On a Decision Method in Restricted Second Order Arithmetic
01 Jan 1990-pp 425-435
TL;DR: The interpreted formalism of SC as mentioned in this paper is a fraction of the restricted second order theory of natural numbers, or of the first-order theory of real numbers, and it is easy to see that SC is equivalent to the first order theory [Re, +, Pw, Nn], whereby Re, + are the sets of non-negative reals, integral powers of 2, and natural numbers.
Abstract: Let SC be the interpreted formalism which makes use of individual variables t, x, y, z,... ranging over natural numbers, monadic predicate variables q( ), r( ), s( ), i( ),... ranging over arbitrary sets of natural numbers, the individual symbol 0 standing for zero, the function symbol ′ denoting the successor function, propositional connectives, and quantifiers for both types of variables. Thus SC is a fraction of the restricted second order theory of natural numbers, or of the first order theory of real numbers. In fact, if predicates on natural numbers are interpreted as binary expansions of real numbers, it is easy to see that SC is equivalent to the first order theory of [Re, +, Pw, Nn], whereby Re, Pw, Nn are, respectively, the sets of non-negative reals, integral powers of 2, and natural numbers.
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TL;DR: Alur et al. as discussed by the authors proposed timed automata to model the behavior of real-time systems over time, and showed that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the non-deterministic case and PSPACE-complete in deterministic case.
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Book•
25 Apr 2008
TL;DR: Principles of Model Checking offers a comprehensive introduction to model checking that is not only a text suitable for classroom use but also a valuable reference for researchers and practitioners in the field.
Abstract: Our growing dependence on increasingly complex computer and software systems necessitates the development of formalisms, techniques, and tools for assessing functional properties of these systems. One such technique that has emerged in the last twenty years is model checking, which systematically (and automatically) checks whether a model of a given system satisfies a desired property such as deadlock freedom, invariants, and request-response properties. This automated technique for verification and debugging has developed into a mature and widely used approach with many applications. Principles of Model Checking offers a comprehensive introduction to model checking that is not only a text suitable for classroom use but also a valuable reference for researchers and practitioners in the field. The book begins with the basic principles for modeling concurrent and communicating systems, introduces different classes of properties (including safety and liveness), presents the notion of fairness, and provides automata-based algorithms for these properties. It introduces the temporal logics LTL and CTL, compares them, and covers algorithms for verifying these logics, discussing real-time systems as well as systems subject to random phenomena. Separate chapters treat such efficiency-improving techniques as abstraction and symbolic manipulation. The book includes an extensive set of examples (most of which run through several chapters) and a complete set of basic results accompanied by detailed proofs. Each chapter concludes with a summary, bibliographic notes, and an extensive list of exercises of both practical and theoretical nature.
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Cites background from "On a Decision Method in Restricted ..."
...188 4.3.4 Generalized Büchi Automata . . . . . . . . . . . . . . . . . . . . ....
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...Research on automata over infinite words (and trees) started in the nineteen sixties with the work of Büchi [73], Trakhtenbrot [392], and Rabin [351] on decision problems for mathematical logics....
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...170 4.3.1 ω-Regular Languages and Properties . . . . . . . . . . . . . . . . . . 170 4.3.2 Nondeterministic Büchi Automata . . . . . . . . . . . . . . . . . . . 173 4.3.3 Deterministic Büchi Automata . . . . . . . . . . . . . . . . . . . . ....
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Book•
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4,478 citations
Book•
01 May 2011
TL;DR: The SPIN Model Checker as mentioned in this paper is used for both teaching software verification techniques, and for validating large scale applications, and it has been estimated that up to three-quarters of the $400 billion spent annually to hire programmers in the United States is ultimately spent on debugging.
Abstract: The SPIN Model Checker is used for both teaching software verification techniques, and for validating large scale applications. The growing number of users has created a need for a more comprehensive user guide and a standard reference manual that describes the most recent version of the tool. This book fills that need. SPIN is used in over 40 countries. The offical SPIN web site, spinroot.com receives between 2500 and 3000 hits per day. It has been estimated that up to three-quarters of the $400 billion spent annually to hire programmers in the United States is ultimately spent on debugging
2,530 citations
02 Jan 1991
TL;DR: This chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree.
Abstract: Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.
1,475 citations
References
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TL;DR: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula.
Abstract: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula*. But in the course of this investigation it is necessary to use certain theorems on combinations which have an independent interest and are most conveniently set out by themselves beforehand.
2,223 citations
TL;DR: Finite automata are considered as instruments for classifying finite tapes as well as generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined.
Abstract: Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
1,930 citations
15 Dec 1951
TL;DR: This memorandum is devoted to an elementary exposition of the problems and of results obtained on the McCulloch-Pitts nerve net during investigations in August 1951.
Abstract: An elementary exposition of the problems and results obtained during investigations in August, 1951, of the kinds of events any finite automation can respond to by assuming one of certain states.
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TL;DR: The formalism of regular expressions was introduced by S. C. Kleene to obtain the following basic theorems.
Abstract: The formalism of regular expressions was introduced by S. C. Kleene [6] to obtain the following basic theorems.
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123 citations