Properties of Set Functors

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Modeling Fresh Names in the π-calculus Using Abstractions

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A Framework for Defining Logics

The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Löf’s system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed

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 Open Logical Framework

The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of \u25a1-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems, which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincar Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP: strong normalization, confluence and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution and reduction in the arguments. Moreover, we provide a canonical presentation of LFP, based on a suitable extension of the notion of \u3b2\u3b7-long normal form, allowing for smooth formulations of adequacy statements. \ua9 The Author, 2013

Uncountable Limits and the Lambda Calculus

In this paper we address the problem of solving recursive domain equations using uncountable limits of domains. These arise for instance, when dealing with the omega_1-continuous function-space constructor and are used in the denotational semantics of programming languages which feature unbounded choice constructs. Surprisingly, the category of cpo’s and omega_1-continuous embeddings is not omega_0-cocomplete. Hence the standard technique for solving reflexive domain equations fails. We give two alternative methods. We discuss also the issue of completeness of the lambda beta eta-calculus w.r.t reflexive domain models. We show that among the reflexive domain models in the category of cpo’s and omega_0-continuous functions there is one which has a minimal theory. We give a reflexive domain model in the category of cpo’s and omega_1-continuous functions whose theory is precisely the lambda beta eta theory. So omega_1-continuous lambda-models are complete for the lambda beta eta-calculus

On the completeness of order-theoretic models of the lambda-calculus

Scott discovered his domain-theoretic models of the \u3bb-calculus, isomorphic to their function space, in 1969. A natural completeness problem then arises: whether any two terms equal in all Scott models are convertible. There is also an analogous consistency problem: whether every equation between two terms, consistent with the \u3bb-calculus, has a Scott model. We consider such questions for wider sets of sentences and wider classes of models, the pointed (completely) partially ordered ones. A negative result for a set of sentences shows the impossibility of finding Scott models for that class; a positive result gives evidence that there might be enough Scott models. We find, for example, that the order-extensional pointed \u3c9-cpo models are complete for \u3a01-sentences with positive matrices, whereas the consistency question for \u3a31-sentences with equational matrices depends on the consistency of certain critical sentences asserting the existence of certain functions analogous to the generalized Mal'cev operators first considered in the context of the \u3bb-calculus by Selinger

Towards a Logical Framework with Intersection and Union Types

International audienceWe present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection , union, and strong (or minimal relevant) implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing to reflect polymorphic features of proofs in formulae. This is in contrast to classical Truth-functional connec-tives where the meaning of a compound formula is only dependent on the truth value of its subformulas. Both Logical Frameworks and proof functional logics consider proofs as first class citizens. But they do it differently namely, explicitly in the former while implicitly in the latter. Their combination opens up new possibilites of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory can be successfully plugged in existing truth-functional proof assistants

Plugging-in Proof Development Environments using Locks in LF

International audienceWe present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for plugging-in, i.e. gluing together, different Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the "CMU School". The first system, CLLFP , is the canonical version of the system LLFP, presented earlier by the authors. The second system, CLLF P? , features the possibility of invoking the oracle to obtain also a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, systems of Light Linear Logic, and partial functions

On Quantitative Algebraic Higher-Order Theories

International audienceWe explore the possibility of extending Mardare et al.’s quantitative algebras to the structures which naturally emerge from Combinatory Logic and the λ-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results clearly delineating to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras