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A Birkhofflike Axiomatizability Result for Hidden Algebra and Coalgebra
, 2000
"... A characterization result for behaviorally definable classes of hidden algebras shows that a class of hidden algebras is behaviorally definable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically ge ..."
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A characterization result for behaviorally definable classes of hidden algebras shows that a class of hidden algebras is behaviorally definable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a final object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the different approaches in the literature.
Coalgebras in Specification and Verification for ObjectOriented Languages
, 1999
"... The aim of this short note is to give an impression of the use of coalgebras in specification and verification for objectoriented languages. Particular emphasis will be given to the rôle of coalgebraic operations in describing statebased systems. At the end some active research topics in coalgebra ..."
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The aim of this short note is to give an impression of the use of coalgebras in specification and verification for objectoriented languages. Particular emphasis will be given to the rôle of coalgebraic operations in describing statebased systems. At the end some active research topics in coalgebra will be sketched, together with pointers to the literature.
Complete Categorical Equational Deduction
, 2001
"... A categorical fourrule deduction system for equational logics is presented. We show that under reasonable niteness requirements this system is complete with respect to equational satisfaction abstracted as injectivity. The generality of the presented framework allows one to derive conditional e ..."
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A categorical fourrule deduction system for equational logics is presented. We show that under reasonable niteness requirements this system is complete with respect to equational satisfaction abstracted as injectivity. The generality of the presented framework allows one to derive conditional equations as well at no extra cost. In fact, our deduction system is also complete for conditional equations, a new result at the author's knowledge.
On Equational Craig Interpolation
, 2000
"... Generalizations of Craig interpolation are investigated for equational logic. Our approach is to do as much as possible at a categorical level, before drawing out the concrete implications. ..."
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Generalizations of Craig interpolation are investigated for equational logic. Our approach is to do as much as possible at a categorical level, before drawing out the concrete implications.
A Birkholike axiomatizability result for hidden algebra and coalgebra
 In Proceedings of CMCS'98, volume 11 of ENTCS
, 1998
"... Acharacterization result for behaviorally de nable classes of hidden algebras shows that a class of hidden algebras is behaviorally de nable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically gener ..."
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Acharacterization result for behaviorally de nable classes of hidden algebras shows that a class of hidden algebras is behaviorally de nable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a nal object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the di erent approaches in the literature. 1
Observational Ultrapowers of Polynomial Coalgebras
, 2001
"... Coalgebras of polynomial functors constructed from set of observable elements have been found useful in modelling various kinds of data types and statetransition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in [6], where it was shown t ..."
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Coalgebras of polynomial functors constructed from set of observable elements have been found useful in modelling various kinds of data types and statetransition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in [6], where it was shown that Boolean combinations of equations between terms of observable type form a natural language of observable formulas for specifying properties of polynomial coalgebras, and for giving a HennessyMilner style logical characterisation of observational indistinguishability (bisimilarity) of states. Here we give a structural characterisation of classes of coalgebras definable by observable formulas. This is an analogue for polynomial coalgebras of Birkhoff's celebrated characterisation of equationally definable classes of abstract algebras as being those closed under homomorphic images, subalgebras, and direct products. The coalgebraic characterisation involves a new notion of an observational ultrapower of a coalgebra, obtained from the usual notion of ultrapower by deleting states that assign "nonstandard" values to terms of observable type. A class of polynomial coalgebras is shown to be the class of all models of a set of observable formulas if, and only if, it is closed under images of bisimilarity relations, disjoint unions and observational ultrapowers.
Modal Operators for Coequations
, 2001
"... this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree ..."
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this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree coalgebra # C : GC ## G 2 C over C. Hence, we can view # as a predicate over GC. In particular, we can form new coequations out of old by means of the logical connectives #, #, etc. Furthermore, we have available a modal operator taking a coequation # to the (carrier of the) largest subcoalgebra # contained in the coequation. As we will see, arises as the formal dual of a familiar operation on sets of equations in categories of algebras. Explicitly, the operator is dual to the closure operation taking a set E of equations over X (i.e., E # UFX UFX , where UFX is the carrier of the free algebra over X) to the least congruence containing E. Hence, is dual to the closure of sets of equations under the first four rules of inference of Birkho#'s equational logic (Birkho#, 1935). Thus, we see that closure under these rules of inferences is dual to the "coalgebra interior" of a set of elements. We introduce a modal operator that is dual to closure under Birkho#'s fifth rule of inference, i.e., substitution of terms for variables. We confirm that is an S4 operator and show that, under certain conditions, commutes with . We then prove the invariance theorem in terms of and . In this way, we develop the coequationsaspredicates view by augmenting the predicates over GC with two modal operators and and show that the partial order of covarieties definable by arbitrary coequations over C is isomorphic to the partial order of predicates # over GC such th...