Alberto Policriti

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on ϕ, ε, =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of ϕ, ε, =, {...}, η, where {...} is the tuple operator and η is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of ϕ, ε, =, {...}, η, obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving η is allowed.

T -theorem proving I

Abstract In this paper we present a theoretical basis justifying the incorporation of decidability results for a first-order theory T into an automated theorem prover for T . We state rules which extend resolution using decidability results relative to T in both the ground and the non-ground case, and prove the correctness and completeness of these rules. This is done by considering the ground case of such theories first, and then by applying a straightforward lifting argument. Examples are given illustrating the inference speed-ups which can be obtained by considering decision procedures with resolution-based inference.

A derived algorithm for evaluating ε-expressions over abstract sets

Four weak theories of pure sets are axiomatically characterized. A decision method is given for checking sentences of the form @?y[emailxa0protected]?y[emailxa0protected]?xp, where n varies over natural numbers and p over unquantified matrices, for provability in each theory. Dually, the method can be used to check @?y[emailxa0protected]?y[emailxa0protected]?x@?p for satisfiability. The completeness proof is fully constructive: this means that given a satisfiable constraint of the form @?y[emailxa0protected]?y[emailxa0protected]?x@?p, a computable model of the axioms which also fulfills the constraint can be synthesized. In this sense, we have a way of automatically generating a concrete representation of the abstract data-type set under varying axioms. The problem is also addressed of how to determine @? x p; i.e., how to find a @z fulfilling p(@x1,...,@xn,@z) in a computable model M of one of our theories, as a function of input M-sets @x1,...,@xn. A partial solution to this problem is supplied, which works when @?y[emailxa0protected]?y[emailxa0protected]?xp is a theorem and M meets a suitable condition which happens to be satisfied by those models that are produced by our automatic synthesis algorithm. A stronger condition on M is also characterized that makes @? x p computable in all cases (at worst through a blind search method). Examples showing the expressive power of the @?*@?-constraints and the usefulness of @?-expressions in set computations are included. Envisaged extensions of the proposed methods to axiomatic set theories antithetic to the classical ones are briefly hinted at.

Witnessing differences without redundancies

We show that n 1 elements suffice to witness the differences of n pairwise distinct sets, and provide sufficient conditions for an infinite family of pairwise distinct sets to have a minimal collection of elements witnessing the differences between any two of its members. By the Extensionality Axiom, the difference between two distinct sets a and b is witnessed by at least one element d such that d E a b or d E b a; in fact any element in the symmetric difference aAb = (a b) U (b a) witnesses such a difference. For that reason we say that aAb is a differentiating set for {a, b}. Since all the elements in aAb but one are redundant for that purpose, unless aAb is a singleton, we say that aAb is a redundant or non-minimal differentiating set for {a, b}, while for any d E aAb, {d} is an irredundant or minimal differentiating set for {a, b}. Suppose now that n pairwise distinct sets ai,. .. , an are given; how many elements do we need to witness their being different from each other? Equivalently, given a differentiating set D for {ai,... , an} , how many redundant elements are to be found in D ? Two extreme cases immediately come under attention. If a,,... ,an can be arranged into an increasing chain with respect to inclusion, or else if a,,... ,an are pairwise disjoint, then obviously we need exactly n 1 elements to witness their differences and any differentiating set for {ai, ... , an} of cardinality m has at least m n + 1 redundant elements. In general it is obvious that we need at most (n) elements to witness the differences of n pairwise distinct sets a1,... ,an . However (n) is by far an excessively large bound; in this note we offer an extremely simple proof that n 1 elements always suffice to witness the differences among n distinct sets (see Proposition 1). For an earlier proof of this result in the special case in which the n sets are subsets of an n-elements domain see [Bon72, Bol86]. Even from the first rough estimate, it is clear that in the case of finitely many pairwise distinct sets a,, ... , an , an irredundant differentiating set can be obtained from any finite differentiating set by suppressing one after the other the elements which are redundant and remain so as the procedure goes on. It is quite natural to enquire whether that holds also for infinite families of pairwise distinct sets. Any sequence of sets densely ordered with respect to inclusion readily provides an example of a family of pairwise distinct sets for which no minimal differentiating set can exist (see Proposition 2 below). However, by making an essential use of the Received by the editors February 7, 1994 and, in revised form, August 28, 1995. 1991 Mathematics Subject Classification. Primary 03E05; Secondary 03E25. This work has been supported by funds MURST 40% and 60% of Italy. (?)1997 American Mathematical Society

Decision Problems and Some Solutions

Viewed as a collection of sentences, any first-order theory of sets is recursively enumerable: this is an advantage ensuing from the axiomatic method. To restate it, a list of all sentences provable in the theory can be generated through an automatic (infinite) process. Moreover, a reliable certification of theoremhood (that is, a formal derivation from the axioms of the theory) can come along with each sentence in the list.

Sets for Problem Solving

This chapter provides many examples of how set theory is used to specify problems and algorithms and to characterize precisely domains formed by mathematical entities (e.g., numeric fields) or formed by algorithmic data structures (e.g., by trees).

Inference Techniques and Methods

This chapter is devoted to the study of inference techniques and methods which have specific potential for set-theoretic reasoning. Actually, the work we have carried out so far in designing axiomatic systems for set theories can be seen as preparation for the implementation of automated reasoning systems. The issues touched upon, and the contributions collected in this chapter are more specific—they relate to areas where sustained efforts have been made to develop techniques suitable for use in the set-theoretic framework.

Set Theory for Nonclassic Logics

This chapter is devoted to showing how the study of nonclassic logic provides a simple yet significant arena where to take advantage of the capability of set theory to provide a common formalism to encode a variety of specific logical languages. Moreover, the axiomatic set-theoretic systems chosen for the various encodings give examples of uses of the automated theorem-proving machinery, developed earlier in this book, for first-order set theories.

Logic Programming with Sets

The advantages ensuing from the availability of set data abstractions in a high-level programming language are widely recognized. In particular, sets can be used conveniently in languages for rapid software prototyping, where the speed of execution—although desirable—is not the most stringent requirement; and in problem-specification languages, where specifications are not even expected to be always executable. Highly representative languages which embody sets are Z [Spi88], SETL [SDDS86], and Godel [HL94].

What Is Computable Set Theory

Set theory, conceived toward the end of the nineteenth century, shaped the language of today’s mathematics. It should provide, essentially for the same reasons, the standard ingredients for the language of computer science as that discipline progresses out of its own infancy. This book aims to bring to light evidence that such a task can indeed be rewarding, by developing tools (algorithmic, as well as conceptual) for better and deeper exploitation of Set theory in computer science.