Towards Combinatory Logic Synthesis

A theory of composition synthesis based on inhabitation in combinatory logic is described and illustrated with examples. Composition synthesis automatically generates applicative combinations of typed combinators inhabiting a goal type. In the intended applications, combinatory type environments model repositories of components, and types are used to specify their interfaces enriched with semantic and behavioural information. The inhabitation relation in combinatory logic is seen to define an abstract logic programming language, by regarding type environments as logic programs whose operational semantics are given by a search procedure for inhabitants. Intersection types are used to specify the intended semantics and behaviour of combinators. The idea of composition synthesis based on combinatory logic has been implemented in a system, CLS (Combinatory Logic Synthesizer).

[1]  Richard Statman,et al.  Intuitionistic Propositional Logic is Polynomial-Space Complete , 1979, Theor. Comput. Sci..

[2]  Giulio Manzonetto,et al.  Loader and Urzyczyn Are Logically Related , 2012, ICALP.

[3]  Alley Stoughton,et al.  Fully Automatic Adaptation of Software Components Based on Semantic Specifications , 2002, AMAST.

[4]  Jakob Rehof,et al.  Bounded Combinatory Logic , 2012, CSL.

[5]  J. Rehof,et al.  Using Inhabitation in Bounded Combinatory Logic with Intersection Types for GUI Synthesis , 2012 .

[6]  Pawel Urzyczyn,et al.  The emptiness problem for intersection types , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[7]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[8]  Frank Pfenning,et al.  Refinement types for ML , 1991, PLDI '91.

[9]  Katalin Bimbó,et al.  On the decidability of implicational ticket entailment , 2013, The Journal of Symbolic Logic.

[10]  A. K. Chandra,et al.  Alternation , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[11]  Mary Katherine Yntema,et al.  A detailed argument for the Post-Linial theorems , 1964, Notre Dame J. Formal Log..

[12]  Martin W. Bunder,et al.  Combinatory abstraction using B, 6' and friends* , 1994 .

[13]  Mariangiola Dezani-Ciancaglini,et al.  Intersection Types for Combinatory Logic , 1992, Theor. Comput. Sci..

[14]  Gopalan Nadathur,et al.  Uniform Proofs as a Foundation for Logic Programming , 1991, Ann. Pure Appl. Log..

[15]  M. Gladstone.  Some ways of constructing a propositional calculus of any required degree of unsolvability , 1965 .

[16]  Mariangiola Dezani-Ciancaglini,et al.  An extension of the basic functionality theory for the λ-calculus , 1980, Notre Dame J. Formal Log..

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

[18]  W. Singletary.  Recursive unsolvability of a complex of problems proposed by post , 1964 .

[19]  Jakob Rehof,et al.  Finite Combinatory Logic with Intersection Types , 2011, TLCA.

[20]  Vincent Padovani,et al.  Ticket Entailment is decidable , 2017 .

[21]  Sylvain Salvati.  Recognizability in the Simply Typed Lambda-Calculus , 2009, WoLLIC.

[22]  Wilson E. Singletary,et al.  Many-one degrees associated with partial propositional calculi , 1974, Notre Dame J. Formal Log..

[23]  J. Roger Hindley,et al.  Lambda-Calculus and Combinators: An Introduction , 2008 .

[24]  G. Pottinger,et al.  A type assignment for the strongly normalizable |?-terms , 1980 .