Proof procedures and axiom set in Petri net models of Horn clause propositional logic

The authors analyze the time complexities of (1) the provability detecting problem, (2) the minimum axiom set problem, and (3) the minimum variable deletion problem in the Horn clause propositional logic. An O(σ) algorithm for the first problem is proposed, where σ is the formula size of a given set of Horn clauses. The second problem is shown to be NP-complete and some approximation algorithms with their experimental evaluation are given. Concerning the third problem, which has been known to be NP-complete, a number of new approximation algorithms with their experimental evaluation are given