Algebraically Enriched Coalgebras Filippo Bonchi 4 Marcello Bonsangue - - PowerPoint PPT Presentation

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Algebraically Enriched Coalgebras Filippo Bonchi 4 Marcello Bonsangue - - PowerPoint PPT Presentation

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Algebraically Enriched Coalgebras Filippo Bonchi 4 Marcello Bonsangue 1 , 2 Jan Rutten 1 , 3 Alexandra Silva 1 1 Centrum Wiskunde en Informatica 2 LIACS - Leiden University 3 Radboud Universiteit Nijmegen 4 INRIA Saclay - LIX, cole Polytechnique

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Algebraically Enriched Coalgebras

Filippo Bonchi4 Marcello Bonsangue1,2 Jan Rutten1,3 Alexandra Silva1

1Centrum Wiskunde en Informatica 2LIACS - Leiden University 3Radboud Universiteit Nijmegen 4INRIA Saclay - LIX, École Polytechnique

CMCS, March 2010

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 1 / 11

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Motivation

One of the nice things about (modelling systems as) coalgebras: The type of the system determines a canonical notion of equivalence. e.g bisimilarity for LTS’s One of the not so nice things about coalgebras: The canonical notion of equivalence is not what one wants. e.g language equivalence for LTS’s Goal of this talk: Show a way of uniformly deriving a new set of canonical equivalences from the type of the system.

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 2 / 11

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Motivation

One of the nice things about (modelling systems as) coalgebras: The type of the system determines a canonical notion of equivalence. e.g bisimilarity for LTS’s One of the not so nice things about coalgebras: The canonical notion of equivalence is not what one wants. e.g language equivalence for LTS’s Goal of this talk: Show a way of uniformly deriving a new set of canonical equivalences from the type of the system.

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 2 / 11

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Example I: Determinizing (coalgebraically)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

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Example I: Determinizing (coalgebraically)

  • (Q) =
  • 1

∃q∈Qo(q) = 1

  • therwise

t(Q)(a) =

  • q∈Q

t(q)(a)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

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Example I: Determinizing (coalgebraically)

  • (Q) =
  • 1

∃q∈Qo(q) = 1

  • therwise

t(Q)(a) =

  • q∈Q

t(q)(a)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

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Example I: Determinizing (coalgebraically)

  • (Q) =
  • 1

∃q∈Qo(q) = 1

  • therwise

t(Q)(a) =

  • q∈Q

t(q)(a) How do we study NDA wrt language equivalence? Ls = [ [ {s} ] ]

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

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Example II: Totalizing

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

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Example II: Totalizing

  • (∗) = 0
  • (s) = o(s)
  • t(∗)(a) = ∗

t(s)(a) = t(s)(a)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

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Example II: Totalizing

  • (∗) = 0
  • (s) = o(s)
  • t(∗)(a) = ∗

t(s)(a) = t(s)(a)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

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Example II: Totalizing

  • (∗) = 0
  • (s) = o(s)
  • t(∗)(a) = ∗

t(s)(a) = t(s)(a) How do we study PA wrt language equivalence? Ls = [ [ i(s) ] ]

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

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Example III: Linearization

S

  • ,t
  • R × (RS

ω)A

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

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Example III: Linearization

S

  • ,t
  • RS
  • ♯,t♯
  • R × (RS

ω)A

  • ♯(

   v1 . . . vn   ) = vi × o(si) t♯(    v1 . . . vn   )(a)(sj) = vi × t(si)(a)(sj)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

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Example III: Linearization

S

  • ,t
  • e

RS

[ [ − ] ]

  • ♯,t♯
  • RA∗

∼ =

  • R × (RS

ω)A

  • R × (RA∗)A

e(si) = s1 . . . si . . . sn         . . . 1 . . .        

  • ♯(

   v1 . . . vn   ) = vi × o(si) t♯(    v1 . . . vn   )(a)(sj) = vi × t(si)(a)(sj)

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

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Example III: Linearization

S

  • ,t
  • e

RS

[ [ − ] ]

  • ♯,t♯
  • RA∗

∼ =

  • R × (RS

ω)A

  • R × (RA∗)A

e(si) = s1 . . . si . . . sn         . . . 1 . . .        

  • ♯(

   v1 . . . vn   ) = vi × o(si) t♯(    v1 . . . vn   )(a)(sj) = vi × t(si)(a)(sj)

How do we study WA wrt weighted languages (linear bisimilarity)? Ls = [ [ e(s) ] ]

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

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Chasing the pattern. . .

How do we capture all the examples (and more) in the same framework?

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 6 / 11

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Chasing the pattern. . .

How do we capture all the examples (and more) in the same framework? The state space was enriched : T monad (P, 1+, . . . ).

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 6 / 11

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Chasing the pattern. . .

How do we capture all the examples (and more) in the same framework? The state space was enriched : T monad (P, 1+, . . . ). Transform an FT-coalgebra (X,f) into an F-coalgebra (T(X), f ♯).

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 6 / 11

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Chasing the pattern. . .

How do we capture all the examples (and more) in the same framework? The state space was enriched : T monad (P, 1+, . . . ). Transform an FT-coalgebra (X,f) into an F-coalgebra (T(X), f ♯). If F has final coalgebra: x1 ≈T

F x2 ⇔ [

[ ηX(x1) ] ] = [ [ ηX(x2) ] ].

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 6 / 11

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In a nutshell. . .

Ingredients: A monad T; A final coalgebra for F (for instance, take F to be bounded); An extension f ♯ of f;

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 7 / 11

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In a nutshell. . .

Ingredients: A monad T; A final coalgebra for F (for instance, take F to be bounded); An extension f ♯ of f; We can require FT(X) to be a T-algebra: (FT(X), h: T(FT(X)) → FT(X)) f ♯ : T(X)

T(f) T(F(T(X))) h

F(T(X))

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 7 / 11

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Bisimilarity implies T-enriched bisimilarity

Theorem

∼FT ⇒ ≈T

F

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 8 / 11

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Bisimilarity implies T-enriched bisimilarity

Theorem

∼FT ⇒ ≈T

F

The above theorem instantiates to well known facts: for NDA (F(X) = 2 × X A, T = P) that bisimilarity implies language equivalence; for PA (F(X) = 2 × X A, T = 1 + −) that equivalences of pair of languages, consisting of defined paths and accepted words, implies equivalence of accepted words; for weighted automata (F(X) = R × X A, T = R−

ω ) that weighted

bisimilarity implies weighted language equivalence.

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 8 / 11

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Examples, Examples, Examples,. . .

Partial Mealy machines S → (B × (1+S))A; Automata with exceptions S → 2 × (E+S)A; Automata with side effects S → EE × ((E×S)E)A; Total subsequential transducers S → O∗ × (O∗×S)A; Probabilistic automata S → [0, 1] × (Dω(X))A; . . .

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 9 / 11

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Conclusions

Lifted powerset construction to the more general framework of FT-coalgebras; Uniform treatment of several types of automata, recovery of known constructions/results; Opens the door to the study of T-enriched equivalences for many types of automata.

Thanks!!

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 10 / 11

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Conclusions

Lifted powerset construction to the more general framework of FT-coalgebras; Uniform treatment of several types of automata, recovery of known constructions/results; Opens the door to the study of T-enriched equivalences for many types of automata.

Thanks!!

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 10 / 11

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The relation with [HJS]

1

Some examples do not fit their framework (e.g., interactive output monad is not commutative, side-effect monad has no ⊥,. . . ); some of our examples might not fit our framework (?);

2

If FT ∼ = TG (e.g 2 × P(−)A ∼ = P(1 + A × −)) then: x ∼tr y ⇐ ⇒ x ≈T

F y

If ρ: TG ⇒ FT then: x ∼tr y ⇒ x ≈T

F y

Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 11 / 11