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Parallel computation with threshold functions

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Book cover Structure in Complexity Theory

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 223))

Abstract

We study two classes of unbounded fan-in parallel computation, the standard one, based on unbounded fan-in ANDs and ORs, and a new class based on unbounded fan-in threshold functions. The latter is motivated by a connectionist model of the brain used in Artificial Intelligence. We are interested in the resources of time and address complexity. Intuitively, the address complexity of a parallel machine is the number of bits needed to describe an individual piece of hardware. We demonstrate that (for WRAMs and uniform unbounded fan-in circuits) parallel time and address complexity is simultaneously equivalent to alternations and time on an alternating Turing machine (the former to within a constant multiple, and the latter a polynomial). In particular, for constant parallel time, the latter equivalence holds to within a constant multiple. Thus, for example, polynomial-processor, constant-time WRAMs recognize exactly the languages in the logarithmic time hierarchy, and polynomial-word-size, constant-time WRAMs recognize exactly the languages in the polynomial time hierarchy. As a corollary, we provide improved simulations of deterministic Turing machines by constant-time shared-memory machines. Furthermore, in the threshold model, the same results hold if we replace the alternating Turing machine with the analogous threshold Turing machine, and replace the resource of alternations with the corresponding resource of thresholds. Threshold parallel computers are much more powerful than the standard models (for example, with only polynomially many processors, they can compute the parity function and sort in constant time, and multiply two integers in O(log*n) time), and appear less amenable to known lower-bound proof techniques.

Research supported by NSF grant DCR-84-07256.

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Authors and Affiliations

  1. Department of Computer Science, The Pennsylvania State University, 16802, University Park, Pa.

    Ian Parberry & Georg Schnitger

Authors
  1. Ian Parberry
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  2. Georg Schnitger
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Alan L. Selman

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© 1986 Springer-Verlag Berlin Heidelberg

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Parberry, I., Schnitger, G. (1986). Parallel computation with threshold functions. In: Selman, A.L. (eds) Structure in Complexity Theory. Lecture Notes in Computer Science, vol 223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16486-3_105

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  • DOI: https://doi.org/10.1007/3-540-16486-3_105

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16486-9

  • Online ISBN: 978-3-540-39825-7

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