivalence e , f ∈ Reg X Rel | = e = f if ∀ S , ∀ σ : X → P ( S × S ) , σ ( e ) = σ ( f ) Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KA Model equivalence e , f ∈ Reg X Rel | = e = f if ∀ S , ∀ σ : X → P ( S × S ) , σ ( e ) = σ ( f ) Theorem Rel | = e = f ⇔ � e � = � f � Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KL Kleene Allegories | e ⋆ | e � e , f ∈ Reg � ∩ � 0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f X Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Introduction KL Kleene Allegories | e ⋆ | e � e , f ∈ Reg � ∩ � 0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f X Rel | = e = f � � e � = � f � Example � a ∩ b � = ∅ = � 0 � = { a } = � a � � � a � { ( x , y ) , ( y , z ) } σ ( a ) = { ( x , y ) } σ ( a ) = σ ( b ) = { ( y , z ) , ( z , t ) } σ ( a � ) = { ( y , x ) } � σ ( a ) σ ( a ∩ b ) = { ( y , z ) } � ∅ = σ (0) A different approach is needed. Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms u , v ∈ W X � 0 | 1 | x ∈ X | u · v | u ∩ v | u ∪ v | u ⋆ | u � Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms G ( u ) G ( v ) G ( u · v ) ≔ G (1) ≔ G ( u ) x G ( x ) ≔ G ( u ∩ v ) ≔ G ( u � ) ≔ G ( u ) G ( v ) Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms G ( u ) G ( v ) G ( u · v ) ≔ G (1) ≔ G ( u ) x G ( x ) ≔ G ( u ∩ v ) ≔ G ( u � ) ≔ G ( u ) G ( v ) Example a b G ( a · b ): Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms G ( u ) G ( v ) G ( u · v ) ≔ G (1) ≔ G ( u ) x G ( x ) ≔ G ( u ∩ v ) ≔ G ( u � ) ≔ G ( u ) G ( v ) Example a b G ( a · b ): a b G (( a · b ) ∩ ( c · b )): c b Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms G ( u ) G ( v ) G ( u · v ) ≔ G (1) ≔ G ( u ) x G ( x ) ≔ G ( u ∩ v ) ≔ G ( u � ) ≔ G ( u ) G ( v ) Example a b G ( a · b ): a b G (( a · b ) ∩ ( c · b )): c b a b c G ((( a ∩ c ) · b ) ∩ d ): d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Graphs/Ground terms G ( u ) G ( v ) G ( u · v ) ≔ G (1) ≔ G ( u ) x G ( x ) ≔ G ( u ∩ v ) ≔ G ( u � ) ≔ G ( u ) G ( v ) Example a b G ( a · b ): G (( a · b ) ∩ 1): a b a G (( a · b ) ∩ ( c · b )): b c b a b c G ((( a ∩ c ) · b ) ∩ d ): d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Graph languages Ground terms Characterization theorem ◭ is a preorder on graphs. Theorem u , v ∈ W X , Rel | = u � v ⇔ G ( u ) ◭ G ( v ) P. J. Freyd and A. Scedrov. Categories, Allegories . NH, 1990 H. Andréka and D. Bredikhin. The equational theory of union-free algebras of relations. Alg. Univ. , 33(4):516–532, 1995 Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Reg ∩ Graph languages X Graphs/Ground terms languages � _ � : Reg � ∩ → P (W X ) X � 0 � ≔ ∅ � 1 � ≔ { 1 } � x � ≔ { x } � e � � ≔ { w � | w ∈ � e � } � e · f � ≔ { w · w ′ | w ∈ � e � and w ′ ∈ � f � } � e ∩ f � ≔ { w ∩ w ′ | w ∈ � e � and w ′ ∈ � f � } � e ∪ f � ≔ � e � ∪ � f � � e ⋆ � ≔ � n ∈ N { w 1 · · · · · w n | ∀ i , w i ∈ � e � } . Graph language of an expression e ∈ Reg � ∩ X , G ( e ) ≔ { G ( w ) | w ∈ � e � } . Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Reg ∩ Graph languages X Characterization theorem ◭ S is the downwards closure of S with respect to ◭ . Theorem e , f ∈ Reg � ∩ X , Rel | = e � f ⇔ ◭ G ( e ) ⊆ ◭ G ( f ) Almost proven in: H. Andréka, S. Mikulás, and I. Németi. The equational theory of Kleene lattices. TCS , 412(52):7099–7108, 2011 Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Examples Example ((( a ∩ c ) · b ) ∩ d ) ∪ a B a b 1 D c A 0 C E d a 2 F Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Examples Example b � · ( a · c ∩ b ) ⋆ · d ∩ a ∪ a � · b � � 1 J 8 d 7 a c 2 C D c d b � B 1 3 4 F b A 0 E b a G a � b 5 H 6 I Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c d Success! Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Failure! Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Reading a graph in an automaton a B b 1 D c A 0 C d E a 2 F a b c b d Failure! Paul Brunet Kleene Allegories & Petri automata April 8, 2015

Petri Automata Recognition Language recognised by an automaton Correctness For any e ∈ Reg � ∩ X , e Paul Brunet Kleene Allegories & Petri automata April 8, 2015