Higher-Order Tableaux

Even though higher-order calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higher-order logic that use higher-order unification as the key inference procedure. These calculi differ in the treatment of the substitutional properties of equivalences. The first calculus is equivalent in deductive power to the machineoriented higher-order refutation calculi known from the literature, whereas the second is complete with respect to Henkin's general models.

[1]  Michael Kohlhase,et al.  A mechanization of sorted higher-order logic based on the resolution principle , 1994 .

[2]  R. Smullyan.  First-Order Logic , 1968 .

[3]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[4]  Peter B. Andrews.  Resolution in type theory , 1971, Journal of Symbolic Logic.

[5]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[6]  Gérard Huet,et al.  Constrained resolution: a complete method for higher-order logic. , 1972 .

[7]  Steve Reeves,et al.  Semantic tableaux as a framework for automated theorem-proving , 1987 .

[8]  Tomasz Pietrzykowski,et al.  Mechanizing omega-Order Type Theory Through Unification , 1976, Theor. Comput. Sci..

[9]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[10]  Wayne Snyder.  Proof theory for general unification , 1993, Progress in computer science and applied logic.

[11]  J. Roger Hindley,et al.  Introduction to Combinators and Lambda-Calculus , 1986 .

[12]  Raymond M. Smullyan,et al.  A unifying principle for quantification theory , 1963 .

[13]  R M Smullyan,et al.  A UNIFYING PRINCIPAL IN QUANTIFICATION THEORY. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Dale A. Miller.  Proofs in Higher-Order Logic , 1983 .

[15]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .