Source: J. Symbolic Logic Volume 75, Issue 1
(2010), 239-254.
The undecidability of first-order logic implies that there is no
computable bound on the length of shortest proofs of valid sentences
of first-order logic. Some valid sentences can only have quite long
proofs. How hard is it to prove such “hard” valid sentences? The
polynomial time tractability of this problem would imply the
fixed-parameter tractability of the parameterized problem that,
given a natural number n in unary as input and a first-order
sentence φ as parameter, asks whether φ has a proof
of length ≤ n. As the underlying classical problem has been
considered by Gödel we denote this problem by p-Gödel. We show that
p-Gödel is not fixed-parameter tractable if DTIME(hO(1))
≠ NTIME(hO(1)) for all time constructible and
increasing functions h. Moreover we analyze the complexity of the
construction problem associated with p-Gödel.
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