A Higher-order Interpretation of Deductive Tableau

The Deductive Tableau of Manna and Waldinger is a formal system with an associated methodology for synthesizing functional programs by existence proofs in classical first-order theories. We reinterpret the formal system in a setting that is higher-order in two respects: higher-order logic is used to formalize a theory of functional programs and higher-order resolution is used to synthesize programs during proof. Their synthesis methodology can be applied in our setting as well as new methodologies that take advantage of these higher-order features. The reinterpretation gives us a framework for directly formalizing and implementing the Deductive Tableau system in standard theorem provers that support the kinds of higher-order reasoning listed above. We demonstrate this, as well as a new development methodology, within a conservative extension of higher-order logic in the system. We report too on a case-study in synthesizing sorting programs.