We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j,...
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We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j, there is a DNR function bounded by j that does not compute a Martin-Lof random real. Hence, there is a real of effective Hausdorff dimension 1 that does not compute a Martin- Lof random real. This answers a question of Reimann and Terwijn.
Raphael Robinson showed that all primitive recursive functions, depending on one argument, and only they could be obtained from two functions s(x) = x + 1 and q(x) = x divided by [root x](2) by using the operations of...
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Raphael Robinson showed that all primitive recursive functions, depending on one argument, and only they could be obtained from two functions s(x) = x + 1 and q(x) = x divided by [root x](2) by using the operations of addition+, superposition*, and iteration i. Julia Robinson proved that, starting from the same two functions and using the operations of addition+, superposition*, and the operation(-1) of function inversion, one could obtain all general recursive functions (under a certain condition on the inversion operation) and all partial recursive functions. On the basis of these results, A.I. Mal'tsev introduced into consideration Raphael Robinson algebra of all unary primitive recursive functions and two of Julia Robinson's algebras: namely, the partial algebra of all unary general recursive functions and the algebra of all unary partial recursive functions, and proposed to study the properties of these algebras, including the existence of finite bases of identities in these algebras. In this paper, we show that there is no finite basis of identities in any of the above algebras.
Abstract: The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to c...
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Abstract: The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to constructions of number-theoretic functions that are not available by deterministic means. A methodology for treating this question is developed, using an effective mapping of the space of infinite sequences over a finite alphabet into itself. The distribution characterizing the random elements, under the mapping, induces a new distribution. The property of a distribution being recursive is defined. The fundamental theorem states that recursive distributions induce only recursive distributions. A function calculated by any probabilistic means is called $\psi$-calculable. For a class of such calculations, these functions are recursive. Relative to Church’s thesis, this leads to an extension of that thesis: Every $\psi$-effectively calculable function is recursive. In further development, a partial order on distributions is defined through the concept of “inducing.” It is seen that a recursive atom-free distribution induces any recursive distribution. Also, there exist distributions that induce, but are not induced by, any recursive distribution. Some open questions are mentioned.
Let ε stand for the set of all numbers (i.e., nonnegative integers), V for the class of all sets (i.e., subcollections of ε) and for the family of all functions (i.e., mappings from a subset of ε into ε). If ƒ is...
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Let ε stand for the set of all numbers (i.e., nonnegative integers), V for the class of all sets (i.e., subcollections of ε) and for the family of all functions (i.e., mappings from a subset of ε into ε). If ƒ is a function, we write δƒ and ρƒ for its domain and range respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊆.
In this paper, we propose a new and elegant definition of the class of recursive functions, which is analogous to Kleene's definition but differs in the primitives taken, thus demonstrating the computational power...
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Let T be Godel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let T* be the subsystem of T in which the iterator and recursor constants are permitted only when immed...
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Let T be Godel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let T* be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schutte-style argument the T*-derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for T* is obtained. Another consequence is that every T*-representable number-theoretic function is elementary recursive. Furthermore, it is shown that, conversely, every elementary recursive function is representable in T*. The expressive weakness of T* compared to the full system T can be explained as follows: In contrast to T, computation steps in T* never increase the nesting-depth of I-rho and R-rho at recursion positions.
Syntactic translations of classical logic C into intuitionistic logic I are well known (see [Kol25], [Gli29], [Göd32], [Kre58b], [M063], [Cel69] and [Lei71]). Harvey Friedman [Fri78] used a translation of a simil...
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Syntactic translations of classical logic C into intuitionistic logic I are well known (see [Kol25], [Gli29], [Göd32], [Kre58b], [M063], [Cel69] and [Lei71]). Harvey Friedman [Fri78] used a translation of a similar nature, from I into itself, to reprove a theorem of Kreisel [Kre58a] that various theories based on I are closed under Markov's rule: if ¬¬∃x.α is a theorem, where x is a numeric variable and α is a primitive recursive relation, then ∃x.α is a theorem. Composing this with Gödel's translation from classical to intuitionistic theories, it follows that the functions provably recursive in the classical version of the theories considered are provably recursive already in their intuitionistic version. This conservation result is important in that it guarantees that no information about the convergence of recursive functions is lost when proofs are restricted to constructive logic, thus removing a potential objection to the use of constructive logic in reasoning about programs (see [C078] for example). Conversely, no objection can be raised by intuitionists to proofs of formulas that use classical reasoning, because such proofs can be converted to constructive proofs (this has been exploited extensively; see [Smo82]).Proofs of closure under Markov's rule had required, until Friedman's proof, a relatively sophisticated mathematical apparatus. The chief method used Godel's “Dialectica” interpretation (see [Tro73, §3]). Other proofs used cut-elimination, provable reflection for subsystems [Gir73], and Kripke models [Smo73]. Moreover, adapting these proofs to new theories had required that the underlying meta-mathematical techniques be adapted first, not always a trivial step.
R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recu...
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R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that λ < α*, where α* is the Σ1-projectum of α and λ is the Σ2-cofinality of α. The theorem is also established for the metarecursive case, α = ω1, where α* = λ = ω.
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