A Complexity Gap for Tree-Resolution

Søren Riis

Abstract


It is shown that any sequence  psi_n of tautologies which expresses the
validity of a fixed combinatorial principle either is "easy" i.e. has polynomial
size tree-resolution proofs or is "difficult" i.e requires exponential
size tree-resolution proofs. It is shown that the class of tautologies which
are hard (for tree-resolution) is identical to the class of tautologies which
are based on combinatorial principles which are violated for infinite sets.
Actually it is shown that the gap-phenomena is valid for tautologies based
on infinite mathematical theories (i.e. not just based on a single proposition).
We clarify the link between translating combinatorial principles (or
more general statements from predicate logic) and the recent idea of using
the symmetrical group to generate problems of propositional logic.
Finally, we show that it is undecidable whether a sequence  psi_n (of the
kind we consider) has polynomial size tree-resolution proofs or requires
exponential size tree-resolution proofs. Also we show that the degree of
the polynomial in the polynomial size (in case it exists) is non-recursive,
but semi-decidable.

Keywords: Logical aspects of Complexity, Propositional proof complexity,
Resolution proofs.

 


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DOI: http://dx.doi.org/10.7146/brics.v6i29.20098
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ISSN: 0909-0878 

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