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Switch to: References
Citations of:
Theory of recursive functions and effective computability
Cambridge, Mass.: MIT Press (1987)
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We consider two players each of whom attempts to predict the behavior of the other, using no more than the history of earlier predictions. Behaviors are limited to a pair of options, conventionally denoted by 0, 1. Such players face the problem of learning to coordinate choices. The present paper formulates their situation recursion theoretically, and investigates the prospects for success. A pair of players build up a matrix with two rows and infinitely many columns, and are said to “learn” (...) |
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As is well known the derivative of a computable and C1 function may not be computable. For a computable and C∞ function f, the sequence {f} of its derivatives may fail to be computable as a sequence, even though its derivative of any order is computable. In this paper we present a necessary and sufficient condition for the sequence {f} of derivatives of a computable and C∞ function f to be computable. We also give a sharp regularity condition on an (...) |
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For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo is recursively enumerably open in dom × ℝ; we call f lower semi-computable of type two, if its closed epigraph Epi is recursively enumerably closed in dom × ℝ; we call f lower semi-computable of type three, if Epi is recursively closed in dom × ℝ. We show that (...) |
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The project was partially supported by a NSF grant of China. The author was grateful to Professor S. Lempp for his encouragement and suggestion while the author was visiting the Department of Mathematics at the University of Wisconsin. |
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We consider structures of the form, where Φ and Ψ are non-empty sets and is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R -images of elements of Ψ. The proof is a very elementary (...) |
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We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...) |
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Vera & Simon (1993a) have argued that the theories and methods known as situated action or situativity theory are compatible with the assumptions and methodology of the physical symbol systems hypothesis and do not require a new approach to the study of cognition. When the central criterion of computational universality is added to the loose definition of a symbol system which Vera and Simon provide, it becomes apparent that there are important incompatibilities between the two approaches such that situativity theory (...) |
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The Church-Turing Thesis (CTT) is often paraphrased as ``every computable function is computable by means of a Turing machine.'' The author has constructed a family of equational theories that are not Turing-decidable, that is, given one of the theories, no Turing machine can recognize whether an arbitrary equation is in the theory or not. But the theory is called pseudorecursive because it has the additional property that when attention is limited to equations with a bounded number of variables, one obtains, (...) |
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We consider the strongest forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities which are counterparts to 1- and i-reducibility, respectively, in the same way that nm- and npm-reducibility are counterparts to m- and pm-reducibility, respectively, we bring out the structure of the strong reducibilities. By further restricting n1- and nm-reducibility we are able to define infinite families of reducibilities which isomorphically embed the r. e. Turing degrees. Thus the many well-known results (...) |
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Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅' of the probability that the output be in some set. |
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In this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is $\Pi _1^1 - complete$ . So, the complexity results in (Horsten and Welch, Synthese 158:41–60,2007) are mistaken for adaptive logics using the minimal abnormality strategy. |
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We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ∃1. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds. |
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We prove a converse to one of the theorems from [F], giving a description in terms of Turing complexity of sets which can be coded into recursive and r.e. quotient Boolean algebras. |
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In this paper we define the concept of a system function language which is a language generated by a system function. We identify system function languages with recursively enumerable sets which are non-simple and co-infinite. We then define restricted system function languages and identify them with recursive sets which are co-infinite. Finally we state and prove some independence and dependence relationships between system function languages and some of the more well-known decision problems. MSC: 03D05, 03D20, 03D25. |
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We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups. |
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In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set. |
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We present some natural examples of countable Borel equivalence relations E, F with E ≤ B F such that there does not exist a continuous reduction from E to F. |
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This paper investigates the ontological presuppositions of quantifier logic. It is seen that the actual infinite, although present in the usual completeness proofs, is not needed for a proper semantic foundation. Additionally, quantifier logic can be given an adequate formulation in which neither the notion of individual nor that of a predicate appears. |
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Simpson introduced the lattice of Π0 1 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π0 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We (...) |
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