Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Interpretations as coalgebra morphisms Manuel A. Martins 1 Alexandre Madeira 2 Luis S. Barbosa 3 CMCS 2010 Paphos, Cyprus, March 2010 1Mathematics Department, Aveiro University, Portugal 2CCTC, Minho University & Mathematics Dep. of Aveiro University & Critical Software S.A., Portugal 3Dep. Informatics & CCTC, Minho University, Portugal Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Outline 1 Starting point Logics as coalgebras Objectives 2 Strict refinement revisited 3 Category of Logics and interpretations Logical interpretation The logics induced by the Frege relation Interpretations as coalgebras morphisms 4 Conclusions Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Outline 1 Starting point Logics as coalgebras Objectives 2 Strict refinement revisited 3 Category of Logics and interpretations Logical interpretation The logics induced by the Frege relation Interpretations as coalgebras morphisms 4 Conclusions Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Abstract definitions of logic Abstract Logic as a consequence relation A = � A , ⊢ A � , where ⊢ A : P ( A ) × A is a consequence relation in A . Abstract Logic as a closure operator A = � A , C A � , where C A is a closure operator, i.e., a mapping C A : P ( A ) → P ( A ) such for that for all X , Y ⊆ A , X ⊆ C A ( X ); 1 X ⊆ Y ⇒ C A ( X ) ⊆ C A ( Y ); 2 C A ( C A ( X )) = C A ( X ). 3 Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Abstract definitions of logic Abstract Logic as a closure system A = � A , T A � where T A is a closure system on A , i.e., a family F of subsets of A closed under arbitrary intersections (here we consider � ∅ = A ). Theorem Let A be a set. For each closure operator C A in A we can associate a closure system T A and, conversely, for each closure system T A a closure operator C A in such way that they are mutually inverses of one another: C A �→ T A := { X ⊆ A | C A ( X ) = X } T A �→ C A ( X ) := � { T ∈ T A | X ⊆ T } Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Logics as coalgebras Palmigiano shows in [Pal02] that an abstract logic can be represented by a coalgebra these coalgebras maps a formula into the set of its theories; the morphisms on that category correspond exactly to the usual morphisms between logics. the class of coalgebras that corresponds to abstract logics of empty signature defines a covariety. Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
� � � � � Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Logics as coalgebras closure system (contravariant) functor: is the functor that maps a set in the set of the closure systems over it and, each function f : A → B , in the map C ( f ) : C ( B ) → C ( A ) { f − 1 [ T ] : T ∈ F} . F �→ Let A = � A , T A � . f � B a A Coalg ( C ): A � η ξ ξ C ( f ) C ( A ) ξ ( a ) = { T ∈ T A | a ∈ T } C ( A ) C ( B ) Fact [Pal02] f is a logical morphism between two abstract logics i ff it is a morphism between its underlying coalgebras. Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Objectives Logical interpretation on software development We introduced in [MMB09a, MMB09b, MMB10] a formalization of refinement on algebraic specifications based on logical interpretations; The formalization is suitable to deal with data encapsulation, decomposition of operations in atomic transactions, and on the reuse of specifications; Aims The work aims to frame logical interpretation on the “logics as coalgebras” perspective; formalize refinement via interpretation on this setting; Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Logics as coalgebras Category of Logics and interpretations Objectives Conclusions Refinement by interpretation [MMB09a, MMB09b] Interpretation τ : Fm ( Σ ) → P ( Fm ( Σ ′ )) interprets SP if there is a specification SP ′ under Σ ′ such that: = ϕ i ff SP ′ | ∀ ϕ ∈ Fm ( Sig ( SP )) , SP | = τ ( ϕ ) SP ′ is a refinement by the interpretation τ of SP if τ interprets SP = ϕ implies SP ′ | ∀ ϕ ∈ Fm ( Sig ( SP )) , SP | = τ ( ϕ ) Theorem (Characterization) SP ⇁ τ SP ′ if there is an interpretation SP 0 of SP such that SP 0 � SP ′ . Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Outline 1 Starting point Logics as coalgebras Objectives 2 Strict refinement revisited 3 Category of Logics and interpretations Logical interpretation The logics induced by the Frege relation Interpretations as coalgebras morphisms 4 Conclusions Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
� � � � Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Strict refinement revisited Definition Let A = � A , C A � and A ′ = � A , C A ′ � be two abstract logics. A � A ′ , if for any X ∪ { x } ∈ A, x ∈ C A ( X ) ⇒ x ∈ C A ′ ( X ) . Theorem A � A ′ i ff T A ′ ⊆ T A . First intuition i A � � A ξ ′ ξ C ( i ) C ( A ) C ( A ) However, this implies that T A ′ = T A and we just need the first inclusion! Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Definition (Forward morphism) A forward morphism between � A , α � and � B , β � with respect to a pre-order ⊑ , is a map h : A → B such that C h ◦ β ◦ h ⊑ α . Theorem A ′ is a strict refinement of A i ff the inclusion map is a forward morphism from � A , ξ � to � A , ξ ′ � wrt ⊆ . Theorem The tuple � Log , ref , i , ◦� , where Log is the class of C -coalgebras induced by abstract logics; ref is the class of its inclusion forward morphisms wrt ⊆ ; i is the class of identical maps; ◦ is the composition of functions, defines a category. Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Strict refinement revisited Category of Logics and interpretations Conclusions Relating logics: Morphisms & Interpretations Definition (Logical morphism) A logical morphism between the logics A = � A , T A � and B = � B , T B � consists of an (algebraic) morphism h : A → B such that { h − 1 [ T ′ ] | T ′ ∈ T B } = T A . Definition (Interpretation) Let A = � A , C A � and B = � B , C B � be two abstract logics. A multifunction f : A ⇒ B is an interpretation (f : A ⇒ B for short), if for any { x } ∪ X ⊆ A, x ∈ C A ( X ) ⇔ f ( x ) ⊆ C B ( f [ X ]) . Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Logical interpretation Strict refinement revisited The logics induced by the Frege relation Category of Logics and interpretations Interpretations as coalgebras morphisms Conclusions Outline 1 Starting point Logics as coalgebras Objectives 2 Strict refinement revisited 3 Category of Logics and interpretations Logical interpretation The logics induced by the Frege relation Interpretations as coalgebras morphisms 4 Conclusions Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
Starting point Logical interpretation Strict refinement revisited The logics induced by the Frege relation Category of Logics and interpretations Interpretations as coalgebras morphisms Conclusions Some preliminaries Let f : A ⇒ B be a multifunction image: f [ X ] = � { f ( a ) | a ∈ X } ; inverse image: f − 1 [ Y ] = { a ∈ A | f ( a ) ⊆ Y } Let A = � A , C A � and B = � B , C B � two abstract logics. The multifunction f : A ⇒ B is said to be continuous wrt A and B if for every X ⊆ A , f [ C A ( X )] ⊆ C B ( f [ X ]) closed if maps closed set wrt A in closed sets wrt B ; Manuel A. Martins , Alexandre Madeira, Luis S. Barbosa Interpretations as coalgebra morphisms
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