Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Encoding industrial hardware verification problems into effectively propositional logic

Abstract: Word-level bounded model checking and equivalence checking problems are naturally encoded in the theory of bit-vectors and arrays. The standard practice of deciding formulas of such theories in the hardware industry is either SAT- (using bit-blasting) or SMT-based methods. These methods perform reasoning on a low level but perform it very efficiently. To find alternative potentially promising model checking and equivalence checking methods, a natural idea is to lift reasoning from the bit and bit-vector levels to higher levels. In such an attempt, in [14] we proposed translating memory designs into the Effectively PRopositional (EPR) fragment of first-order logic. The first experiments with using such a translation have been encouraging but raised some questions. Since the high-level encoding we used was incomplete (yet avoiding bit-blasting) some equivalences could not be proved. Another problem was that there was no natural correspondence between models of EPR formulas and bit-vector based models that would demonstrate non-equivalence and hence design errors. This paper addresses these problems by providing more refined translations of equivalence checking problems arising from hardware verification into EPR formulas. We provide three such translations and formulate their properties. All three translations are designed in such a way that models of EPR problems can be translated into bit-vector models demonstrating non-equivalence. We also evaluate the best EPR solvers on industrial equivalence checking problems and compare them with SMT solvers designed and tuned for such formulas specifically. We present empirical evidence demonstrating that EPR-based methods and solvers are competitive.

Complexity vs energy: theory of computation and theoretical physics

Abstract: This paper is a survey based upon the talk at the satellite QQQ conference to ECM6, 3Quantum: Algebra Geometry Information, Tallinn, July 2012. It is dedicated to the analogy between the notions of complexity in theoretical computer science and energy in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.

Propositional computability logic I

Abstract: In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at this http URL and this http URL .

Computability and Numberings

Abstract: The theory of computable numberings is one of the main parts of the theory of numberings. The papers of H. Rogers [36] and R. Friedberg [21] are the starting points in the systematical investigation of computable numberings. The general notion of a computable numbering was proposed in 1954 by A.N. Kolmogorov and V.A. Uspensky (see [40, p. 398]), and the monograph of Uspensky [41] was the first textbook that contained several basic results of the theory of computable numberings. The theory was developed further by many authors, and the most important contribution to it and its applications was made by A.I. Malt’sev, Yu.L. Ershov, A. Lachlan, S.S. Goncharov, S.A. Badaev, A.B. Khutoretskii, V.L. Selivanov, M. Kummer, M.B. Pouer-El, I.A. Lavrov, S.D. Denisov, and many other authors.

Tally Languages Accepted by Monte Carlo Pushdown Automata

Abstract: Rather often difficult (and sometimes even undecidable) problems become easily decidable for tally languages, i.e. for languages in a single-letter alphabet. For instance, the class of languages recognizable by 1-way nondeterministic pushdown automata equals the class of the context-free languages, but the class of the tally languages recognizable by 1-way nondeterministic pushdown automata, contains only regular languages [LP81]. We prove that languages over one-letter alphabet accepted by randomized one-way 1-tape Monte Carlo pushdown automata are regular. However Monte Carlo pushdown automata can be much more concise than deterministic 1-way finite state automata.