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Computational Foundations of Basic Recursive Function Theory
 Theoretical Computer Science
, 1988
"... The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on complet ..."
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The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the ...
Partial Objects in Type Theory
, 1989
"... computability theory : : : : : : : : : : : : : : : : : : : : : : 82 5.5 Building a partial object type theory : : : : : : : : : : : : : : : : : : 83 5.5.1 Expressing computational induction : : : : : : : : : : : : : : : 83 5.5.2 Expressing the fixedpoint principle : : : : : : : : : : : : : : : 84 ..."
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Cited by 19 (6 self)
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computability theory : : : : : : : : : : : : : : : : : : : : : : 82 5.5 Building a partial object type theory : : : : : : : : : : : : : : : : : : 83 5.5.1 Expressing computational induction : : : : : : : : : : : : : : : 83 5.5.2 Expressing the fixedpoint principle : : : : : : : : : : : : : : : 84 A A partial object Nuprl 85 A.1 The rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 Bibliography 93 vii Chapter 1 Introduction Type theories such as Nuprl [CAB + 86] or MartinLof's CMCP [Mar82] are foundational theories for constructive mathematics; MartinLof had philosophical motivations for developing his intuitionistic type theory. Nuprl incorporates many of the ideas of CMCP, but it is also designed to be of practical use as the logic in a powerful theoremproving system and as a foundation for reasoning about computing. Type theories have an extensive collection of types, far surpassing those found in programming languages. Like programming languages, ...
A Limiting First Order Realizability Interpretation
"... Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics ..."
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Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit.
A COMBINATORY ACCOUNT OF INTERNAL STRUCTURE
"... Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. S ..."
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Cited by 5 (4 self)
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Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. Some of this expressive power is captured by adding a factorisation combinator. It supports structural equality, and more generally, a large class of generic queries for updating of, and selecting from, arbitrary structures. The resulting combinatory logic is structure complete in the sense of being able to represent patternmatching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λcalculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combinators themselves that cannot be so represented, as they examine the internal structure of their arguments.
Towards Limit Computable Mathematics
"... The notion of LimitComputable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to 1 0 2 formulas. We can give an accountable computational interpretation to the proofs of LCM. The computational content of LCMp ..."
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The notion of LimitComputable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to 1 0 2 formulas. We can give an accountable computational interpretation to the proofs of LCM. The computational content of LCMproofs is given by Gold's limiting recursive functions, which is the fundamental notion of learning theory. LCM is expected to be a right means for "Proof Animation," which was introduced by the first author [10]. LCM is related not only to learning theory and recursion theory, but also to many areas in mathematics and computer science such as computational algebra, computability theories in analysis, reverse mathematics, and many others.
Complexity and Mixed Strategy Equilibria ∗
"... Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strateg ..."
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Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.