17 Mar 2005
CS 3243 - Logical Inference
45
Reduction con’td.
Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the
propositionalized KB
Idea: For n = 0 to ∞ do
create a
propositional KB by instantiating with depth-n terms
see if α is entailed by this KB
Problem: works if α is entailed,
loops if α is not entailed
Theorem: Turing (1936), Church (1936) Entailment for FOL
is
semi-decidable (algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every non-entailed sentence.)
semi-decidable (algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every non-entailed sentence.)