Petri automata for Kleene Allegories Talk at the Rapido meeting Paul - PowerPoint PPT Presentation

Petri automata for Kleene Allegories Talk at the Rapido meeting Paul Brunet & Damien Pous Plume team – LIP, CNRS, ENS de Lyon, Inria, UCBL, Université de Lyon, UMR 5668 June 19, 2015 Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , � ( i , k ) ∈ ( R ∩ S ) there is k such that ( k , j ) ∈ T Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , � ( i , k ) ∈ ( R ∩ S ) there is k such that ( k , j ) ∈ T  ( i , k ) ∈ R  ( i , k ) ∈ S thus ( k , j ) ∈ T  Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , � ( i , k ) ∈ ( R ∩ S ) there is k such that ( k , j ) ∈ T  ( i , k ) ∈ R  ( i , k ) ∈ S thus ( k , j ) ∈ T  � ( i , j ) ∈ R ◦ T thus ( i , j ) ∈ S ◦ T Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , � ( i , k ) ∈ ( R ∩ S ) there is k such that ( k , j ) ∈ T  ( i , k ) ∈ R  ( i , k ) ∈ S thus ( k , j ) ∈ T  � ( i , j ) ∈ R ◦ T thus ( i , j ) ∈ S ◦ T hence ( i , j ) ∈ ( R ◦ T ) ∩ ( S ◦ T ). Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Motivation : Relation Algebra ( R ∩ S ) ◦ T ⊆ ( R ◦ T ) ∩ ( S ◦ T ) Let ( i , j ) ∈ ( R ∩ S ) ◦ T , � ( i , k ) ∈ ( R ∩ S ) there is k such that ( k , j ) ∈ T  ( i , k ) ∈ R  ( i , k ) ∈ S thus ( k , j ) ∈ T  � ( i , j ) ∈ R ◦ T thus ( i , j ) ∈ S ◦ T hence ( i , j ) ∈ ( R ◦ T ) ∩ ( S ◦ T ). Simple and boring : could it be done automatically ? Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Expressions Outline Expressions 1 Kleene Algebra Kleene Allegories Graph languages 2 Ground terms Regular expressions with intersection and converse Petri Automata 3 Examples Recognition by Petri automata Comparing automata 4 Conclusions 5 Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Expressions Kleene Algebra Regular expressions e , f ∈ Reg X � 0 | 1 | x ∈ X | e · f | e ∪ f | e ⋆ Interpretations Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Expressions Kleene Algebra 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 June 19, 2015

Expressions Kleene Algebra 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 June 19, 2015

Expressions Kleene Algebra 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 June 19, 2015

Expressions Kleene Algebra Relational 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 June 19, 2015

Expressions Kleene Algebra Relational 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 � Corollary Relational equivalence is decidable for regular expressions. Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Expressions 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 June 19, 2015

Expressions KL Kleene Allegories | e ⋆ | e � e , f ∈ Reg � ∩ � 0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f X Rel | = e = f ⇐ ⇒ � e � = � f � ? Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Expressions 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 June 19, 2015

Expressions KL Outline Expressions 1 Kleene Algebra Kleene Allegories Graph languages 2 Ground terms Regular expressions with intersection and converse Petri Automata 3 Examples Recognition by Petri automata Comparing automata 4 Conclusions 5 Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Graph languages Outline Expressions 1 Kleene Algebra Kleene Allegories Graph languages 2 Ground terms Regular expressions with intersection and converse Petri Automata 3 Examples Recognition by Petri automata Comparing automata 4 Conclusions 5 Paul Brunet Kleene Allegories & Petri automata June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 2015

Graph languages Ground terms Preorder Preorder on graphs G ◭ H if there exists a graph morphism from H to G . a b (( a ∩ c ) · b ) ∩ d G : c d a b ( a · b ) ∩ ( c · b ) H : c b Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Graph languages Ground terms Example: Modularity law Rel | = ( a · b ) ∩ c ≤ a · ( b ∩ a � · c ) (1) c a b a c a b Paul Brunet Kleene Allegories & Petri automata June 19, 2015

Graph languages Ground terms Characterization theorem 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 June 19, 2015