 
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 bitstrings, 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
