An extended logic programming language embodying sets is developed in successive stages, introducing at each stage simple set dictions and operations, and discussing their operational as well as declarative semantics. First, by means of special set terms added to definite Horn Clause logic, one is enabled to define enumerated sets. A new unification algorithm which can cope with set terms is developed and proved to terminate. Moreover, distinguished predicates representing set membership and equality are added to the base language along with their negative counterparts, and SLD resolution is modified accordingly. It is shown that the resulting language allows restricted universal quantifiers in goals and clause bodies to be defined quite simply within the language itself. Finally, abstraction set terms are made available as intensional designations of sets. It is shown that also such terms become directly definable within the language, provided the latter is endowed with negation, which may occur in goals and clause bodies.

Abstract. A way of introducing simple (finite) set designations and operations as first-class objects of an (unrestricted) logic programming language is discussed from both the declarative and the operational semantics viewpoint. First, special set terms are added to definite Horn clause logic and an extended Herbrand Universe based on an axiomatic characterization of the kind of sets we are dealing with is defined accordingly. Moreover, distinguished predicates representing set membership and equality are added to the base language along with their negative counterparts (π and). A new unification algorithm which can cope with set terms is developed and proved to terminate. Usual SLD resolution is modified so as to incorporate the new unification algorithm and to properly manage the distinguished predicates for set operations (in particular, conjunctions of atoms containing π and are dealt with as constraints, first reduced to a canonical form through a suitable canonization algorithm). Finally, the application of the resulting language to the definition of Restricted Universal Quantifiers is discussed. 1

A program development methodology based on verified program transformations is described and illustrated through derivations of a high level bisimulation algorithm and an improved minimum-state DFA algorithm. Certain doubts that were raised about the correctness of an initial paper-and-pencil derivation of the DFA minimizationalgorithm were laid to rest by machine-checked formal proofs of the most difficult derivational steps. Although the protracted labor involved in designing and checking these proofs was almost overwhelming, the expense was somewhat offset by a successful reuse of major portions of these proofs. In particular, the DFA minimization algorithm is obtained by specializing and then extending the last step in the derivation of the high level bisimulation algorithm. Our experience suggests that a major focus of future research should be aimed towards improving the technology of machine checkable proofs --- their construction, presentation, and reuse. This paper demonstrat...

Hereditarily finite sets and hypersets are characterized both as an algorithmic data structure and by means of a first-order axiomatization which, although rather weak, suffices to make the following two problems decidable: (1) Establishing whether a conjunction r of formulae of the form 8 y 1 \Delta \Delta \Delta 8 y m ((y 1 2 w 1 & \Delta \Delta \Delta & y m 2 wm ) ! q), with q quantifier-free and involving only the relators =; 2 and propositional connectives, and each y i distinct from all w j 's, is satisfiable. (2) Establishing whether a formula of the form 8 y q, q quantifier-free, is satisfiable. Concerning (1), an explicit decision algorithm is provided; moreover, significantly broad sub-problems of (1) are singled out in which a classification ---named the `syllogistic decomposition' of r--- of all possible ways of satisfying the input conjunction r can be obtained automatically. For one of these sub-problems, carrying out the decomposition results in producing a fi...

The aim of this paper is to extend the Constructive Negation technique to the case of CLP (SET ), a Constraint Logic Programming (CLP ) language based on hereditarily (and hybrid) finite sets. The challenging aspects of the problem originate from the fact that the structure on which CLP (SET ) is based is not admissible closed, and this does not allow to reuse the results presented in the literature concerning the relationships between CLP and constructive negation. We propose a new constraint satisfaction algorithm, capable of correctly handling constructive negation for large classes of CLP (SET ) programs; we also provide a syntactic characterization of such classes of programs. The resulting algorithm provides a novel constraint simplification procedure to handle constructive negation, suitable to theories where unification admits multiple most general unifiers. We also show, using a general result, that it is impossible to construct an interpreter...

New successes in dealing with set theories by means of state-of-the-art theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) Tarski-Givant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments. Key words: Set theory, relation algebras, first-order theorem-proving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domain-knowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theorem-proving systems. Still today such experiments pose consider...

: the paper analyzes an approach for integrating set theoretical constructs into a logic programming language. The focus is on describing a new abstract machine, based on the classical WAM, designed to support these new features. A major part of the paper concentrates on the implementation of the new unification algorithm and the set-constraints management support. 1 Introduction Set theory represents a universally accepted mean for representing various forms of knowledge, both in formal and commonsense reasoning. Nevertheless their use as a programming tool has been quite limited, due to the inherent complexity of computing with sets and the gap existing between unordered (sets) and ordered (computer memory) structures. In recent years, the evolution of declarative programming paradigms (like functional and logic programming) resulted in more effort being put towards integrating set theory in these new programming languages. Various proposals [2,5,31] describe the problems intr...

In this paper we present a fast tableau saturation strategy which can be used as an optimized decision procedure for some fragments of set theory. Such a strategy is based on the use of a model checking technique which guides the saturation process. As a result, it turns out that the saturation process converges much faster than previous decision algorithms either to a closed tableau or to a model satisfying the input formula. For the sake of simplicity, the strategy is illustrated for the extension MLSS of MLS with the finite enumeration operator...

Tarski-Givant’s map calculus is briefly reviewed, and a plan of research is outlined aimed at investigating applications of this ground equational formalism in the theorem-proving field. The main goal is to create synergy between first-order predicate calculus and the map calculus. Techniques for translating isolated sentences, as well as entire theories, from first-order logic into map calculus are designed, or in some cases simply brought nearer through the exercise of specifying properties of a few familiar structures (natural numbers, nested lists, finite sets, lattices). It is also highlighted to what extent a state-of-the-art theorem-prover for first-order logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus. Issues regarding ’safe ’ forms of map reasoning are singled out, in sight of possible generalizations to the database area. 1

This paper introduces formative processes, composed by transitive partitions. Given a family F of sets, a formative process ending in the Venn partition of F is shown to exist. Sufficient criteria are also singled out for a transitive partition to model (via a function from set variables to unions of sets in the partition) all set-literals modeled by . On the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by C(j j), with C a specific computable function. As a by-product, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satis ability of unquanti ed set-theoretic formulae involving Boolean operators, the singleton-former, and the powerset operator. The method described can be extended to decide the satisfiability problem for broader fragments of set theory.