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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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01 Feb 2000
TL;DR: A global and comprehensive view of the properties of subclasses of two counters automata for which counters are only accessed through the following operations: increment (+1), decrement (-1), reset (c := 0), transfer (the whole content of counter c is transfered into counter c′), and testing for zero is presented.
Abstract: We present a global and comprehensive view of the properties of subclasses of two counters automata for which counters are only accessed through the following operations: increment (+1), decrement (-1), reset (c := 0), transfer (the whole content of counter c is transfered into counter c′), and testing for zero. We first extend Hopcroft-Pansiot's result (an algorithm for computing a finite description of the semilinear set post*) to two counters automata with only one test for zero (and one reset and one transfer operations). Then, we prove the semilinearity and the computability of pre* for the subclass of 2 counters automata with one test for zero on c1, two reset operations and one transfer from c1 to c2. By proving simulations between subclasses, we show that this subclass is the maximal class for which pre* is semilinear and effectively computable. All the (effective) semilinearity results are obtained with the help of a new symbolic reachability tree algorithm for counter automata using an Acceleration function. When Acceleration has the so-called stability property, the constructed tree computes exactly the reachability set.
46 citations
TL;DR: This article gives a detailed exposition of a soundness and completeness proof for the rather new type of a deductive propositional system CL1, the logical vocabulary of which contains operators for the so called parallel and choice operations, and the atoms of which represent elementary problems, that is, predicates in the standard sense.
Abstract: In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of “always computabl ” problems. Computability logic has been introduced semantically, and now among its main technical goals is to axiomatize the set of valid formulas or various natural fragments of that set. The present contribution signifies a first step towards this goal. It gives a detailed exposition of a soundness and completeness proof for the rather new type of a deductive propositional system CL1, the logical vocabulary of which contains operators for the so called parallel and choice operations, and the atoms of which represent elementary problems, that is, predicates in the standard sense.This article is self-contained as it explains all relevant concepts. While not technically necessary, familiarity with the foundational paper “Introduction to Computability Logi ” [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic---the context that determines the value of the present results.
46 citations
TL;DR: In this paper, a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure, was proposed and applied to branched polymers, Brownian trees, and self-avoiding random walks.
Abstract: We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and self-avoiding random walks.
46 citations
TL;DR: The present article explains how the notion of “Object with Effective Homology” on the contrary allows the user to recursively obtain all the components of the Serre and Eilenberg‐Moore spectral sequences, when the data are objects with effective homology.
46 citations
TL;DR: The present work compares and unifies different relaxed notions of computability to cover also discontinuous functions based on the concept of the jump of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation and a formalization of revising computation in the sense of Shoenfield.
45 citations