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 beta-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 pattern-matching in the style of Cirstea, Kirchner, and Liquori. The second is the Closed Logical Framework (CLF) which features, besides standard beta-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

An Ontology Based Approach Towards A Universal Description Framework for Home Networks

Current home networks typically involve two or more machines sharing network resources. The vision for the home network has grown from a simple computer network, to every day appliances embedded with network capabilities. In this environment devices and services within the home can interoperate, regardless of protocol or platform. Network clients can discover required resources by performing network discovery over component descriptions. Common approaches to this discovery process involve simple matching of keywords or attribute/value pairings. Interest emerging from the Semantic Web community has led to ontology languages being applied to network domains, providing a logical and semantically rich approach to both describing and discovering network components. In much of the existing work within this domain, developers have focused on defining new description frameworks in isolation from existing protocol frameworks and vocabularies. This work proposes an ontology-based description framework which takes the ontology approach to the next step, where existing description frameworks are in- corporated into the ontology-based framework, allowing discovery mechanisms to cover multiple existing domains. In this manner, existing protocols and networking approaches can participate in semantically-rich discovery processes. This framework also includes a system architecture developed for the purpose of reconciling existing home network solutions with the ontology-based discovery process. This work also describes an implementation of the approach and is deployed within a home-network environment. This implementation involves existing home networking frameworks, protocols and components, allowing the claims of this work to be examined and evaluated from a ‘real-world’ perspective

Frameworks for logically classifying polynomial-time optimisation problems.

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems

Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance

Refinement types sharpen systems of simple and dependent types by offering expressive means to more precisely classify well-typed terms. We present a system of refinement types for LF in the style of recent formulations where only canonical forms are well-typed. Both the usual LF rules and the rules for type refinements are bidirectional, leading to a straightforward proof of decidability of typechecking even in the presence of intersection types. Because we insist on canonical forms, structural rules for subtyping can now be derived rather than being assumed as primitive. We illustrate the expressive power of our system with examples and validate its design by demonstrating a precise correspondence with traditional presentations of subtyping. Proof irrelevance provides a mechanism for selectively hiding the identities of terms in type theories. We show that LF refinement types can be interpreted as predicates using proof irrelevance, establishing a uniform relationship between two previously studied concepts in type theory. The interpretation and its correctness proof are surprisingly complex, lending support to the claim that refinement types are a fundamental construct rather than just a convenient surface syntax for certain uses of proof irrelevance