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29
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.
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.
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.
Inherently nonfinitely based lattices
, 2002
"... We give a general method for constructing lattices L whose equational theories are inherently nonfinitely based. This means that the equational class (that is, the variety) generated by L is locally finite and that L belongs to no locally finite finitely axiomatizable equational class. We also provi ..."
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We give a general method for constructing lattices L whose equational theories are inherently nonfinitely based. This means that the equational class (that is, the variety) generated by L is locally finite and that L belongs to no locally finite finitely axiomatizable equational class. We also provide an example of a lattice which fails to be inherently nonfinitely based but whose
Logic with equality: . . .
, 1999
"... Herbrand's theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a ..."
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Herbrand's theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a quantifier free formula, one seeks a valid disjunction of m instantiations of that formula. In the presence of equality, which is the case in this paper, this problem was recently shown to be undecidable. The main contributions of this paper are two theorems. The first, the Partisan Corroboration Theorem, relates corroboration problems with dierent multiplicities. The second, the Shifted Pairing Theorem, is a finite tree automata formalization of a technique for proving undecidability results through direct encodings of valid Turing machine computations. These theorems are used in the paper to explain and sharpen several recent undecidability results related to the corroboration problem, the simultaneous rigid Eunification problem and the prenex fragment of intuitionistic logic with equality.