The primary aim of this work is to provide new tools for machine learning and reasoning within a framework of computing with holistic data representations. Specifically, we demonstrate recursive construction of mappin...
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ISBN:
(纸本)9781509006212
The primary aim of this work is to provide new tools for machine learning and reasoning within a framework of computing with holistic data representations. Specifically, we demonstrate recursive construction of mappings and functions in high-dimensional computing with random vectors - a connectionist computing model which provides benefits similar to neural networks, but allows compositional learning. The computational operations considered in this work are applicable in learning by generalizing from small sets of example data. We demonstrate their use by constructing a simple reasoning model which learns modulo 10 addition and subtraction from a minimal set of examples. Finally, we give examples on how the presented computational operations can be used to compose and manipulate more complex structures in computing with high-dimensional random vectors.
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicit...
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In this paper a variant of transition P systems with external output designed to compute partial functions on natural numbers is presented. These P systems are stable under composition, iteration and unbounded minimiz...
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Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal...
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Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.
A natural example of a function algebra is R(T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kin...
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The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are rep...
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Partial functions are a key concept in programming. Without partiality a programming language has limited expressiveness - it is not Turing- complete, hence, it excludes some constructs such as while-loops. In functio...
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Partial functions are a key concept in programming. Without partiality a programming language has limited expressiveness - it is not Turing- complete, hence, it excludes some constructs such as while-loops. In functional programming languages, partiality mostly originates from the non-termination of recursive functions. Corecursive functions are another source of partiality: here, the issue is not termination, but the inability to produce arbitrary large, finite approximations of a theoretically infinite output. Partial functions have been formally studied in the branch of theoretical computer science called domain theory. In this paper we propose to step up the level of formality by using the Coq proof assistant. The main difficulty is that Coq requires all functions to be total, since partiality would break the soundness of its underlying logic. We propose practical solutions for this issue, and others, which appear when one attempts to define and reason about partial (co)recursive functions in a total functional language.
A language is considered in which the reader can express such properties of block-structured programs with recursive functions as correctness and partial correctness. The semantics of this language is fully described ...
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A language is considered in which the reader can express such properties of block-structured programs with recursive functions as correctness and partial correctness. The semantics of this language is fully described by a set of schemes of axioms and inference rules. The completeness theorem and the soundness theorem for this axiomatization are proved.
We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions ...
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We aim to reason about the correctness of behaviour-preserving transformations of Erlang programs. Behaviour preservation is characterised by semantic equivalence. Based upon our existing formal semantics for Core Erl...
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