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Journal of Logic and Computation Advance Access originally published online on July 24, 2008
Journal of Logic and Computation 2009 19(1):217-242; doi:10.1093/logcom/exn025
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article appears in the following Journal of Logic and Computation issue: Special Issue: Logic and Computation in the Real World: CiE 2007 [View the issue table of contents]

Original Articles

Logical and Complexity-theoretic Aspects of Models of Computation with Restricted Access to Arrays

Iain A. Stewart

Department of Computer Science, University of Durham, Science Labs, Durham DH1 3LE, UK.

E-mail: i.a.stewart{at}durham.ac.uk

Received 17 May 2008.


  Abstract

We study a class of program schemes, NPSB, in which, aside from basic assignments, non-deterministic guessing and while loops, we have access to arrays; but where these arrays are binary write-once in that they are initialized to ‘zero’ and can only ever be set to ‘one’. We show, amongst other results, that: NPSB can be realized as a vectorized Lindström logic; there are problems accepted by program schemes of NPSB that are not definable in the bounded-variable infinitary logic Formula; all problems accepted by the program schemes of NPSB have asymptotic probability 1 and on ordered structures, NPSB captures the complexity class LNP. We give equivalences (on the class of all finite structures) of the complexity-theoretic question ‘Does NP equal PSPACE?’, where the logics and classes of program schemes involved in the equivalent statements define or accept only problems with asymptotic probability 0 or 1 and so do not cover many computationally trivial problems. The class of program schemes NPSB is actually the union of an infinite hierarchy of classes of program schemes. Finally, when we amend the semantics of our program schemes slightly, we find that the classes of the resulting hierarchy capture the complexity classes {Sigma}Formula (where i≥2) of the Polynomial Hierarchy PH.

Keywords: Finite model theory; descriptive complexity; program schemes


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