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Theory of representations
 Theoretical Computer Science
, 1985
"... Abstract. An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on F = {f: N 3 N} has been developed in a previous paper. By representations 6: B * M this theory is extende ..."
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Cited by 20 (3 self)
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Abstract. An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on F = {f: N 3 N} has been developed in a previous paper. By representations 6: B * M this theory is extended to other sets M. Topological and recursion theoretical properties of representations are studied, where the final topology of a representation plays an essential role. It is shown that for any separable T,space an (up to equivalence) unique admissible representation can be defined which reflects the topological properties correctly. 1.
Semantics vs. Syntax vs. Computations  Machine Models For Type2 . . .
 JOURNAL OF COMPUTER AND SYSTEM SCIENCE
, 1997
"... This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic po ..."
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Cited by 10 (0 self)
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This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic polynomialtime functionals. We show that if P = NP, these polynomialtime effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomialtime effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomialtime effective operations, the analog of the KreiselLacombeShoenfield is shown to holdtheir power matches that of the basic feasible functionals on R.
1996], Computation on abstract data types. The extensional approach, with an application to streams
 Annals of Pure and Applied Logic
"... In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definiti ..."
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Cited by 7 (2 self)
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In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1
Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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Cited by 1 (1 self)
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that
Computable Real Functions: Type 1 Computability Versus Type 2 Computability
, 1996
"... Based on the Turing machine model there are essentially two different notions of computable functions over the real numbers. The effective functions are defined only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The effectively conti ..."
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Based on the Turing machine model there are essentially two different notions of computable functions over the real numbers. The effective functions are defined only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The effectively continuous functions may be defined on arbitrary real nunbers. They are exactly those functions which are Type 2 computable with respect to an appropriate representation of the real numbers. We characterize the Type 2 computable functions on computable real numbers as exactly those Type 1 computable functions which satisfy a certain additional condition concerning their domain of definition. Our result is a sharp strengthening of the wellknown continuity result of Tseitin and Moschovakis for effective functions. The result is presented for arbitrary computable metric spaces. 1 Introduction In this paper we compare two approaches for defining computability of functions between computable metric ...
Uniformity in Computable Structure Theory
, 2000
"... We investigate the eects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity (of a structure) and intrinsic computability (of a relation on a computable structure). We consider and compare two dierent notions of uniformity, and discuss conn ..."
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We investigate the eects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity (of a structure) and intrinsic computability (of a relation on a computable structure). We consider and compare two dierent notions of uniformity, and discuss connections with the relative computable structure theory of Ash, Knight, Manasse, and Slaman and Chisholm and with previous work of Ash, Knight, and Slaman on uniformity in a general computable structuretheoretical setting.
Explicit Mathematics with Positive Existential Stratified Comprehension, Join and Uniform Monotone Inductive Definitions
, 2008
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