Lambda-calculus combinators and functional programming
Lambda-calculus combinators and functional programming
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
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The need to solve structural constraints arises when we investigate computational solutions to the question: "in which logics is a given formula deducible?" This question is posed when one wants to learn the structural permissions for a Categorial Grammar deduction. This paper is part of a project started at [10], to deal with the question above. Here, we focus on structural constraints, a form of Structurally-Free Theorem Proving, that deal with an unknown transformation X which, when applied to a given set of components P1...Pn, generates a desired structure Q. The constraint is treated in the framework of the combinator calculus as XP1...Pn ↠ Q, where the transformation X is a combinator, the components Pi and Q are terms, and ↠ reads "reduces to". We show that in the usual combinator system not all admissible constraints have a solution; in particular, we show that a structural constraint that represents right-associativity cannot be solved in it nor in any consistent extension of it. To solve this problem, we introduce the notion of a restricted combinator system, which can be consistently extended with complex combinators to represent right-associativity. Finally, we show that solutions for admissible structural constraints always exist and can be efficiently computed in such extension.