Abstract
We give a finer analysis of the difficulty of proof search in classical first-order logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees.
In classical first-order logic without interpreted symbols, we prove that for all these measures, the search for a proof of bounded difficulty (i.e, for a simple proof) is σ p2 -complete. We also show that the same problem when the initial formula is a set of Horn clauses is only NP-complete, and examine the case of first-order logic modulo an equational theory. These results allow us not only to give estimations of the inherent difficulty of automated theorem proving problems, but to gain some insight into the computational relevance of several automated theorem proving methods.
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Goubault, J. (1994). The complexity of resource-bounded first-order classical logic. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_131
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DOI: https://doi.org/10.1007/3-540-57785-8_131
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