Human-oriented and machine-oriented reasoning: Remarks on some problems in the history of Automated Theorem Proving

Examples in the history of Automated Theorem Proving are given, in order to show that even a seemingly mechanical activity, such as deductive inference drawing, involves special cultural features and tacit knowledge. Mechanisation of reasoning is thus regarded as a complex undertaking in cultural pruning of human-oriented reasoning. Sociological counterparts of this passage from human- to machine-oriented reasoning are discussed, by focusing on problems of man-machine interaction in the area of computer-assisted proof processing.     

Axioms and models of linear logic

Girard's recent system of linear logic is presented in a way that avoids the two-level structure of formulae and sequents, and that minimises the number of primitive function symbols. A deduction theorem is proved concerning the classical implication as embedded in linear logic. The Hilbert-style axiomatisation is proved to be equivalent to the sequent formalism. The axiomatisation leads to a complete class of algebraic models. Various models are exhibited. On the meta-level we use Dijkstra's method of explicit equational proofs.     

Structured development of a virtual shared memory system

In this paper we formalise three different views of a virtual shared memory system and show that they are equivalent. The formalisation starts with five basic component processes specified in the language of CSP [Hoa85], which can be adapted as necessary by two operations called labelling and clamping, and are combined in two basic ways: either they are chained, so that the output of one component becomes the input of the next, or they are put in parallel, so that their communications are arbitrarily interleaved. Using the laws of CSP we show that these basic processes and operators satisfy a number of algebraic equivalences, which enable us to prove equivalence of the different models of the memory system by reasoning entirely at the level of processes, instead of at the lower and more complicated level of events. As a result the proofs of equivalence of the different models are purely algebraic and very simple.The specification is intended to provide a general framework for any architecture using an interconnection network, such as the on-chip interconnect between macrocells or the networks of processor nodes connected by bit-serial interconnect which are described in [Jon93]. It addresses architecture independent design issues such as access transparency, connectivity, addressing models and serialisability. By structuring it as a hierarchy of models it is hoped that the treatment of these many issues is made as clear and tractable as possible, whilst the proofs of equivalence ensure consistency.Funded by Esprit Project 7267/ OMI-Standards.     

The equational theory of regular words

Courcelle introduced the study of regular words, i.e., words isomorphic to frontiers of regular trees. Heilbrunner showed that a nonempty word is regular iff it can be generated from the singletons by the operations of concatenation, omega power, omega-op power, and the infinite family of shuffle operations. We prove that the algebra of nonempty regular words on the set A, equipped with these operations, is freely generated by A in a variety which is axiomatizable by an infinite collection of some natural equations. We also show that this variety has no finite equational basis and that its equational theory is decidable in polynomial time.     

A perfect zero-knowledge proof system for a problem equivalent to the discrete logarithm

An interactive proof system is calledperfect zero-knowledge if the probability distribution generated by any probabilistic polynomial-time verifier interacting with the prover on input theoremϕ, can be generated by another probabilistic polynomial-time machine which only getsϕ as input (and interacts with nobody!). In this paper we present aperfect zero-knowledge proof system for a decision problem which is computationally equivalent to the Discrete Logarithm Problem. Doing so we provide additional evidence to the belief thatperfect zero-knowledge proof systems exist in a nontrivial manner (i.e., for languages not inBPP). Our results extend to the logarithm problem in any finite Abelian group. This research was partially supported by the Fund for Basic Research Administered by the Israeli Academy of Sciences and Humanities. An early version of this paper appeared inAdvances in Cryptology —Crypto 88 (Proceedings), S. Goldwasser (ed.), pp. 57–70, Lecture Notes in Computer Science, vol. 403, Springer-Verlag, Berlin, 1990.     

Philosophical aspects of program verification

A debate over the theoretical capabilities of formal methods in computer science has raged for more than two years now. The function of this paper is to summarize the key elements of this debate and to respond to important criticisms others have advanced by placing these issues within a broader context of philosophical considerations about the nature of hardware and of software and about the kinds of knowledge that we have the capacity to acquire concerning their performance.     

The impact of detergent phosphorus bans on receiving water quality

One of the chemicals most clearly exemplifying scientific and political controversy concerning efforts to control its discharge to surface waters is phosphorus and its complexes. These materials are discharged as natural components of domestic wastewaters and include phosphorus from human waste and food waste as well as residual detergent phosphorus. Significant amounts of phosphorus also reach surface waters from non-point sources such as agricultural and urban runoff. This paper presents results of several field and laboratory investigations designed to position the impact of detergent phosphorus contributions to surface water quality. In a number of areas where legislation banned the sale of phosphorus detergents, limnological investigations were carried out to assess the impact of the ban upon receiving water quality. Field studies in natural lakes demonstrate that reductions of phosphorus in wastewaters, even up to 50%, may not substantially improve the trophic status of lakes. The consistent conclusion emerging from these studies is that the elimination of detergent phosphorus has not measurably improved lake water quality.     

OTTER and the Moufang identity problem

This article provides additional evidence of the value of using an automated reasoning program as a research assistant. Featured is the use of Bill McCune's program OTTER to find proofs of theorems taken from the study of Moufang loops, but not just any proofs. Specifically, the proofs satisfy the property of purity. In particular, when given, say, four equivalent identities (which is the case in this article), one is asked to prove the second identity from the first, the third from the second, the fourth from the third, and the first from the fourth. If the proof that 1 implies 2 does not rely on 3 or 4, then by definition the proof is pure with respect to 3 and 4, or simply the proof is pure. If for the four identities one finds four pure proofs showing that 1 implies 2, 2 implies 3, 3 implies 4, and 4 implies 1, then by definition one has found a circle of pure proofs. By finding the needed twelve pure proofs, this article shows that there does exist a circle of pure proofs for the four equivalent identities for Moufang loops and for all orderings of the identities; however, for much of this article, the emphasis is on the first three identities. In addition — in part to promote the use of automated reasoning programs and to answer questions concerning the choice of options — featured here is the methodology that was employed and a discussion of some of the obstacles, some of which are subtle. The approach relies on paramodulation (which generalizes equality substitution), on demodulation, and — so crucial for attacking deep questions and hard problems — on various strategies, most important of which are the hot list strategy, the set of support strategy, and McCune's ratio strategy. To permit verification of the results presented here, extension of them, and application of the methodology to other unrelated fields, a sample input file and four proofs (relevant to a circle of pure proofs for the four identities) are included. Research topics and challenges are offered at the close of this article.This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Pointers and data abstractions in high level languages—II: Correctness proofs

Recognizing the problems that the use of pointers pose to the construction of reliable software, this two-part paper proposes a scheme by which pointers may be used in a controlled manner to build data abstractions without being used as abstractions in their own right. Part I, published in Computer Languages 2, 135–148, presents the language constructs facilitating the proposal. Part II, in this Issue, attempts to show, by use of a fairly complex example, that proving the correctness of an implementation of an abstraction built in this manner from pointers need not be more difficult than other implementation correctness proofs.