The use of abduction and recursion-editor techniques for the correction of faulty conjectures

The synthesis of programs, as well as other synthetic tasks, often ends up with an unprovable, partially false conjecture. A successful subsequent synthesis attempt depends on determining why the conjecture is faulty and how it can be corrected. Hence, it is highly desirable to have an automated means for detecting and correcting fault conjectures. We introduce a method for patching faulty conjectures. The method is based on abduction and performs its task during an attempt to prove a given conjecture. On input ∀X.G(X), the method builds a definition for a corrective predicate, P(X), such that ∀X.P(X)→G(X) is a theorem. The synthesis of a corrective predicate is guided by the constructive principle of “formulae as types”, relating inference to computation. We take the construction of a corrective predicate as a program transformation task. The method consists of a collection of construction commands. A construction command is a small program that makes use of one or more program editing commands, geared towards building recursive, equational procedures. A synthesised corrective predicate is guaranteed to be correct, turning a faulty conjecture into a theorem. If conditional, it will be well-defined. If recursive, it will also be terminating