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Mathematical logic

About: Mathematical logic is a research topic. Over the lifetime, 2842 publications have been published within this topic receiving 89917 citations. The topic is also known as: logic & formal logic.


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Book
01 Jun 1972
TL;DR: The aim of the book is to advance the understanding of how humans think by putting forth a theory of human problem solving, along with a body of empirical evidence that permits assessment of the theory.
Abstract: : The aim of the book is to advance the understanding of how humans think. It seeks to do so by putting forth a theory of human problem solving, along with a body of empirical evidence that permits assessment of the theory. (Author)

10,770 citations

01 Jan 2019
TL;DR: The book presents a thorough treatment of the central ideas and their applications of Kolmogorov complexity with a wide range of illustrative applications, and will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics.
Abstract: The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

3,361 citations

ReportDOI
17 Feb 1966
TL;DR: This book contains the collected and unified material necessary for the presentation of such branches of modern cybernetics as the theory of electronic digital computers, Theory of discrete automata, theory of discrete self-organizing systems, automation of thought processes, theoryof image recognition, etc.
Abstract: : This book contains the collected and unified material necessary for the presentation of such branches of modern cybernetics as the theory of electronic digital computers, theory of discrete automata, theory of discrete self-organizing systems, automation of thought processes, theory of image recognition, etc. Discussions are given of the fundamentals of the theory of boolean functions, algorithm theory, principles of the design of electronic digital computers and universal algorithmical languages, fundamentals of perceptron theory, some theoretical questions of the theory of self-organizing systems. Many fundamental results in mathematical logic and algorithm theory are presented in summary form, without detailed proofs, and in some cases without any proof. The book is intended for a broad audience of mathematicians and scientists of many specialties who wish to acquaint themselves with the problems of modern cybernetics.

2,922 citations

Journal ArticleDOI
TL;DR: In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation.
Abstract: The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilbert's school in the 1920's. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this “decision” problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions. However, recently there has been a revival of interest in the whole question. Specifically, it has been realized that while no decision procedure exists for quantification theory there are many proof procedures available—that is, uniform procedures which will ultimately locate a proof for any formula of quantification theory which is valid but which will usually involve seeking “forever” in the case of a formula which is not valid—and that some of these proof procedures could well turn out to be feasible for use with modern computing machinery.Hao Wang [9] and P. C. Gilmore [3] have each produced working programs which employ proof procedures in quantification theory. Gilmore's program employs a form of a basic theorem of mathematical logic due to Herbrand, and Wang's makes use of a formulation of quantification theory related to those studied by Gentzen. However, both programs encounter decisive difficulties with any but the simplest formulas of quantification theory, in connection with methods of doing propositional calculus. Wang's program, because of its use of Gentzen-like methods, involves exponentiation on the total number of truth-functional connectives, whereas Gilmore's program, using normal forms, involves exponentiation on the number of clauses present. Both methods are superior in many cases to truth table methods which involve exponentiation on the total number of variables present, and represent important initial contributions, but both run into difficulty with some fairly simple examples.In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation. The superiority of the present procedure over those previously available is indicated in part by the fact that a formula on which Gilmore's routine for the IBM 704 causes the machine to computer for 21 minutes without obtaining a result was worked successfully by hand computation using the present method in 30 minutes. Cf. §6, below.It should be mentioned that, before it can be hoped to employ proof procedures for quantification theory in obtaining proofs of theorems belonging to “genuine” mathematics, finite axiomatizations, which are “short,” must be obtained for various branches of mathematics. This last question will not be pursued further here; cf., however, Davis and Putnam [2], where one solution to this problem is given for ele

2,743 citations

Book
01 Jan 1967

2,116 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202222
202142
202081
201961
201873