Theorem Proving for Classical Logic with Partial Functions by Reduction to Geometric Kleene Logic

Partial functions are abundant in mathematics and program specifications. Unfortunately, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into sets of formulas in Kleene logic. For proof search in Kleene logic, we use a threevalued adaptation of geometric resolution. All procedures described in this paper generate explicit proofs. We show that a proof of a well-typed formula, found by geometric resolution in Kleene logic, can be translated back into a PCL proof. Although the geometric reasoning procedure is intended for reasoning on problems that originate from PCL, it can be directly used for Kleene logic as well.