J. Richard Büchi

Bio: J. Richard Büchi is an academic researcher from University of Michigan. The author has contributed to research in topics: Natural number & Second-order arithmetic. The author has an hindex of 6, co-authored 7 publications receiving 1790 citations.

On a Decision Method in Restricted Second Order Arithmetic

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.

Symposium on Decision Problems: On a Decision Method in Restricted Second Order Arithmetic

Abstract: Publisher Summary In this chapter, SC is a fraction of the restricted second order theory of natural numbers or of the first order theory of real numbers. This chapter discusses the definability in SC and outlines an effective method for deciding the truth of sentences in SC. A congruence of finite rank on words is in congruence with the finite partition of concatenation; a multi-periodic set of words is a union of congruence classes of a congruence of finite rank. These concepts are related to that of a finite automaton and turn out to be the key to an investigation of SC. The results concerning SC may be viewed as an application of the theory of finite automata to logic. In turn, SC arises quite naturally as a condition-language on finite automata or sequential circuits and “sequential calculus” is an appropriate name for SC. The significance of the decision method for SC is that it provides a method for deciding whether or not the input (i)-to-output (u) transformation of a proposed circuit A (i, r, u) satisfies a condition C (i, u) stated in SC.

Representation of Complete Lattices by Sets

Abstract: Throughout this paper L denotes a complete lattice. That is a set of elements a, b, c, ⋯ partly ordered by a relation «⊂» such that to every class (aν) of elements aν in L there exists a greatest lower bound \( \mathop{ \cup }\limits_{ u } \) aν in L. Then also the least upper bound \( \mathop{ \cup }\limits_{ u } \)aν exists in L and furthermore there is a greatest element e in L, namely the g. l. b. of the empty subclass of L, and there exists the smallest element 0 in L, the g. l. b. of all elements of L.

Principles of Model Checking

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.

The SPIN Model Checker: Primer and Reference Manual

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

On the synthesis of a reactive module

Abstract: @(x, y) is valid over all tree models. For the restricted case that all variables range over finite domains, the validity problem is decidable, and we present an algorithm for constructing the program whenever it exists. The algorithm is based on a new procedure for checking the emptiness of Rabin automata on infinite trees in time exponential in the number of pairs, but only polynomial in the number of states. This leads to a synthesis algorithm whose complexity is double exponential in the length of the given specification.