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

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

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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

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Introduction KA

Regular expressions

e, f ∈ RegX 0 | 1 | x ∈ X | e · f | e ∪ f | e⋆

Interpretations

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Introduction KA

Regular expressions

e, f ∈ RegX 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

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Introduction KA

Regular expressions

e, f ∈ RegX 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), ∅, IdS, composition, union Rationnal languages correspond to _ : X → P (X ⋆) x → {x} .

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Introduction KA

Regular expressions

e, f ∈ RegX 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), ∅, IdS, composition, union Rationnal languages correspond to _ : X → P (X ⋆) x → {x} .

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Introduction KA

Model equivalence

e, f ∈ RegX Rel |= e = f if ∀S, ∀σ : X → P (S × S) , σ(e) = σ(f )

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Introduction KA

Model equivalence

e, f ∈ RegX 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

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Introduction KL

Kleene Allegories

e, f ∈ Reg∩

X

0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f | e⋆ | e

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Introduction KL

Kleene Allegories

e, f ∈ Reg∩

X

0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f | e⋆ | e Rel |= e = f e = f

Example

a ∩ b = ∅ = σ(a) = {(x, y), (y, z)} σ(b) = {(y, z), (z, t)} σ (a ∩ b) = {(y, z)} ∅ = σ (0) a = {a} = a σ(a) = {(x, y)} σ (a) = {(y, x)} σ(a) A different approach is needed.

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

u, v ∈ WX 0 | 1 | x ∈ X | u · v | u ∩ v | u ∪ v | u⋆ | u

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

G(1) ≔ G(x) ≔ x G(u) ≔ G(u) G(u · v) ≔ G(u) G(v) G(u ∩ v) ≔ G(u) G(v)

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

G(1) ≔ G(x) ≔ x G(u) ≔ G(u) G(u · v) ≔ G(u) G(v) G(u ∩ v) ≔ G(u) G(v)

Example

G (a · b): a b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

G(1) ≔ G(x) ≔ x G(u) ≔ G(u) G(u · v) ≔ G(u) G(v) G(u ∩ v) ≔ G(u) G(v)

Example

G (a · b): a b G ((a · b) ∩ (c · b)): a c b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

G(1) ≔ G(x) ≔ x G(u) ≔ G(u) G(u · v) ≔ G(u) G(v) G(u ∩ v) ≔ G(u) G(v)

Example

G (a · b): a b G ((a · b) ∩ (c · b)): a c b b G (((a ∩ c) · b) ∩ d): a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Graphs/Ground terms

G(1) ≔ G(x) ≔ x G(u) ≔ G(u) G(u · v) ≔ G(u) G(v) G(u ∩ v) ≔ G(u) G(v)

Example

G (a · b): a b G ((a · b) ∩ 1): G ((a · b) ∩ (c · b)): a c b b a b G (((a ∩ c) · b) ∩ d): a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Ground terms

Characterization theorem

◭ is a preorder on graphs.

Theorem

u, v ∈ WX, 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

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Graph languages Reg∩ X

Graphs/Ground terms languages

_ : Reg∩

X

→ P (WX) 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 {w1 · · · · · wn | ∀i, wi ∈ e} .

Graph language of an expression

e ∈ Reg∩

X ,

G(e) ≔ {G(w) | w ∈ e} .

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Graph languages Reg∩ 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

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Petri Automata Examples

Example

(((a ∩ c) · b) ∩ d) ∪ a A B C D E F 1 2 a c d b a

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Examples

Example

  • b · (a · c ∩ b)⋆ · d
  • ∩ a ∪ a · b

A B C D E F G H I J 1 7 2 3 4 5 6 8 b a c b a c b d a b 1 d

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b Success!

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b Failure!

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Reading a graph in an automaton

A B C D E F 1 2 a c d b a a c d b b Failure!

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Language recognised by an automaton

Correctness

For any e ∈ Reg∩

X ,

e

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Language recognised by an automaton

Correctness

For any e ∈ Reg∩

X ,

A (e)

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Language recognised by an automaton

Correctness

For any e ∈ Reg∩

X ,

L (A (e))

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Language recognised by an automaton

Correctness

For any e ∈ Reg∩

X ,

L (A (e)) = ◭G (e).

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Petri Automata Recognition

Language recognised by an automaton

Correctness

For any e ∈ Reg∩

X ,

L (A (e)) = ◭G (e).

This far:

e, f ∈ Reg∩

X

Rel |= e f ⇔

◭G(e) ⊆ ◭G(f ) ⇔ L (A (e)) ⊆ L (A (f )).

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Comparing automata

Restriction: identity-free lattice terms

G ((a · b) ∩ 1): G (a ∩ b): a b a b

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Comparing automata

Restriction: identity-free lattice terms

G ((a · b) ∩ 1): G (a ∩ b): a b a b u, v ∈ W−

X 0 | 1 | x ∈ X | u · v | u ∩ v | u ∪ v | u⋆ | u

Identity-free Kleene Lattice

e, f ∈ Reg∩−

X

0 | 1 | x ∈ X | e · f | e ∩ f | e ∪ f | e+ | e

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages.

Ongoing/Future work

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata.

Ongoing/Future work

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata. Decidability of simple automata equivalence, thus of relational equivalence for Identity-free Kleene Lattices.

Ongoing/Future work

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata. Decidability of simple automata equivalence, thus of relational equivalence for Identity-free Kleene Lattices. Simple Petri automata equivalence is EXPSPACE-complete. This decision procedure was implemented in OCaml, and is available as an online application.

Ongoing/Future work

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata. Decidability of simple automata equivalence, thus of relational equivalence for Identity-free Kleene Lattices. Simple Petri automata equivalence is EXPSPACE-complete. This decision procedure was implemented in OCaml, and is available as an online application.

Ongoing/Future work

Converting back Petri automata into expressions.

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata. Decidability of simple automata equivalence, thus of relational equivalence for Identity-free Kleene Lattices. Simple Petri automata equivalence is EXPSPACE-complete. This decision procedure was implemented in OCaml, and is available as an online application.

Ongoing/Future work

Converting back Petri automata into expressions. Decidability with 1 and/or _.

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Conclusion and future work

Results

Reduction of relational equivalence to equality of closed graph languages. Representation of closed graph languages through Petri automata. Decidability of simple automata equivalence, thus of relational equivalence for Identity-free Kleene Lattices. Simple Petri automata equivalence is EXPSPACE-complete. This decision procedure was implemented in OCaml, and is available as an online application.

Ongoing/Future work

Converting back Petri automata into expressions. Decidability with 1 and/or _. Completeness.

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

That’s it!

Thank you ! The content presented here has been accepted for publication in LICS 2015. http://perso.ens-lyon.fr/paul.brunet/rklm.

Paul Brunet Kleene Allegories & Petri automata April 8, 2015

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Conclusions

Plan

1

Introduction Kleene Algebra Kleene Allegories

2

Graph languages Ground terms Regular expressions with intersection and converse

3

Petri Automata Examples Recognition by Petri automata

4

Comparing automata

5

Conclusions

Paul Brunet Kleene Allegories & Petri automata April 8, 2015