On a uniform representation of combinators, arithmetic, lambda terms and types

Abstract

A uniform representation, as binary trees with empty leaves, is given to expressions built with Rosser's X-combinator, natural numbers, lambda terms and simple types. Type inference, normalization of combinator expressions and lambda terms in de Bruijn notation, ranking/unranking algorithms and tree-based natural numbers are described as a literate Prolog program.With sound unification and compact expression of combinatorial generation algorithms, logic programming is shown to conveniently host a declarative playground where interesting properties and behaviors emerge from the interaction of heterogenous but deeply connected computational objects.

On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms

“…However, the concepts of closable and typable skeletons of lambda terms and their uniquely closable and typable variants are new and have not been studied previously. The second author has used extensively Prolog as a metalanguage for the study of combinatorial and computational properties of lambda terms in papers like [16,11] covering different families of terms and properties, but not in combination with the precise analytic methods as developed in this paper.…”

“…We will follow the interleaving of term generation, checking for closedness and type inference steps shown in [11], but split it into a two stage program, with the first stage generating code to be executed, via Prolog's metacall by the second, while also ensuring that the terms generated by the second stage are closed.…”

“…Note that this is a specialization of a general type inferencer to the case when exactly one type variable is available for each leaf node. in papers like [16,11] covering different families of terms and properties, but not in combination with the precise analytic methods as developed in this paper. It has been a long tradition in logic programming to use step-by-step program transformations to derive semantically simpler as well as more efficient code, going back as far as [17], that we have informally followed.…”

“…5,11,26,65,163,417,1086,2858,7599,20391,55127,150028,410719, ... unclosable: 1,0,1,2,4,10,25,62,160,418,1102,2940,7912,21444,58507,160544,442748, ...…”

A Hiking Trip Through the Orders of Magnitude: Deriving Efficient Generators for Closed Simply-Typed Lambda Terms and Normal Forms

“…As a fortunate synergy, Prolog's sound unification of logic variables, backtracking and definite clause grammars have been shown to provide compact combinatorial generation algorithms for various families of lambda terms [11,12,13,14].…”