Nominal logic programming

Abstract

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, α-equivalence). This article investigates logic programming based on nominal logic. This technique is especially well-suited for prototyping type systems, proof theories, operational semantics rules, and other formal systems in which bound names are present. In many cases, nominal logic programs are essentially literal translations of "paper" specifications. As such, nominal logic programming provides an executable specification language for prototyping, communicating, and experimenting with formal systems.We describe some typical nominal logic programs, and develop the model-theoretic, prooftheoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via two examples.

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

“…Section 4 introduces the main contribution of the paper, a typed higher-order functional programming language that extends core ML with name-binding types, a type of "semi-decidable propositions" and existential quantification for types in a class of equality types coinciding with the arities of a user-declared nominal signature. This language, which we call αML, draws upon the ideas of αProlog [7], extending them to higher-order functional programming. αML is a simplification of both our first attempt to do this [16] and of αProlog itself, in that it avoids the use of concrete names and name-permutations in programs (see Remark 2.2 and Sect.…”

“…At the very least we need name-inequality constraints x = x , where x, x ∈ Var (N ) with N a name sort. Experience with nominal logic [23,11] and nominal logic programming [7] shows that it is useful to generalize name-inequality to name freshness constraints, x # p where x ∈ Var (N ) and p ∈ Pat Σ (A), even though these can be inductively defined in terms of name-inequality (cf. Fig.…”

“…The underlined freshness constraint in (5) enforces the usual "capture-avoiding" property when substituting under a λ-binder (cf. [7,Example 2.3]). …”

“…In any case, in the Introduction we advocated leaving the HOAS mainstream and pursuing a nominal approach, for reasons to do with name-aliasing. As far as we know the first such approach was Cheney and Urban's first-order logic programming language αProlog [7]. Our first attempt to combine αProlog's computational mechanism (resolution based on nominal unification [28]) with higher-order typed functional programming was influenced by the work on FreshML [26] and produced MLSOS [16].…”

“…One approach is to identify restrictions on α-inductive definitions that do not limit their applicability for specifying operational semantics too much, but for which the associated α-tree constraint problems are in P rather than NP; cf. Cheney and Urban [7,Sect. 5.3].…”