2 results
Turing and the development of computational complexity
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- By Steven Homer, Boston University, Alan L. Selman, State University of New York
- Edited by Rod Downey, Victoria University of Wellington
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- Book:
- Turing's Legacy
- Published online:
- 05 June 2014
- Print publication:
- 01 May 2014, pp 299-328
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Summary
Abstract. Turing's beautiful capture of the concept of computability by the “Turing machine” linked computability to a device with explicit steps of operations and use of resources. This invention led in a most natural way to build the foundations for computational complexity.
§1. Introduction. Computational complexity provides mechanisms for classifying combinatorial problems and measuring the computational resources necessary to solve them. The discipline provides explanations of why no practical solutions to certain problems have been found, and provides a way of anticipating difficulties involved in solving these problems. The classification is quantitative and is intended to investigate what resources are necessary, lower bounds, and what resources are sufficient, upper bounds, to solve various problems.
This classification should not depend on a particular computational model but rather should measure the intrinsic difficulty of a problem. Precisely for this reason, as we will explain, the basic model of computation for our study is the multitape Turing machine.
Computational complexity theory today addresses issues of contemporary concern, for example, parallel computation, circuit design, computations that depend on random number generators, and development of efficient algorithms. Above all, computational complexity is interested in distinguishing problems that are efficiently computable. Algorithms whose running times are n2 in the size of their inputs can be implemented to execute efficiently even for fairly large values of n, but algorithms that require an exponential running time can be executed only for small values of n.
Turing machines and the spectra of first-order formulas
- Neil D. Jones, Alan L. Selman
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- Journal:
- The Journal of Symbolic Logic / Volume 39 / Issue 1 / March 1974
- Published online by Cambridge University Press:
- 12 March 2014, pp. 139-150
- Print publication:
- March 1974
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H. Scholz [11] defined the spectrum of a formula φ of first-order logic with equality to be the set of all natural numbers n for which φ has a model of cardinality n. He then asked for a characterization of spectra. Only partial progress has been made. Computational aspects of this problem have been worked on by Gunter Asser [1], A. Mostowski [9], and J. H. Bennett [2]. It is known that spectra include the Grzegorczyk class and are properly included in . However, no progress has been made toward establishing whether spectra properly include , or whether spectra are closed under complementation.
A possible connection with automata theory arises from the fact that contains just those sets which are accepted by deterministic linear-bounded Turing machines (Ritchie [10]). Another resemblance lies in the fact that the same two problems (closure under complement, and proper inclusion of ) have remained open for the class of context sensitive languages for several years.
In this paper we show that these similarities are not accidental—that spectra and context sensitive languages are closely related, and that their open questions are merely special cases of a family of open questions which relate to the difference (if any) between deterministic and nondeterministic time or space bounded Turing machines.
In particular we show that spectra are just those sets which are acceptable by nondeterministic Turing machines in time 2cx, where c is constant and x is the length of the input. Combining this result with results of Bennett [2], Ritchie [10], Kuroda [7], and Cook [3], we obtain the “hierarchy” of classes of sets shown in Figure 1. It is of interest to note that in all of these cases the amount of unrestricted read/write memory appears to be too small to allow diagonalization within the larger classes.