Implementing the ‘Fool's model’ of combinatory logic

This paper studies ‘Fool's models’ of combinatory logic, and relates them to Hindley's ‘D-completeness’ problem. A ‘fool's model’ is a family of sets of → formulas, closed under condensed detachment. Alternatively, it is a ‘model’ ofCL in naive set theory. We examine Resolution; and the P-W problem. A sequel shows T→ is D-complete; also, its extensions. We close with an implementation FMO of these ideas.

[1]  Mariangiola Dezani-Ciancaglini,et al.  A filter lambda model and the completeness of type assignment , 1983, Journal of Symbolic Logic.

[2]  Mariangiola Dezani-Ciancaglini,et al.  A new type assignment for λ-terms , 1978, Arch. Math. Log..

[3]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[4]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[5]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[6]  J. A. Robinson,et al.  Logic: Form and function - The Mechanization of Deductive Reasoning , 1979 .

[7]  Richard Sylvan,et al.  The semantics of entailment—II , 1972, Journal of Philosophical Logic.

[8]  Michael A. McRobbie,et al.  Automated theorem proving for non-classical logics , 1988, Research Notes in Theoretical Computer Science.

[9]  Errol Peter Martin.  The P-W problem , 1979, Bulletin of the Australian Mathematical Society.

[10]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[11]  Haskell B. Curry,et al.  Foundations of Mathematical Logic , 1964 .

[12]  J. A. Kalman.  Condensed detachment as a rule of inference , 1983 .

[13]  Larry Wos.  Searching for Fixed Point Combinators by Using Automated Theorem Proving : a Preliminary Report , 1988 .

[14]  Robert K. Meyer,et al.  New axiomatics for relevant logics, I , 1974, J. Philos. Log..

[15]  Alasdair Urquhart,et al.  The undecidability of entailment and relevant implication , 1984, Journal of Symbolic Logic.

[16]  J. Roger Hindley,et al.  Principal type-schemes and condensed detachment , 1990, Journal of Symbolic Logic.

[17]  Coppo Mario,et al.  A new type assignment for lambda-terms , 1978 .

[18]  David Meredith,et al.  In memoriam: Carew Arthur Meredith (1904-1976) , 1977, Notre Dame J. Formal Log..

[19]  R. Meyer,et al.  Algebraic analysis of entailment I , 1972 .

[20]  Maria Paola Bonacina,et al.  Automated Reasoning , 2014, Lecture Notes in Computer Science.

[21]  P. Sallé.  Une extension de la theorie des types en λ-calcul , 1978 .