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

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

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

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

Example I: Determinizing (coalgebraically) Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

Example I: Determinizing (coalgebraically) � 1 ∃ q ∈ Q o ( q ) = 1 � o ( Q ) = t ( Q )( a ) = t ( q )( a ) 0 otherwise q ∈ Q Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

Example I: Determinizing (coalgebraically) � 1 ∃ q ∈ Q o ( q ) = 1 � o ( Q ) = t ( Q )( a ) = t ( q )( a ) 0 otherwise q ∈ Q Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

Example I: Determinizing (coalgebraically) � 1 ∃ q ∈ Q o ( q ) = 1 � o ( Q ) = t ( Q )( a ) = t ( q )( a ) 0 otherwise q ∈ Q How do we study NDA wrt language equivalence? L s = [ [ { s } ] ] Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 3 / 11

Example II: Totalizing Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

Example II: Totalizing � � o ( ∗ ) = 0 t ( ∗ )( a ) = ∗ o ( s ) = o ( s ) t ( s )( a ) = t ( s )( a ) Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

Example II: Totalizing � � o ( ∗ ) = 0 t ( ∗ )( a ) = ∗ o ( s ) = o ( s ) t ( s )( a ) = t ( s )( a ) Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

Example II: Totalizing � � o ( ∗ ) = 0 t ( ∗ )( a ) = ∗ o ( s ) = o ( s ) t ( s )( a ) = t ( s )( a ) How do we study PA wrt language equivalence? L s = [ [ i ( s ) ] ] Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 4 / 11

� Example III: Linearization S � o , t � R × ( R S ω ) A Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

� Example III: Linearization R S S � ������������������� � o , t � � o ♯ , t ♯ � R × ( R S ω ) A     v 1 v 1  ) = � v i × o ( s i )  )( a )( s j ) = � v i × t ( s i )( a )( s j ) . . . . o ♯ ( t ♯ (  .   .    v n v n Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

� � � � Example III: Linearization [ [ − ] ] e � R S R A ∗ S � � � � � � � � � ������������������� 0   s 1 . . . . . .     ∼ � o , t � = 1   e ( s i ) = s i � o ♯ , t ♯ �   . .   . . .  .    0 s n R × ( R A ∗ ) A R × ( R S ω ) A � � � � � � � � � � � � � �     v 1 v 1  ) = � v i × o ( s i )  )( a )( s j ) = � v i × t ( s i )( a )( s j ) . . . . o ♯ ( t ♯ (  .   .    v n v n Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

� � � � Example III: Linearization [ [ − ] ] e � R S R A ∗ S � � � � � � � � � ������������������� 0   s 1 . . . . . .     ∼ � o , t � = 1   e ( s i ) = s i � o ♯ , t ♯ �   . .   . . .  .    0 s n R × ( R A ∗ ) A R × ( R S ω ) A � � � � � � � � � � � � � �     v 1 v 1  ) = � v i × o ( s i )  )( a )( s j ) = � v i × t ( s i )( a )( s j ) . . . . o ♯ ( t ♯ (  .   .    v n v n How do we study WA wrt weighted languages (linear bisimilarity)? L s = [ [ e ( s ) ] ] Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 5 / 11

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

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

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

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: x 1 ≈ T ] . F x 2 ⇔ [ [ η X ( x 1 ) ] ] = [ [ η X ( x 2 ) ] Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 6 / 11

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

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 )) T ( f ) � T ( F ( T ( X ))) f ♯ : T ( X ) h � F ( T ( X )) Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 7 / 11

Bisimilarity implies T -enriched bisimilarity Theorem ≈ T ∼ FT ⇒ F Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 8 / 11

Bisimilarity implies T -enriched bisimilarity Theorem ≈ T ∼ FT ⇒ 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

Examples, Examples, Examples,. . . Partial Mealy machines S → ( B × ( 1 + S )) A ; Automata with exceptions S → 2 × ( E + S ) A ; Automata with side effects S → E E × (( E × S ) E ) A ; Total subsequential transducers S → O ∗ × ( O ∗ × S ) A ; Probabilistic automata S → [ 0 , 1 ] × ( D ω ( X )) A ; . . . Alexandra Silva (CWI) Algebraically Enriched Coalgebras CMCS, March 2010 9 / 11

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

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

The relation with [HJS] Some examples do not fit their framework (e.g., interactive output 1 monad is not commutative, side-effect monad has no ⊥ ,. . . ); some of our examples might not fit our framework (?); If FT ∼ = TG (e.g 2 × P ( − ) A ∼ = P ( 1 + A × − ) ) then: 2 ⇒ x ≈ T x ∼ tr y ⇐ 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