Regular Expressions for Polynomial Coalgebras

Marcello Bonsangue1,2, Jan Rutten1,3 and Alexandra Silva1

1Centrum voor Wiskunde en Informatica2LIACS - Leiden University

3Vrije Universiteit Amsterdam

CoCoCo 2008

Pisa, January 21-23, 2008

Alexandra and Marcello (CWI and LIACS) Regular Expressions for Polynomial Coalg. CoCoCo 2008 1 / 43

MotivationRegular expressions for polynomial coalgebras

Regular ExpressionsWell known for classical automataThey provide a concise description of the accepted languageAutomaton ! Regular expressions

b∗a(a + bb∗a)∗

Polynomial coalgebrasGeneralizations of deterministic automataPolynomial coalgebras: set of states S + t : S → GS

G ::= Id | B | G ×G | G + G | GA

G = 2× IdA – Deterministic automataSet of states S + trans. function t : S → 2× SA

G ::= Id | B | G ×G | G + G | GA

G = Id × A× Id – Binary tree automataSet of states S + trans. function t : S → S × A× S

G ::= Id | B | G ×G | G + G | GA

G = (B × Id)A – Mealy machines

Set of states S + trans. function t : S → (B × S)A

0|11|0

The story

Logic for Mealy machines[BRS08] Coalgebraic Logic and Synthesis of Mealy Machines. FoSSaCS 2008

Generalization to polynomial coalgebras[BRS08-2] Regular expressions for polynomial coalgebras. Submitted

The story

Part I

Logic for Mealy machines

Part II

Regular expressions for polynomial coalgebras

intuition

Part I

Motivation

Design of Sequential systems(Binary) Mealy machines ↔ digital circuitsThere is no formal way of specifying Mealy machinesTypically they are “defined” in a natural language, such as English

Source of ambiguitiesWe need a formal way of specifying Mealy machines

Motivation

What will we show?

Specification

Mealy Machine

φ1a1 | b1

a2 | b2

Digital Circuit

What will we show?

Specification

Mealy Machine

φ1a1 | b1

a2 | b2

Digital Circuit

But first. . .

What do we mean by Binary Mealy Machine?

Mealy machines = Deterministic Mealy machinesMealy machine = set of states S + transition function f

f : S → (B × S)A

f (s)(a) = < b, s′ >

sa|b // s′

A is the input alphabet and B is the output alphabetBinary Mealy machines : A = 2 and >

B = B = 1 0

⊥||||

But first. . .

What do we mean by Binary Mealy Machine?

Mealy machines = Deterministic Mealy machinesMealy machine = set of states S + transition function f

f : S → (B × S)A

f (s)(a) = < b, s′ >

sa|b // s′

A is the input alphabet and B is the output alphabetBinary Mealy machines : A = 2 and >

B = B = 1 0

⊥||||

Information order

B = 1 0

⊥||||

> – abstraction (under-specification)⊥ – inconsistency (over-specification)0 and 1 – concrete output values.

Mealy automata are coalgebras

Observation:A Mealy machine is a coalgebra of the functor

M : Set → SetM(X ) = (B × X )A

(Almost) for free:Notion of (bi)simulation : equivalence between states,minimizationSemantics in terms of final coalgebra – causal functionsSpecification language for Mealy machines

Mealy automata are coalgebras

Observation:A Mealy machine is a coalgebra of the functor

M : Set → SetM(X ) = (B × X )A

(Almost) for free:Notion of (bi)simulation : equivalence between states,minimizationSemantics in terms of final coalgebra – causal functionsSpecification language for Mealy machines

A coalgebraic logic for Mealy machines

Based on the work by Marcello and Alexander Kurz, we derive aspecification language for M(X ) = (B × X )A:

Remark Simple but expressive language: Every finite Mealy machinecorresponds to a finite formula in our language.

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

ψ guarded

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

ψ guarded

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

ψ guarded

Simple properties

Output 0 at each input of 1

νx .(

1 ↓ 0

∧ 1(x) ∧ 0(x))

Output 0 at each input of two consecutive 1’s

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

Output 0 at each second input of 1

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(

1 ↓ 0

∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(1 ↓ 0 ∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(1 ↓ 0 ∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(1 ↓ 0 ∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(1 ↓ 0 ∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Simple properties

νx .(1 ↓ 0 ∧ 1(x) ∧ 0(x))

νx .(1(1 ↓ 0 ∧ 1(x) ∧ 0(x)) ∧ 0(x))

νx .(0(x) ∧ 1(νy .0(y) ∧ 1 ↓ 0 ∧ 1(x))

Formulae are coalgebras

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(tt)(a) =< >, tt >

1 | T, 0 | T

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(a′(φ))(a) =

{< >, φ > a = a′

< >, tt > otherwise

a'(φ) φa' | T

ora'(φ) tt

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(a′ ↓ b)(a) =

{< b, tt > a = a′

< >, tt > otherwise

a' b tta' | b

ora' b tt

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(φ1 ∧ φ2)(a) = λ(φ1)(a) u λ(φ2)(a)

φ1 φ'1b1

φ2 φ'2b2

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(φ1 ∧ φ2)(a) = λ(φ1)(a) u λ(φ2)(a)

φ1 φ'1b1

φ2 φ'2b2

φ1 ∧ φ2 φ'1 ∧ φ'2b1 ∧B b2

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(νx .φ)(a) = λ(φ[νx .φ/x ])(a)

φ[νx.φ/x] φ'a | b

λ : L → (B × L)A

φ ::= tt | φ ∧ φ | a(φ) | a ↓ b | x | νx .ψ

λ(νx .φ)(a) = λ(φ[νx .φ/x ])(a)

φ[νx.φ/x] φ'a | b

νx.φ φ'a | b

Why a coalgebra structure?

Two advantages1 Semantics

L[[ · ]] //

λ��

γ��

(B × L)A // (B × Γ)A

2 Satisfaction relation in terms of simulation

s |= φ⇔ s . φ

L[[ · ]] //

λ��

γ��

(B × L)A // (B × Γ)A

s |= φ⇔ s . φ

L[[ · ]] //

λ��

γ��

(B × L)A // (B × Γ)A

s |= φ⇔ s . φ

Logic is expressive

Theorem1 For all states s, s′ of a Mealy machine (S, f ),

s ∼ s′ iff ∀φ∈L.s |= φ⇔ s′ |= φ.

2 If S is finite then there exists for any s ∈ S a formula φs such thats ∼ φ.

We also want:For every formula φ construct a finite Mealy machine (S, f ) such that∃s∈S s ∼ φ.

Logic is expressive

Theorem1 For all states s, s′ of a Mealy machine (S, f ),

s ∼ s′ iff ∀φ∈L.s |= φ⇔ s′ |= φ.

2 If S is finite then there exists for any s ∈ S a formula φs such thats ∼ φ.

We also want:For every formula φ construct a finite Mealy machine (S, f ) such that∃s∈S s ∼ φ.

But. . .

Easy answer: Apply λ repeatedly!

Repeat for new states

φ1a1 | b1

a2 | b2

λ will not deliver a finite automata.

φ = νx .1(x ∧ (νy .1(y)))

λ(φ)(1) =< >, φ ∧ (νy .1(y)) >λ(φ ∧ (νy .1(y)))(1) =< >, φ ∧ (νy .1(y)) ∧ (νy .1(y)) >...

We need normalization!Alexandra and Marcello (CWI and LIACS) Regular Expressions for Polynomial Coalg. CoCoCo 2008 16 / 43

But. . .

Easy answer: Apply λ repeatedly!

φ1a1 | b1

a2 | b2

λ will not deliver a finite automata.

φ = νx .1(x ∧ (νy .1(y)))

λ(φ)(1) =< >, φ ∧ (νy .1(y)) >λ(φ ∧ (νy .1(y)))(1) =< >, φ ∧ (νy .1(y)) ∧ (νy .1(y)) >...

We need normalization!Alexandra and Marcello (CWI and LIACS) Regular Expressions for Polynomial Coalg. CoCoCo 2008 16 / 43

Normalization

norm(tt) = ttnorm(a(φ)) = a(norm(φ))norm(a ↓ b) = a ↓ bnorm(φ1 ∧ φ2) = conj(rem(flatten(norm(φ1) ∧ norm(φ2))))norm(νx .φ) = νx .norm(φ)

norm(a(tt) ∧ a ↓ b ∧ tt ∧ a ↓ b) = norm(a(tt) ∧ a ↓ b)

Normalization

norm(tt) = ttnorm(a(φ)) = a(norm(φ))norm(a ↓ b) = a ↓ bnorm(φ1 ∧ φ2) = conj(rem(flatten(norm(φ1) ∧ norm(φ2))))norm(νx .φ) = νx .norm(φ)

norm(a(tt) ∧ a ↓ b ∧ tt ∧ a ↓ b) = norm(a(tt) ∧ a ↓ b)

Synthesis

Specification

Mealy Machine

φ1a1 | b1

a2 | b2

Synthesis

φ One step synthesis

φ1a1 | b1

a2 | b2

λ with normalization

One-step synthesis

δ : L → (B × L)A

δ(tt) = < >, tt >

δ(a′(φ))(a) =

{< >,norm(φ) > a = a′

< >, tt > otherwise

δ(a′ ↓ b)(a) =

{< b, tt > a = a′

< >, tt > otherwiseδ(φ1 ∧ φ2)(a) = δ(φ1)(a) u δ(φ2)(a)δ(νx .φ)(a) = < b,norm(φ′) >

where < b, φ′ >= δ(φ[νx .φ/x ])(a)

< b1, φ1 > u < b2, φ2 > = < b1 ∧B b2,norm(φ1 ∧ φ2) >

Examples

φ = 1 ↓ 0 ∧ (νx .1(x))

δ(φ)(0) = < >, tt >δ(φ)(1) = δ(1 ↓ 0)(1) u δ(νx .1(x))(1)

= < 0, tt > u < >B, νx .1(x) >= < 0, νx .1(x) >

Remark: Not minimal.

Examples

φ = 1 ↓ 0 ∧ (νx .1(x))

δ(φ)(0) = < >, tt >δ(φ)(1) = δ(1 ↓ 0)(1) u δ(νx .1(x))(1)

= < 0, tt > u < >B, νx .1(x) >= < 0, νx .1(x) >

Examples

φ = 1 ↓ 0 ∧ (νx .1(x))

δ(φ)(0) = < >, tt >δ(φ)(1) = δ(1 ↓ 0)(1) u δ(νx .1(x))(1)

= < 0, tt > u < >B, νx .1(x) >= < 0, νx .1(x) >

φ νx.1(x)

Examples

φ = 1 ↓ 0 ∧ (νx .1(x))

δ(φ)(0) = < >, tt >δ(φ)(1) = δ(1 ↓ 0)(1) u δ(νx .1(x))(1)

= < 0, tt > u < >B, νx .1(x) >= < 0, νx .1(x) >

φ νx.1(x)

1 | T, 0 | T

Examples

φ = 1 ↓ 0 ∧ (νx .1(x))

δ(φ)(0) = < >, tt >δ(φ)(1) = δ(1 ↓ 0)(1) u δ(νx .1(x))(1)

= < 0, tt > u < >B, νx .1(x) >= < 0, νx .1(x) >

φ νx.1(x)

1 | T, 0 | T

Remark: Not minimal.Alexandra and Marcello (CWI and LIACS) Regular Expressions for Polynomial Coalg. CoCoCo 2008 20 / 43

Examples

φ2 = νx .(1(1 ↓ 0) ∧ 1(x))

δ(φ2)(0) = < >, tt >δ(φ2)(1) = δ(1(1 ↓ 0))(1) u δ(νx .1(x))(1)

= < >,1 ↓ 0 > u < >, νx .1(x) >= < >,1 ↓ 0 ∧ (νx .1(x)) >

Examples

φ2 = νx .(1(1 ↓ 0) ∧ 1(x))

δ(φ2)(0) = < >, tt >δ(φ2)(1) = δ(1(1 ↓ 0))(1) u δ(νx .1(x))(1)

= < >,1 ↓ 0 > u < >, νx .1(x) >= < >,1 ↓ 0 ∧ (νx .1(x)) >

Examples

φ2 = νx .(1(1 ↓ 0) ∧ 1(x))

δ(φ2)(0) = < >, tt >δ(φ2)(1) = δ(1(1 ↓ 0))(1) u δ(νx .1(x))(1)

= < >,1 ↓ 0 > u < >, νx .1(x) >= < >,1 ↓ 0 ∧ (νx .1(x)) >

φ 1 0 ∧ νx.1(x)

Examples

φ2 = νx .(1(1 ↓ 0) ∧ 1(x))

δ(φ2)(0) = < >, tt >δ(φ2)(1) = δ(1(1 ↓ 0))(1) u δ(νx .1(x))(1)

= < >,1 ↓ 0 > u < >, νx .1(x) >= < >,1 ↓ 0 ∧ (νx .1(x)) >

φ 1 0 ∧ νx.1(x)

1 | T, 0 | T

νx.1(x)0 | T

Examples

φ3 = νx .1(x ∧ (νy .1(y) ∧ 1 ↓ 0))

δ(φ3)(0) = < >, tt >δ(φ3)(1) = < >, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples

φ3 = νx .1(x ∧ (νy .1(y) ∧ 1 ↓ 0))

δ(φ3)(0) = < >, tt >δ(φ3)(1) = < >, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples

φ3 = νx .1(x ∧ (νy .1(y) ∧ 1 ↓ 0))

δ(φ3)(0) = < >, tt >δ(φ3)(1) = < >, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

δ(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0))(1)

= δ(φ3)(1) u δ(νy .1(y) ∧ 1 ↓ 0)(1)= < >, φ ∧ (νy .1(y) ∧ 1 ↓ 0) > u < 0, νy .1(y) ∧ 1 ↓ 0 >= < 0,norm(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) ∧ (νy .1(y) ∧ 1 ↓ 0)) >= < 0, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

δ(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0))(1)= δ(φ3)(1) u δ(νy .1(y) ∧ 1 ↓ 0)(1)

= < >, φ ∧ (νy .1(y) ∧ 1 ↓ 0) > u < 0, νy .1(y) ∧ 1 ↓ 0 >= < 0,norm(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) ∧ (νy .1(y) ∧ 1 ↓ 0)) >= < 0, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

δ(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0))(1)= δ(φ3)(1) u δ(νy .1(y) ∧ 1 ↓ 0)(1)= < >, φ ∧ (νy .1(y) ∧ 1 ↓ 0) > u < 0, νy .1(y) ∧ 1 ↓ 0 >

= < 0,norm(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) ∧ (νy .1(y) ∧ 1 ↓ 0)) >= < 0, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

δ(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0))(1)= δ(φ3)(1) u δ(νy .1(y) ∧ 1 ↓ 0)(1)= < >, φ ∧ (νy .1(y) ∧ 1 ↓ 0) > u < 0, νy .1(y) ∧ 1 ↓ 0 >= < 0,norm(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) ∧ (νy .1(y) ∧ 1 ↓ 0)) >

= < 0, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

δ(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0))(1)= δ(φ3)(1) u δ(νy .1(y) ∧ 1 ↓ 0)(1)= < >, φ ∧ (νy .1(y) ∧ 1 ↓ 0) > u < 0, νy .1(y) ∧ 1 ↓ 0 >= < 0,norm(φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) ∧ (νy .1(y) ∧ 1 ↓ 0)) >= < 0, φ3 ∧ (νy .1(y) ∧ 1 ↓ 0) >

1 | T φ3∧ (νy.1(y) ∧ 1 0)φ3

Examples (cont.)

φ3∧ (νy.1(y) ∧ 1 0)φ3

0 | T0 | T

Examples (cont.)

φ3∧ (νy.1(y) ∧ 1 0) ∧ (νy.1(y) ∧ 1 0)φ3∧ (νy.1(y) ∧ 1 0)φ3

0 | T0 | T

1 | 0 ...

Conclusions (Part I)

Coalgebraic approach : bisimulation and logicsNew logic for Mealy machinesSynthesis algorithm that produces a finite machine

End of Part I

Part II

Generalizing

Specification Language + Synthesis for systems of type

S → (B × S)A

G ::= Id | B | G ×G | G + G | GA

B semilattice

Notation: We will write F C G whenever F is an ingredient of G.

Generalizing

S → GS

G ::= Id | B | G ×G | G + G | GA

B semilattice

Generalizing

S → GS

G ::= Id | B | G ×G | G + G | GA

B semilattice

Generalizing

S → GS

G ::= Id | B | G ×G | G + G | GA

B semilattice

Consequences

Everything will be parameterized in GSpecification Language

Untyped expressions

G-typed expressions

Type system parametrized with G

Untyped expressions

ε ::= ∅ | x | ε⊕ ε | µx .γ | b | l(ε) | r(ε) | l[ε] | r [ε] | a(ε)

Untyped expressions

Type System

F 6= F1 + F2

` ∅ : F C G

` ε1 : F C G ` ε2 : F C G

` ε1 ⊕ ε2 : F C G ` x : F C G

` ε : F C G

` µx .ε : F C G

` b : B C G

` ε : G C G

` ε : Id C G

` ε : F1 C G

` l(ε) : F1 × F2 C G

` ε : F2 C G

` r(ε) : F1 × F2 C G

` ε : F1 C G

` l[ε] : F1 + F2 C G

` ε : F2 C G

` r [ε] : F1 + F2 C G

` ε : F C G

` a(ε) : F A C G

ExpFCG = {ε ∈ Exp | ` ε : F C G} .

ExpG = ExpGCG.

Type System

F 6= F1 + F2

` ∅ : F C G

` ε1 : F C G ` ε2 : F C G

` ε1 ⊕ ε2 : F C G ` x : F C G

` ε : F C G

` µx .ε : F C G

` b : B C G

` ε : G C G

` ε : Id C G

` ε : F1 C G

` l(ε) : F1 × F2 C G

` ε : F2 C G

` r(ε) : F1 × F2 C G

` ε : F1 C G

` l[ε] : F1 + F2 C G

` ε : F2 C G

` r [ε] : F1 + F2 C G

` ε : F C G

` a(ε) : F A C G

ExpFCG = {ε ∈ Exp | ` ε : F C G} .

ExpG = ExpGCG.

ExpG has a coalgebra structure

Goal: λG : ExpG → G(ExpG)

λFCG : ExpFCG → F (ExpG)

λFCG(∅) =?

λFCG(∅) : F (ExpG)

λFCG(∅) = EmptyFCG : F (ExpG)

λFCG(∅) =?

EmptyFCG : F (ExpG)

EmptyIdCG : ExpGEmptyIdCG = ∅

But recall:F 6= F1 + F2

` ∅ : F C G

EmptyIdCG : 1 + ExpG

EmptyIdCG =

{∅ G 6= G1 + G2? otherwise

Thus :EmptyFCG : F?(ExpG), F? = 1 + F

EmptyFCG : F (ExpG)

` ∅ : F C G

EmptyIdCG =

EmptyFCG : F (ExpG)

` ∅ : F C G

EmptyIdCG =

EmptyFCG : F (ExpG)

` ∅ : F C G

EmptyIdCG =

EmptyFCG : F?(ExpG)

EmptyIdCG =

EmptyBCG = ⊥BEmptyF1×F2CG =< EmptyF1CG,EmptyF2CG >EmptyF1+F2CG = ?EmptyF ACG = λa.EmptyFCG

λFCG : ExpFCG → F?(ExpG)

λFCG(∅) = EmptyFCGλF1×F2CG(l(ε)) =< λF1CG(ε),EmptyF2CG >λF1×F2CG(r(ε)) =< EmptyF1CG, λF2CG(ε) >λF1+F2CG(l[ε]) = κ1(λF1CG(ε))λF1+F2CG(r [ε]) = κ2(λF2CG(ε))...

Regular expressions

RFCG = {ε ∈ ExpFCG | λFCG(ε) 6= ?}

Real coalgebra structure

λFCG : RFCG → F (RFCG)

Advantage: We can define a notion of satisfaction

s 6|=G ε⇔ ε . s

Regular expressions

s 6|=G ε⇔ ε . s

Regular expressions

s 6|=G ε⇔ ε . s

Language expressive w.r.t bisimulation

Theorem

Let G be a polynomial functor G and (S,g) a G-coalgebra.1 For all states s, s′ ∈ S, s ∼ s′ if and only if∀ε ∈ RG . s |= ε⇔ s′ |= ε.

2 If S is finite then there exists for any s ∈ S an expression εs ∈ RGsuch that εs ∼ s.

Proof system

(Idempotency) ε⊕ ε = ε(Commutativity) ε1 ⊕ ε2 = ε2 ⊕ ε1(Associativity) ε1 ⊕ (ε2 ⊕ ε3) = (ε1 ⊕ ε2)⊕ ε3(Empty) ∅ ⊕ ε = ε

Proof system (cont.)

(B − ∅) ∅ = ⊥B (B −⊕) b1 ⊕ b2 = b1 ∨B b2

(×− ∅ − L) l(∅) = ∅ (×−⊕− L) l(ε1 ⊕ ε2) = l(ε1)⊕ l(ε2)(×− ∅ − R) r(∅) = ∅ (×−⊕− R) r(ε1 ⊕ ε2) = r(ε1)⊕ r(ε2)

l[∅] = ∅ r[∅] = ∅

(+−⊕− R) r [ε1 ⊕ ε2] = r [ε1]⊕ r [ε2](+−⊕− L) l[ε1 ⊕ ε2] = l[ε1]⊕ l[ε2]

Proof system (cont.)

(B − ∅) ∅ = ⊥B (B −⊕) b1 ⊕ b2 = b1 ∨B b2

(×− ∅ − L) l(∅) = ∅ (×−⊕− L) l(ε1 ⊕ ε2) = l(ε1)⊕ l(ε2)(×− ∅ − R) r(∅) = ∅ (×−⊕− R) r(ε1 ⊕ ε2) = r(ε1)⊕ r(ε2)

l[∅] = ∅ r[∅] = ∅

(+−⊕− R) r [ε1 ⊕ ε2] = r [ε1]⊕ r [ε2](+−⊕− L) l[ε1 ⊕ ε2] = l[ε1]⊕ l[ε2]

Soundness

Theorem (Soundness)The above equational system is sound, that is, for every polynomialfunctor G, if `G ε1 = ε2 then ε1 ∼G ε2.

Completeness (modal fragment)

Theorem (Completeness)For every polynomial functor G and modal regular expressionsε1, ε2 ∈ RG, for all G-coalgebra (S, f ), if ε1 ∼G ε2 then ` ε1 = ε2.

Synthesis

Normalization easily generalizesSynthesis is as before : λ+ normalization

Theorem (Kleene)1 If S is finite then there exists for any s ∈ S an expression εs ∈ RG

such that εs ∼ s.2 For all ε ∈ RG, there exists a finite G-coalgebra (S,g) such that∃s∈S s ∼ ε.

Synthesis

Normalization easily generalizesSynthesis is as before : λ+ normalization

Theorem (Kleene)1 If S is finite then there exists for any s ∈ S an expression εs ∈ RG

such that εs ∼ s.2 For all ε ∈ RG, there exists a finite G-coalgebra (S,g) such that∃s∈S s ∼ ε.

Conclusions and Future work

ConclusionsLanguage of regular expressions for polynomial coalgebrasEquational system sound and complete (for the modal fragment)Generalization of Kleene theorem

Future workMove from Set to pSet : better definiton of regularityEnlarge the class of functors treated: add PCompleteness of the full logicModel checking

Conclusions and Future work

ConclusionsLanguage of regular expressions for polynomial coalgebrasEquational system sound and complete (for the modal fragment)Generalization of Kleene theorem

Future workMove from Set to pSet : better definiton of regularityEnlarge the class of functors treated: add PCompleteness of the full logicModel checking

End of Part II

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