In this paper we focus on two powerful techniques to obtain compact clause normal forms: Renaming of formulae and refined Skolemization methods. We illustrate their effect on various examples. By an exhaustive experiment of all first-order TPTP problems, it shows that our clause normal form transformation yields fewer clauses and fewer literals than the methods known and used so far. This often allows for exponentially shorter proofs and, in some cases, it makes it even possible for a theorem prover to find a proof where it was unable to do so with more standard clause normal form transformations. 1

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Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIME-complete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order two-variable logic L 2 is NEXPTIME-complete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in non-deterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of two-variables logic L 2 by a week access to car...

The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resource-oriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive P-specific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.

structures

this paper, we reserve the term

This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

In the early 1890’s, Richard Dedekind was working on a revised and enlarged edition of Dirichlet’s Vorlesungen über Zahlentheorie, and asked himself the following question: Given three subgroups A, B, C of an abelian group G, how many different subgroups can you get by taking intersections and sums, e.g., A + B, (A+B)∩C,

Abstract. We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks w.r.t. the base theory). Many theories important for computer science or mathematics fall into this class (typical examples are theories of data structures, theories of free or monotone functions, but also functions occurring in mathematical analysis). In fact, it is often necessary to consider complex extensions, in which various types of functions or data structures need to be taken into account at the same time. We show how such local theory extensions can be identified and under which conditions locality is preserved when combining theories, and we investigate possibilities of efficient modular reasoning in such theory combinations. We present several examples of application domains where local theories and local theory extensions occur in a natural way. We show, in particular, that various phenomena analyzed in the verification literature can be explained in a unified way using the notion of locality. 1

In this paper, we continue to develop our approach to theorem proof search in the EA-style, that is theorem proving in the framework of integrated processing mathematical texts written in a 1st-order formal language close to the natural language used in mathematical papers. This framework enables constructing a sound and complete goal-oriented sequent-type calculus with "large-block " inference rules. In particular, it contains the formal analogs of such natural proof search techniques as definition handling and auxiliary proposition application. The calculus allows to incorporate symbolic computations in an inference search.