Focusing on Binding and Computation

Abstract

Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with the notion of a scoped inference rule, which permits the adequate representation of binding via higher-order abstract syntax. On the other hand, the usual computational arrow classifies recursive functions over such higher-order data. Unlike many previous approaches, both binding and computation can mix freely. Following Zeilberger [POPL 2008], the computational function space admits a form of openendedness, in that it is represented by an abstract map from patterns to expressions. As we demonstrate with Coq and Agda implementations, this has the important practical benefit that we can reuse the pattern coverage checking of these tools.

Unembedding domain-specific languages

“…In the binop operation we use the function type computationally as a host language function. Licata et al [2008] define a new logical system based on a proof theoretic analysis of focussing to mix the computational and representation function spaces. Using parametric polymorphism, we get the same functionality for free.…”

Resolving Inductive Definitions with Binders in Higher-Order Typed Functional Programming

“…We think it is both useful and interesting to study executable operational semantics also using functional logic programming [14]. It has proved harder to integrate HOAS representations with functional programming: see [18] for a recent view on this. In any case, in the Introduction we advocated leaving the HOAS mainstream and pursuing a nominal approach, for reasons to do with name-aliasing.…”

A universe of binding and computation

“…This distinguishes certain types C which may be populated by variables from other types D which may not. This definition of VarType permits only variables of base type, rather than the full language of higher-order rules that we considered in previous work (Licata et al, 2008 The only subtlety in this definition is that we represent the bodies of ∀c and ∃c by computational functions in Agda. This choice has some trade-offs: on the one hand, it means that the bodies of quantifiers can be specified by any Agda computation (e.g.…”

“…In our previous work (Licata et al, 2008), we investigated the logical foundations of a pronominal approach to mixing binding and computation. In the present paper, we give an implementation of (a slight variant of) our framework, and we demonstrate the viability of our approach by programming some standard difficult test cases from the literature.…”