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

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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

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T)

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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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

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T) Let (i, j) ∈ (R ∩ S) ◦ T, there is k such that (i, k) ∈ (R ∩ S) (k, j) ∈ T

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T) Let (i, j) ∈ (R ∩ S) ◦ T, there is k such that (i, k) ∈ (R ∩ S) (k, j) ∈ T thus    (i, k) ∈ R (i, k) ∈ S (k, j) ∈ T

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T) Let (i, j) ∈ (R ∩ S) ◦ T, there is k such that (i, k) ∈ (R ∩ S) (k, j) ∈ T thus    (i, k) ∈ R (i, k) ∈ S (k, j) ∈ T thus

  • (i, j) ∈ R ◦ T

(i, j) ∈ S ◦ T

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T) Let (i, j) ∈ (R ∩ S) ◦ T, there is k such that (i, k) ∈ (R ∩ S) (k, j) ∈ T thus    (i, k) ∈ R (i, k) ∈ S (k, j) ∈ T thus

  • (i, j) ∈ R ◦ T

(i, j) ∈ S ◦ T hence (i, j) ∈ (R ◦ T) ∩ (S ◦ T).

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Motivation : Relation Algebra

(R ∩ S) ◦ T ⊆ (R ◦ T) ∩ (S ◦ T) Let (i, j) ∈ (R ∩ S) ◦ T, there is k such that (i, k) ∈ (R ∩ S) (k, j) ∈ T thus    (i, k) ∈ R (i, k) ∈ S (k, j) ∈ T thus

  • (i, j) ∈ R ◦ T

(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

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Expressions

Outline

1

Expressions 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 June 19, 2015

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Expressions Kleene Algebra

Regular expressions

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

Interpretations

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Expressions Kleene Algebra

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 June 19, 2015

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Expressions Kleene Algebra

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 June 19, 2015

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Expressions Kleene Algebra

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 June 19, 2015

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Expressions Kleene Algebra

Relational equivalence

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

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Expressions Kleene Algebra

Relational equivalence

e, f ∈ RegX 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

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Expressions 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 June 19, 2015

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Expressions KL

Kleene Allegories

e, f ∈ Reg∩

X

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

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Expressions 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 June 19, 2015

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Expressions KL

Outline

1

Expressions 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 June 19, 2015

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Graph languages

Outline

1

Expressions 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 2015

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Graph languages Ground terms

Preorder

Preorder on graphs

G ◭ H if there exists a graph morphism from H to G. G : H : a c d b a c b b ((a ∩ c) · b) ∩ d (a · b) ∩ (c · b)

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Graph languages Ground terms

Example: Modularity law

Rel |= (a · b) ∩ c ≤ a · (b ∩ a · c) (1) a a c b a c b

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Graph languages Ground terms

Characterization theorem

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 June 19, 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 June 19, 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 ) Follows easily from:

  • 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 June 19, 2015

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Petri Automata

Outline

1

Expressions 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 2015

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Comparing automata

Outline

1

Expressions 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 June 19, 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 June 19, 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 June 19, 2015

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Comparing automata

Decision procedure

e, f ∈ Reg∩−

X

Rel |= e f ⇔

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

Problem:

How to compare two Petri automata? . . . not that easily! L (A1) ⊆ L (A2) if and only if there is a simulation relation ⊆ P (P1) × P (P2 P1) between the configurations of A1 and the partial maps from the places of A2 to the places of A1.

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Decision procedure

e, f ∈ Reg∩−

X

Rel |= e f ⇔

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

Problem:

How to compare two Petri automata? . . . not that easily! L (A1) ⊆ L (A2) if and only if there is a simulation relation ⊆ P (P1) × P (P2 P1) between the configurations of A1 and the partial maps from the places of A2 to the places of A1.

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Decision procedure

e, f ∈ Reg∩−

X

Rel |= e f ⇔

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

Problem:

How to compare two Petri automata? . . . not that easily! L (A1) ⊆ L (A2) if and only if there is a simulation relation ⊆ P (P1) × P (P2 P1) between the configurations of A1 and the partial maps from the places of A2 to the places of A1.

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulations - Non-deterministic Finite Automata

⊆ Q1 × P (Q2) p0 p1 p2 p3 a b a q0 q1 q2 q3 q4 q5 a b b b a

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulations - Non-deterministic Finite Automata

⊆ Q1 × P (Q2) p0 p1 p2 p3 a b a q0 q1 q2 q3 q4 q5 a b b b a p0 { q0 }

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulations - Non-deterministic Finite Automata

⊆ Q1 × P (Q2) p0 p1 p2 p3 a b a q0 q1 q2 q3 q4 q5 a b b b a p0 { q0 } p1 { q1 }

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulations - Non-deterministic Finite Automata

⊆ Q1 × P (Q2) p0 p1 p2 p3 a b a q0 q1 q2 q3 q4 q5 a b b b a p0 { q0 } p1 { q1 } p2 {q2, q4}

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulations - Non-deterministic Finite Automata

⊆ Q1 × P (Q2) p0 p1 p2 p3 a b a q0 q1 q2 q3 q4 q5 a b b b a p0 { q0 } p1 { q1 } p2 {q2, q4} p3 { q1 }

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulation - Petri Automata

⊆ P (P1) × P (P2 P1) 1 2 3 a b b A B C D G H I b a c b a b

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulation - Petri Automata

⊆ P (P1) × P (P2 P1) 1 2 3 a b b A B C D G H I b a c b a b

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulation - Petri Automata

⊆ P (P1) × P (P2 P1) 1 2 3 a b b A B C D G H I b a c b a b

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulation - Petri Automata

⊆ P (P1) × P (P2 P1) 1 2 3 a b b A B C D G H I b a c b a b

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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Comparing automata

Simulation - Petri Automata

⊆ P (P1) × P (P2 P1) 1 2 3 a b b A B C D G H I b a c b a b

Paul Brunet Kleene Allegories & Petri automata June 19, 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 June 19, 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 June 19, 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 June 19, 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 June 19, 2015

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SLIDE 75

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 June 19, 2015

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SLIDE 76

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 June 19, 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 _. Complete axiomatization.

Paul Brunet Kleene Allegories & Petri automata June 19, 2015

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SLIDE 78

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 June 19, 2015

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SLIDE 79

Conclusions

Plan

1

Expressions 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 June 19, 2015