The main ideas underlying work on the model-theoretic foundations of algebraic specification and formal program development are presented in an informal way. An attempt is made to offer an overall view, rather than new results, and to focus on the basic motivation behind the technicalities presented elsewhere.

Abstract † We introduce a new paradigm for the integration of functional and logic programming. Unlike most current research, our approach is not based on extending unification to general-purpose equation solving. Rather, we propose a computation delaying mechanism called residuation. This allows a clear distinction between functional evaluation and logical deduction. The former is based on the λ-calculus, and the latter on Horn clause resolution. In clear contrast with equation-solving approaches, our model supports higher-order function evaluation and efficient compilation of both functional and logic programming expressions, without being plagued by non-deterministic term-rewriting. In addition, residuation lends itself naturally to process synchronization and constrained search. Besides unification (equations), other residuations may be any ground-decidable goal, such as mutual exclusion (inequations), and comparisons (inequalities). We describe an implementation of the residuation paradigm as a prototype language called Le Fun—Logic, equations, and Functions.

We study topological queries over two-dimensional spatial databases. First, we show that the topological properties of semi-algebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invariant characterizing semi-algebraic regions up to homeomorphism. All topological queries on semi-algebraic regions can be answered by queries on the invariant whose complexity is polynomially related to the original. Also, we show that for the purpose of answering topological queries, semi-algebraic regions can always be represented simply as polygonal regions. We then study query languages for topological properties of two-dimensional spatial databases, starting from the topological relationships between pairs of planar regions introduced by Egenhofer. We show that the closure of these relationships under appropriate logical operators yields languages which are complete for topological prope...

. The two main directions pursued in the present paper are the following. The first direction was (perhaps) started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic (with a single unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable (and in strongly nice) logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics will be discus...

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in 1957 for the first order logic was a source of a lot of investigations devoted to interpolation problem in various logical theories [1, 3]. Interpolation is considered as a desirable and "nice " property; also it has important practical applications in computer science [3].

. In this paper we present a powerful uniform first-order framework for representing and reasoning with complex forms of default knowledge. This is achieved by extending first-order predicate logic with a new generalized quantifier, anchored in the quasi-probabilistic ranking measure paradigm [Weydert 94], which subsumes and refines the original, propositional notion of a default conditional [Delgrande 88, Weydert 91, Boutilier 94]. 1. INTRODUCTION In recent times, default conditionals interpreted by ranking measures or order-of-magnitude probability distributions have become an increasingly popular tool for encoding defeasible relationships. Conceptual adequacy, semantic transparency, probabilistic justifications, the existence of proof-theoretic characterizations, the correct, natural handling of specificity and the availability of promising nonmonotonic inference relations [Weydert 96] are strong arguments in favour of this approach. However, like most formalisms for plausible reaso...

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The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a rst order sentence in a generalized product of rst order structures by reducing this computation to the computation of truth values of other rst order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Lauchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and de nability theory. Here we give a uni ed presentation, including some new results, of how to use the Feferman– Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is de nable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.