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94
Experience with FS 0 as a framework theory
, 1993
"... Feferman has proposed a system, FS 0 , as an alternative framework for encoding logics and also for reasoning about those encodings. We have implemented a version of this framework and performed experiments that show that it is practical. Specifically, we describe a formalisation of predicate calcul ..."
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Cited by 16 (4 self)
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Feferman has proposed a system, FS 0 , as an alternative framework for encoding logics and also for reasoning about those encodings. We have implemented a version of this framework and performed experiments that show that it is practical. Specifically, we describe a formalisation of predicate calculus and the development of an admissible rule that manipulates formulae with bound variables. This application will be of interest to researchers working with frameworks that use mechanisms based on substitution in the lambda calculus to implement variable binding and substitution in the declared logic directly. We suggest that metatheoretic reasoning, even for a theory using bound variables, is not as difficult as is often supposed, and leads to more powerful ways of reasoning about the encoded theory. x 1 Introduction: why metamathematics? A logical framework is a formal theory that is designed for the purpose of describing other formal theories in a uniform way, and for making the work ...
Computational Logic and Human Thinking: How to be Artificially Intelligent
, 2011
"... The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prosp ..."
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Cited by 14 (7 self)
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The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prospect with horror. Partly because these controversies attract so much attention, one of the most important accomplishments of AI has gone largely unnoticed: the fact that many of its advances can also be used directly by people, to improve their own human intelligence. Chief among these advances is Computational Logic. Computational Logic builds upon traditional logic, which was originally developed to help people think more effectively. It employs the techniques of symbolic logic, which has been used to build the foundations of mathematics and computing. However, compared with traditional logic, Computational Logic is much more powerful; and compared with symbolic logic, it is much simpler and more practical. Although the applications of Computational Logic in AI require the use of mathematical notation, its human applications do not. As a consequence, I have written the main part of this book informally, to reach as wide an audience as possible. Because human thinking is also the subject of study in many other fields, I have drawn upon related studies in Cognitive Psychology, Linguistics, Philosophy, Law, Management Science and English
Contrasting applications of logic in natural language syntactic description
 Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress
, 2005
"... Abstract. Formal syntax has hitherto worked mostly with theoretical frameworks that take grammars to be generative, in Emil Post’s sense: they provide recursive enumerations of sets. This work has its origins in Post’s formalization of proof theory. There is an alternative, with roots in the semanti ..."
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Abstract. Formal syntax has hitherto worked mostly with theoretical frameworks that take grammars to be generative, in Emil Post’s sense: they provide recursive enumerations of sets. This work has its origins in Post’s formalization of proof theory. There is an alternative, with roots in the semantic side of logic: modeltheoretic syntax (MTS). MTS takes grammars to be sets of statements of which (algebraically idealized) wellformed expressions are models. We clarify the difference between the two kinds of framework and review their separate histories, and then argue that the generative perspective has misled linguists concerning the properties of natural languages. We select two elementary facts about natural language phenomena for discussion: the gradient character of the property of being ungrammatical and the open nature of natural language lexicons. We claim that the MTS perspective on syntactic structure does much better on representing the facts in these two domains. We also examine the arguments linguists give for the infinitude of the class of all expressions in a natural language. These arguments turn out on examination to be either unsound or lacking in empirical content. We claim that infinitude is an unsupportable claim that is also unimportant. What is actually needed is a way of representing the structure of expressions in a natural language without assigning any importance to the notion of a unique set with definite cardinality that contains all and only the expressions in the language. MTS provides that.
A Recursion Removal Theorem  Proof and Applications
, 1999
"... In this paper we briey introduce a Wide Spectrum Language and its transformation theory and describe a recent success of the theory: a general recursion removal theorem. This theorem includes as special cases the two techniques discussed by Knuth [12] and Bird [7]. We describe some applications of t ..."
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Cited by 11 (8 self)
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In this paper we briey introduce a Wide Spectrum Language and its transformation theory and describe a recent success of the theory: a general recursion removal theorem. This theorem includes as special cases the two techniques discussed by Knuth [12] and Bird [7]. We describe some applications of the theorem to cascade recursion, binary cascade recursion, Gray codes, the Towers of Hanoi problem, and an inverse engineering problem. 1 Introduction In this paper we briey introduce some of the ideas behind the transformation theory we have developed over the last eight years at Oxford and Durham Universities and describe a recent result: a general recursion removal theorem. We use a Wide Spectrum Language (called WSL), developed in [19,20,21] which includes lowlevel programming constructs and highlevel abstract specications within a single language. Working within a single language means that the proof that a program correctly implements a specication, or that a specication correct...
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Toward Learning Systems That Integrate Different Strategies and Representations

, 1993
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EFFECTIVE MATCHMAKING (RECURSION THEORETIC ASPECTS OF A THEOREM OF Philip Hall)
, 1971
"... Given a set B of boys and a set 0 of girls, we call a subset S of BxO a society and we say that b knows g when (b,gy e S. The marriage problem for the society S is said to be solvable if it is possible to marry, in the traditional onetoone manner, each boy to a girl whom he knows. We are concerned ..."
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Cited by 8 (0 self)
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Given a set B of boys and a set 0 of girls, we call a subset S of BxO a society and we say that b knows g when (b,gy e S. The marriage problem for the society S is said to be solvable if it is possible to marry, in the traditional onetoone manner, each boy to a girl whom he knows. We are concerned here with the computable analogues of these notions. Thus a society is recursive if there exists an algorithm which, when presented with a boy 6 and a girl g, effectively determines whether 6 knows g. Similarly, the marriage problem for the society S is said to be recursively solvable if there exists a onetoone algorithm which, when presented with a boy 6, effectively marries him to a girl whom he knows. We first show that, even if (the marriage problem for) a recursive society is solvable, it need not be recursively solvable. We then consider several conditions on solvable recursive societies; for each we determine whether such a society must be recursively solvable and, if not, how computationally complex its solutions need be. We also discuss some sociological variations of the marriage problem and indicate how our results can be applied to them. We have drawn upon ideas from two branches of mathematics— combinatorics and recursive function theory. The combinatorial motivation has its source in a famous theorem of Philip Hall ([4]) which implies that if there are only a finite number of boys, then the society S is solvable if and only if, for each natural number k, any k distinct boys know among them at least k different girls. Using a compactness argument one can show that this same condition is necessary and sufficient even if there are an infinite number of boys, so long as no boy knows infinitely many girls. (See either [5] for a combinatorial argument or [1], p. 47, for a proof based on the propositional calculus. This generalization was first proved by M. Hall ([3]). L. Mirsky's new book ([10]) contains an exhaustive
Dna Splicing Systems And Post Systems
, 1996
"... This paper concerns the formal study on the generative powers of extended splicing ..."
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Cited by 8 (2 self)
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This paper concerns the formal study on the generative powers of extended splicing
Computing and Information Compression: A Reply
 AI Communications
, 1994
"... An earlier article [25] discusses the proposition that the storage and processing of information in computers and in brains may often be understood as information compression. A subsequent article [15] criticises the computing aspects of [25] and research on the more specific conjecture that all for ..."
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Cited by 7 (7 self)
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An earlier article [25] discusses the proposition that the storage and processing of information in computers and in brains may often be understood as information compression. A subsequent article [15] criticises the computing aspects of [25] and research on the more specific conjecture that all forms of computing and formal reasoning may usefully be understood as information compression. The present article, which is intended to be intelligible without recourse to earlier articles, answers the main points in [15], tries to correct the many inaccuracies and misconceptions in that article, and discusses related issues. Topics which are discussed include: the way theories are or should be developed; the role of evidence in motivating research; apparent shortcomings in the Turing machine concept as a reason for seeking new principles of computing; the apparent conflict between the idea of `computing as compression' and the fact that computers may create redundancy  and how the contradict...