Predicate Synthesis from Inductive Proof Attempt of Faulty Conjectures

Abstract

We present a method for patching faulty conjectures in automatic theorem proving. The method is based on well-known folding /unfolding deduction rules. The conjectures we are interested in here are implicative formulas that are of the following form: $\forall \overline{x} \phi( \overline{ x})=\forall \overline{x} \exists \overline{Y} \Gamma(\overline{x},\overline{Y}) \leftarrow \Delta(\overline{x})$. A faulty conjecture is a statement $\forall \overline{x} \phi( \overline{x})$, which is not provable in some given program ${\cal T}$, defining all the predicates occurring in φ, i.e, ${\cal M(T)}\not \models \forall \overline{x} \phi( \overline{ x})$, where ${\cal M(T)}$ means the least Herbrand model of ${\cal T}$, but it would be if enough conditions, say P, were assumed to hold, i.e., ${\cal M(T\cup P)}\models \forall \overline{x} \phi( \overline{x})\leftarrow P$, where ${\cal P}$ is the definition of P. The missing hypothesis P is called a corrective predicate for φ. To construct P, we use the abduction mechanism that is the process of hypothesis formation. In this paper, we use the logic based approach because it is suitable for the application of deductive rules.