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In this paper we describe a simple semi-decision algorithm applicable to a wide class of quantified formulas. The formulas we consider are built using the propositional connectives from prenex formulas in a language for which a decision algorithm for the corresponding quantifier-free theory T is available. In general the algorithm to be presented is not complete, but for certain formulas whose matrix belongs to one of several special sublanguages of set theory the algorithm is complete. Proper choice of the quantifier-free theory one starts from will allow many significant set theoretic constructs to be expressed by a quantified formula of the class for which our algorithm is complete. In particular, we shall show that various elementary but useful statements concerning simple set theoretic operations and relations, ordinals, and maps can be verified automatically by the technique to be described. A Naive Unsatisfibility TestLet T be a quantifier-free theory (cf. [ 131) for which a satisfiability algorithm Aprenex formula is a formula of the form is available.where n is an integer greater than or equal to 0, Q, is either (Vx,) or (ax,), and p is a quantifier-free formula of T. We shall describe a semi-decision algorithm for formulas which have been built from prenex formulas by means of the usual logical connectives 1, &, V , -+, t) .1. Let cp be a formula of this type. Our algorithm is as follows: Bring cp into disjunctive normal form, i.e., rewrite it as a disjunction ' P O V C P I .

We describe a simple language for an unquantified set theory, and give a decision procedure for the formulas of this language. The language includes all the operators and relators (intersection, union, difference, equality, inclusion, etc.) of elementary Boolean set theory, and also the membership relator. We show that a few favorable set-theoretic constructs expressible by existentially quantified formulas, can be introduced without losing decidability. Finally we show that decidability is not lost even if the language is extended with the singleton and cardinality operators and also with integer terms.Very little knowledge about general set theory is used by the decision procedure for our zero-order theory. Indeed, the only nonelementary principle that we need is the following mild version of the foundation axiom: the membership predicate admits no finite cycles Extensions of the decision procedure to handle additional predicates and operations are currently under study, in an attempt to discover how far one can push an unquantified theory of sets without crossing the boundary of the undecidable. LanguageThe symbols of our language are: the constant 8; infinitely many variables uo, u l , 0 2 , * -. ; the binary operators U and \; the binary relators E and = ;

As is well-known, the Bernays-Schönfinkel-Ramsey class of all prenex ∃*∀*-sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem for the BSR class is proved by following a purely semantic approach, the remaining part of the decidability result being postponed to a forthcoming paper.

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To Jack with gratitude, in the sixteenth anniversary year of the inception of common work on multilevel syllogistic. AbstractHereditarily finite sets and hypersets are characterized both as algorithmic data structures and by means of a first-order axiomatization which, despite being rather weak, susces to make the following two problems decidable:(I) Establishing whether a conjunction r of formulae of the form:with q quantifier-free and involving only the relators =, E and propositional connectives, and each yi distinct from all w,'s, is satisfiable. (2) Establishing whether a formula of the form V y q. q quantifier-free, is satisfiable.Concerning (l), an explicit decision algorithm is provided moreover, significantly broad subproblems of ( I ) are singled out in which a classification -that we call the 'syllogistic decomposition' of r -of all possible ways of satisfying the input conjunction r can be obtained automatically. For one of these subproblems, carrying out the decomposition results in a finite family of syntactic substitutions that generate the space of all solutions to r. In this sense, one has a unification algorithm.Concerning (2), a technique is provided for reducing it to a subproblem of (1) for which a decomposition method is available. The algorithmic complexity of the problems under study is highlighted a generalization of the decidability results to a theory where sets are blended with free Herbrand functors is announced.

We report on the formalization of two classical results about claw-free graphs, which have been verified correct by Jacob T. Schwartz’s proof-checker Referee. We have proved formally that every connected claw-free graph admits (1) a near-perfect matching, (2) Hamiltonian cycles in its square. To take advantage of the set-theoretic foundation of Referee, we exploited set equivalents of the graph-theoretic notions involved in our experiment: edge, source, square, etc. To ease some\ud proofs, we have often resorted to weak counterparts of well-established notions such as cycle, claw-freeness, longest directed path, etc