Reversible computing is both forward and backward deterministic. This means that a uniquely determined step exists from the previous computational configuration (backward determinism) to the next one (forward determin...
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Reversible computing is both forward and backward deterministic. This means that a uniquely determined step exists from the previous computational configuration (backward determinism) to the next one (forward determinism) and vice versa. We present the reversible primitive recursive functions (RPRF), a class of reversible (endo-)functions over natural numbers which allows to capture interesting extensional aspects of reversible computation in a formalism quite close to that of classical primitive recursive functions. The class RPRF can express bijections over integers (not only natural numbers), is expressive enough to admit an embedding of the primitive recursive functions and, of course, its evaluation is effective. We also extend RPRF to obtain a new class of functions which are effective and Turing complete, and represent all Kleene's -recursivefunctions. Finally, we consider reversible recursion schemes that lead outside the reversible endo-functions.
Reversible computing is bi-deterministic which means that its execution is both forward and backward deterministic, i.e. next/previous computational step is uniquely determined. Various approaches exist to catch its e...
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Reversible computing is bi-deterministic which means that its execution is both forward and backward deterministic, i.e. next/previous computational step is uniquely determined. Various approaches exist to catch its extensional or intensional aspects and properties. We present a class RPRF of reversible functions which holds at bay intensional aspects and emphasizes the extensional side of the reversible computation by following the style of Dedekind-Robinson primitive recursive functions. The class RPRF is closed by inversion, can only express bijections on integers - not only natural numbers -, and it is expressive enough to simulate primitive recursive functions, of course, in an effective way.
We focus on total functions in the theory of reversible computational models. We define a class of recursive permutations, dubbed Reversible primitive Permutations (RPP) which are computable invertible total endo-func...
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We focus on total functions in the theory of reversible computational models. We define a class of recursive permutations, dubbed Reversible primitive Permutations (RPP) which are computable invertible total endo-functions on integers, so a subset of total reversible computations. RPP is generated from five basic functions (identity, sign-change, successor, predecessor, swap), two notions of composition (sequential and parallel), one functional iteration and one functional selection. RPP is closed by inversion and it is expressive enough to encode Cantor pairing and the whole class of primitive recursive functions. (C) 2019 Elsevier B.V. All rights reserved.
Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics' safe core. This approach to finitism has often admitted primitiverecursive function definitions as ...
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Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics' safe core. This approach to finitism has often admitted primitiverecursive function definitions as obviously finitistic, but some have advocated the inclusion of additional variants of recurrence, while others argued that, to the contrary, primitive recursion exceeds finitism. In a landmark essay, William Tait contested the finitistic nature of these extensions, due to their impredicativity, and advocated identifying finitism with primitiverecursive arithmetic, a stance often referred to as Tait's Thesis. However, a problem with Tait's argument is that the recurrence schema has itself impredicative and non-finitistic facets, starting with an explicit reference to the functions being defined, which are after all infinite objects. It is therefore desirable to buttress Tait's Thesis on grounds that avoid altogether any trace of concrete infinities or impredicativity. We propose here to do just that, building on the generic framework of [13]. We provide further evidence for Tait's Thesis by outlining a proof of a purely finitistic version of Parsons' theorem, whose intuitive gist is that finitistic reasoning is equivalent to finitistic computing.
A notion of strictly primitiverecursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest ...
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A notion of strictly primitiverecursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest to study the corresponding predicate logic. It was argued by Park in 2003 that the predicate logic of strictly primitiverecursive realizability is not arithmetical. Park's argument is essentially based on a claim of Damnjanovic that intuitionistic logic is sound with respect to strictly primitiverecursive realizability, but that claim was disproved by the author of this article in 2006. The aim of this paper is to present a correct proof of the result of Park.
The finitistic philosophy of mathematics, critical of referencing infinite totalities, has been associated from its inception with primitive recursion. That kinship was not initially substantiated, but is widely assum...
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ISBN:
(纸本)9783030367558;9783030367541
The finitistic philosophy of mathematics, critical of referencing infinite totalities, has been associated from its inception with primitive recursion. That kinship was not initially substantiated, but is widely assumed, and is supported by Parson's Theorem, which may be construed as equating finitistic reasoning with finitistic computing. In support of identifying PR with finitism we build on the generic framework of [7] and articulate a finitistic theory of finite partialstructures, and a generic imperative programming language for modifying them, equally rooted in finitism. The theory is an abstract generalization of primitiverecursive Arithmetic, and the programming language is a generic generalization of first-order recurrence (primitive recursion). We then prove an abstract form of Parson's Theorem that links the two.
Shannon's general purpose analog computer (GPAC) is an elegant model of analog computation in continuous Lime. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, t...
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Shannon's general purpose analog computer (GPAC) is an elegant model of analog computation in continuous Lime. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) is an element of G there is a function F(x, t) is an element of G such that F(x, t) = f(t)(x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x(k)theta (x) that sense inequalities in a differentiable way, the resulting class, which we call G+theta (k), is closed under iteration. Furthermore. G+theta (k) includes all primitive recursive functions and has the additional closure property that if nl) is in G+theta (k), then any function of x computable by a Turing machine in T(x) time is also. (C) 2000 Academic Press.
Let III2- denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free II2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable ...
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Let III2- denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free II2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of III2- are, precisely, the primitiverecursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to III2-. In this way, we obtain a more direct answer to Kaye's question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev's original proof. Our methods are model-theoretic and allow for a general study of IIIn+1- for all n >= 0. In particular, we derive a new conservation result for these theories, namely that IIIn+1- is IIn+2-conservative over I Sigma(n), for each n >= 1. (C) 2014 Elsevier B.V. All rights reserved.
By slightly modifying the original definition of loop-programs by Meyer and Ritchie, a modified hierarchy of loop(n) programs, is obtained, with the following characteristics. Letbe the class of functions defined by p...
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By slightly modifying the original definition of loop-programs by Meyer and Ritchie, a modified hierarchy of loop(n) programs, is obtained, with the following characteristics. Letbe the class of functions defined by programs in, be the Grzegorzcyk hierarchy. Then, where in particularhas a natural counterpart in loop programs. It also turns out thatcan be characterized as the class of all recursivefunctions that are computable with a polynomial number of steps by modified programs.
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