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Propositional Logic for Circuit Classes

  • Klaus Aehlig
  • Arnold Beckmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4646)

Abstract

By introducing a parallel extension rule that is aware of independence of the introduced extension variables, a calculus for quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding an uninterpreted predicate on bit-strings (analog to an oracle in relativised complexity classes) this statement can be made precise in the sense that the height of the most shallow proof that a circuit can be evaluated is, up to an additive constant, the height of that circuit.

The main tool for showing lower bounds on proof heights is a variant of an iteration principle studied by Takeuti. This reformulation might be of independent interest, as it allows for polynomial size formulae in the relativised language that require proofs of exponential height.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Klaus Aehlig
    • 1
  • Arnold Beckmann
    • 2
  1. 1.Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, ON M5S 3G4Canada
  2. 2.Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea, SA2 8PPUnited Kingdom

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