A Framework for Defining Logical Frameworks

In this paper, we introduce a General Logical Framework, called GLF, for defining Logical Frameworks, based on dependent types, in the style of the well known Edinburgh Logical Framework LF. The framework GLF features a generalized form of lambda abstraction where β-reductions fire provided the argument satisfies a logical predicate and may produce an n-ary substitution. The type system keeps track of when reductions have yet to fire. The framework GLF subsumes, by simple instantiation, LF as well as a large class of generalized constrained-based lambda calculi, ranging from well known restricted lambda calculi, such as Plotkin’s call-by-value lambda calculus, to lambda calculi with patterns. But it suggests also a wide spectrum of completely new calculi which have intriguing potential as Logical Frameworks. We investigate the metatheoretical properties of the calculus underpinning GLF and illustrate its expressive power. In particular, we focus on two interesting instantiations of GLF. The first is the Pattern Logical Framework (PLF), where applications fire via patternmatching in the style of Cirstea, Kirchner, and Liquori. The second is the Closed Logical Framework (CLF) which features, besides standard β-reduction, also a reduction which fires only if the argument is a closed term. For both these instantiations of GLF we discuss standard metaproperties, such as subject reduction, confluence and strong normalization. The GLF framework is particularly suitable, as a metalanguage, for encoding rewriting logics and logical systems, where rules require proof terms to have special syntactic constraints, e.g. logics with rules of proof, in addition to rules of derivations, such as, e.g., modal logics, and call-by-value lambda calculus. Category D.3.1: Formal Definitions and Theory: Syntax, Semantics. Category D.3.2: Language Classifications: Applicative (functional) languages, Constraint and logic languages ∗ University of Udine, Italy, e-mail: [lenisa,honsell]@dimi.uniud.it † Research partially supported by the EU Project FP6-IST-510996 TYPES, and by the French CNRS grant ACI Modulogic. ‡ Inria Sophia Antipolis, France, e-mail: [email protected] 2 Honsell & Lenisa & Liquori Category F.4.1: Mathematical Logic: Lambda calculus and related systems, Logic and constraint programming, Mechanical theorem proving. Key-words: Edinburgh’s Logical Framework, Lambda calculus, Patterns, Pattern matching, Dependent-type systems, Curry-Howard, Logics.