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Parallel Time and Proof Complexity Klaus Aehlig
 

Summary: Parallel Time and Proof Complexity
Klaus Aehlig
ii
Abstract
Consider the following variant of quantified propositional logic. A
new, parallel extension rule is introduced. This rule is aware of inde-
pendence of the introduced variables. The obtained calculus has the
property that heights of derivations correspond to heights of Boolean
circuits. Adding an uninterpreted predicate on bit-strings, akin to an
oracle in relativised complexity classes, this statement can be made
precise. Consider proofs of the statement that a given circuit can be
evaluated. The most shallow of these proofs in the said calculus has
a height that is, up to an additive constant, the height of the circuit
considered.
The main tool for showing lower bounds on proof heights is a vari-
ant of an iteration principle studied by Takeuti. This reformulation
might be of independent interest, as it allows for polynomial size for-
mulae in the relativised language that require proofs of exponential
height.
An arithmetical formulation of the iteration principle yields a

  

Source: Aehlig, Klaus T. - Institut für Informatik, Ludwig-Maximilians-Universität München

 

Collections: Mathematics; Computer Technologies and Information Sciences