Larry Wos

Unit Refutations and Horn Sets

The key concepts for this automated theorem-proving paper are those of Horn set and strictly-unit refutation. A Horn set is a set of clauses such that none of its members contains more than one positive literal. A strictly-unit refutation is a proof by contradiction in which no step is justified by applying a rule of inference to a set of clauses all of which contain more than one literal. Horn sets occur in many fields of mathematics such as the theory of groups, rings, Moufang loops, and Henkin models. The usual translation into first-order predicate calculus of the axioms of these and many other fields yields a set of Horn clauses. The striking feature of the Horn property for finite sets of clauses is that its presence or absence can be determined by inspection. Thus, the determination of the applicability of the theorems and procedures of this paper is immediate. In Theorem 1 it is proved that, if S is an unsatisfiable Horn set, there exists a strictly-unit refutation of S employing binary resolution alone, thus eliminating the need for factoring; moreover, one of the immediate ancestors of each step of the refutation is in fact a positive unit clause. A theorem similar to Theorem 1 for paramodulation-based inference systems is proven in Theorem 3 but with the inclusion of factoring as an inference rule. In Section 3 two reduction procedures are discussed. For the first, Changs splitting, a rule is provided to guide both the choice of clauses and the way in which to split. The second reduction procedure enables one to refute a Horn set by refuting but one of a corresponding family of simpler subproblems.

Job-shop scheduling using automated reasoning: a case study of the car-sequencing problem

The ‘job-shop scheduling problem’ is known to be NP-complete. The version of interest in this paper concerns an assembly line designed to produce various cars, each of which requires a (possibly different) set of options. The combinatorics of the problem preclude seeking a maximal solution. Nevertheless, because of the underlying economic considerations, an approach that yields a ‘good’ sequence of cars, given the specific required options for each, would be most valuable. In this paper, we focus on an environment for seeking, studying, and evaluating approaches for yielding good sequences. The environment we discuss relies on the automated reasoning program ITP. Automated reasoning programs of this type offer a wide variety of ways to reason, strategies for controlling the reasoning, and auxiliary procedures that contribute to the effective study of problems of this kind. We view the study presented in this paper as a prototype of what can be accomplished with the assistance of an automated reasoning program.

Set theory in first-order logic: clauses for Go¨del's axioms

In this paper we present a set of clauses for set theory, thus developing a foundation for the expression of most theorems of mathematics in a form acceptable to a resolution-based automated theoren prover. Because Gödels formulation of set theory permits presentation in a finite number of first-orde formulas, we employ it rather than that of Zermelo-Fraenkel. We illustrate the expressive power of thi formulation by providing statements of some well-known open questions in number theory, and give some intuition about how the axioms are used by including some sample proofs. A small set of challeng problems is also given.

Otter - The CADE-13 Competition Incarnations

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving System Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

A fascinating country in the world of computing: your guide to automated reasoning

The menu, the map, and the magic learning logic by example automated reasoning in full logic circuit design logic circuit validation research in mathematics research in formal logic the formal treatment of automated reasoning Woss biased guide for the effective use of OTTER an authors appraisal of his papers open questions, hard problems, intriguing challenges epilogue and after-dinner liqueur.

Experiments in Automated Deduction with Condensed Detachment

This paper contains the results of experiments with several search strategies on 112 problems involving condensed detachment. The problems are taken from nine different logic calculi: three versions of the two-valued sentential calculus, the many-valued sentential calculus, the implicational calculus, the equivalential calculus, the R calculus, the left group calculus, and the right group calculus. Each problem was given to the theorem prover Otter and was run with at least three strategies: (1) a basic strategy, (2) a strategy with a more refined method for selecting clauses on which to focus, and (3) a strategy that uses the refined selection mechanism and deletes deduced formulas according to some simple rules. Two new features of Otter are also presented: the refined method for selecting the next formula on which to focus, and a method for controlling memory usage.

Short Single Axioms for Boolean Algebra

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.

The Linked Inference Principle, II: The User's Viewpoint

In the field of automated reasoning, the search continues for useful representations of information, for powerful inference rules, for effective canonlcallzatlon procedures, and for intelligent strategies. The practical objective of this search is, of course, to produce ever more powerful automated reasoning programs. In this paper, we show how the power of such programs can be sharply increased by employing inference rules called linked inference rules. In particular, we focus on linked UR-resolutlon, a generalization of standard UR-resolution [2], and discuss ongoing experiments that permit comparison of the two inference rules. The intention is to present the results of those experiments at the Seventh Conference on Automated Deduction. Much of the treatment of linked inference rules given in this paper is from the user’s viewpoint, with certain abstract considerations reserved for Section 3.

A new use of an automated reasoning assistant: open questions in equivalential calculus and the study of infinite domains

Abstract The field of automated reasoning is an outgrowth of the field of automated theorem proving. The difference in the two fields is not so much in the procedures on which they rest, but rather in the way the corresponding programs are used. Here we present a comprehensive treatment of the use of an automated reasoning program to answer certain previously open questions from equivalential calculus. The questions are answered with a uniform method that employs schemata to study the infinite domain of theorems deducible from certain formulas. We include sufficient detail both to permit the work to be duplicated and to enable one to consider other applications of the techniques. Perhaps more important than either the results or the methodology is the demonstration of how an automated reasoning program can be used as an assistant and a colleague. Precise evidence is given of the nature of this assistance.

The resonance strategy

Abstract Especially in mathematics and in logic, lemmas (basic truths) play a key role for proving theorems. In ring theory, for example, a useful lemma asserts that, for all elements x , the product in either order of 0 and x is 0; in two-valued sentential (or propositional) calculus, a useful lemma asserts that, for all x , x implies x . Even in algorithm writing and in circuit design, lemmas play a key role: minus ( minus ( x )) = x in the former and NOT(AND( x , y )) = OR(NOT( x ),NOT( y )) in the latter. Whether the object is to prove a theorem, write an algorithm, or design a circuit, and whether the assignment is given to a person or (preferably) to an automated reasoning program, the judicious use of lemmas often spells the difference between success and failure. In this article, we focus on what might be thought of as a generalization of the concept of lemma, namely, the concept of resonator , and on a strategy, the resonance strategy , that keys on resonators. For example, where in Boolean groups—those in which the square of every x is the identity element e —the lemmas yzyz = e and yyzz = e are such that neither generalizes the other, the resonator (formula schema) ∗ ∗ ∗∗ = e , by using each occurrence of “star” to assert the presence of some variable, generalizes and captures (in a manner that is discussed in this article) both lemmas. Note that the cited resonator, if viewed as a lemma with star replaced by some chosen variable, captures neither cited lemma as an instance. Lemmas of a theory are provably “true” in the theory and, therefore, can be used to complete an assignment. In contrast, resonators, which capture collections of equations or collections of formulas that may or may not include truths, are used by the resonance strategy to direct the search for the information needed for assignment completion. In addition to discussing how one finds useful resonators, we detail various successes, in some of which the resonance strategy played a key role in obtaining a far better proof and in some of which the resonance strategy proved indispensable. The successes are taken from group theory, Robbins algebra, and various logic calculi.