Focusing and higher-order abstract syntax

This article aims to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. It was presented at the 36th International Conference on Logic Programming (ICLP).

Recently there has been a great deal of interest in higher-order syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding. The canonical example studied in the literature is that of the untyped ?-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore et al. [13]. Another important syntactic construction is that of explicit substitutions which are used to model local definitions and to implement reduction in the ?-calculus. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Thus explicit substitutions are a natural test case for the further development of the theory and applications of syntax with variable binding. This paper shows that a language containing explicit substitutions and a first-order signature ? is naturally modelled as the initial algebra of the Id + F ??_ +_ ? _ endofunctor. We derive a similar formula for adding explicit substitutions to the untyped ?-calculus and then show these initial algebras provide useful datatypes for manipulating abstract syntax by implementing two reduction machines. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages.

Higher-order abstract syntax is a natural way to formalize programming languages with binders, like the π-calculus, because α-conversion, instantiations and capture avoidance are delegated to the meta-level of the provers, making tedious substitutions superfluous. However, such formalizations usually lack structural induction, which makes syntax-analysis impossible. Moreover, when applied in logical frameworks with object-logics, like Isabelle/HOL or standard extensions of Coq, exotic terms can be defined, for which important syntactic properties become invalid.The paper presents a formalization of the π-calculus in Isabelle/HOL, using well-formedness predicates which both eliminate exotic terms and yield structural induction. These induction-principles are then used to derive the Theory of Contexts fully within the mechanization.

Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus. Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders. Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction --a nondeterministic strategy having a central role in the theory of linear logic-- is standard.

Developing a general component system for a statically typed, object-oriented language is a challenging design problem for two reasons. First, mutually recursive references across components are common in object-oriented programs-an issue that has proven troublesome in the context of component systems for functional and procedural languages. Second, inheritance across component boundaries can cause accidental method overrides. Our recent research shows that a component framework can be constructed for a nominally typed object-oriented language supporting first-class generic types simply by adding appropriate annotations, syntactic sugar, and component-level type-checking. The fundamental semantic building blocks for constructing, type-checking and manipulating components are provided by the underlying first-class generic type system. To demonstrate the simplicity and utility of this approach we have designed and implemented an extension of Java called Component NEXTGEN (CGEN). CGEN, which is based on the Sun Java 5.0 javac compiler, is backwards compatible with existing Java binary code and runs on current Java Virtual Machines. The primary contribution of this paper is a technical analysis of the subtle design issues involved in building a component framework for a nominally typed object-oriented language supporting first-class generics. In contrast to component systems for structurally typed languages, mutual recursion among components is accommodated in the type system and semantics without incorporating any special machinery. Our analysis includes a presentation of Core CGEN (CCG), a small, core language modeling the CGEN framework. It is based on Featherweight GJ and incorporates some ideas from MIXGEN. CCG adds the essential features to support components, but nothing more. Our discussion includes the type rules and semantics for CCG, as well as a proof of type safety.

Interaction nets form a local and strongly confluent model of computation that is per se parallel. We introduce a Curry–Howard correspondence between well-formed interaction nets and a deep-inference deduction system based on linear logic. In particular, linear logic itself is easily expressed in the system and its computational aspects materialise though the correspondence. The system of interaction nets obtained is a typed variant of already well-known sharing graphs. Due to a strong confluence property, strong normalisation for this system follows from weak normalisation. The latter is obtained via an adaptation of Girard's reducibility method. The approach is modular, readily gives rise to generalisations (e.g. second order, known as polymorphism to the programmer) and could therefore be extended to various systems of interaction nets.

On the unity of duality

This paper shows how PILLY(Polymorphic Intuitionistic/Linear Lambda calculus with a fixed point combinatorY) with parametric polymorphism can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using Plotkin's encodings of recursive types in PILLY, we show how parametric models of PILLYgive rise to models of FPC, which is a simply typed lambda calculus with recursive types and an operational call-by-value semantics, reflecting a classical result from domain theory. Essentially, this interpretation is an interpretation of intuitionistic logic into linear logic first discovered by Girard, which in this paper is extended to deal with recursive types. Of particular interest is a model based on ‘admissible’ pers over a reflexive domain, the theory of which can be seen as a domain theory for (impredicative) polymorphism. We show how this model gives rise to a parametric and computationally adequate model of PolyFPC, an extension of FPC with impredicative polymorphism. This is to the author's knowledge the first denotational model of a non-linear language with parametric polymorphism and recursive types.

An implementation model of the typed λ-calculus based on Linear Chemical Abstract Machine

A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard's linear logic with the differences attributable to its focus on behavior instead of proof. As with MLL the multiplicative part can be construed via the Curry-Howard isomorphism as an enrichment of Boolean algebra. The additives cater for independent concurrency or parallel play. The traditional sequential operations of sequence and choice exploit process-specific state information catering for notions of transition and cancellation.

In this paper we deal the recursion and corecursion in functional programming. We discuss about the morphisms which express the recursion or corecursion, resp. We apply the linear logic which provides a logical perspective on computational issues such as control of resources and order of evaluation. The most important feature of linear logic is that formulae are considered as actions and its truth value depends on an internal state of a dynamic system. In this paper we present an alternative way of computation based on algebras and coalgebras. The correctness of our approaches we show by Curry-Howard correspondence.

When designing languages for functional reactive programming (FRP) the main challenge is to provide the user with a simple, flexible interface for writing programs on a high level of abstraction while ensuring that all programs can be implemented efficiently in a low-level language. To meet this challenge, a new family of modal FRP languages has been proposed, in which variants of Nakano's guarded fixed point operator are used for writing recursive programs guaranteeing properties such as causality and productivity. As an apparent extension to this it has also been suggested to use Linear Temporal Logic (LTL) as a language for reactive programming through the Curry-Howard isomorphism, allowing properties such as termination, liveness and fairness to be encoded in types. However, these two ideas are in conflict with each other, since the fixed point operator introduces non-termination into the inductive types that are supposed to provide termination guarantees. In this paper we show that by regarding the modal time step operator of LTL a submodality of the one used for guarded recursion (rather than equating them), one can obtain a modal type system capable of expressing liveness properties while retaining the power of the guarded fixed point operator. We introduce the language Lively RaTT, a modal FRP language with a guarded fixed point operator and an `until' type constructor as in LTL, and show how to program with events and fair streams. Using a step-indexed Kripke logical relation we prove operational properties of Lively RaTT including productivity and causality as well as the termination and liveness properties expected of types from LTL. Finally, we prove that the type system of Lively RaTT guarantees the absence of implicit space leaks.

Primitive recursion for higher-order abstract syntax

Linear logic has recently been introduced by Girard as a logic of actions that seems well suited for concurrent computation. In this paper, we establish a systematic correspondence between Petri nets, linear logic theories, and linear categories. Such a correspondence sheds new light on the relationships between linear logic and concurrency, and on how both areas are related to category theory. Categories are here viewed as concurrent systems the objects of which are states, and the morphisms of which are transitions. This is an instance of the Lambek-Lawvere correspondence between logic and category theory that cannot be expressed within the more restricted framework of the Curry-Howard correspondence.

The linear prewhitening matched filter (PWMF) is the optimal decision function for discriminating exactly-specified signals in additive Gaussian noise. When the signals are less well specified the optimal decision function contains higher than linear terms in the data. Several examples of detection and discrimination tasks are presented for which only the linear term is required and other examples for which higher order terms are necessary. Even in the nonlinear case decision functions can often be approximated by linear operations on the data followed by logical operations. This combination of linear weights plus logic operations is typical of neural network models, which are thought to be elementary models of human processing mechanisms. A number of experiments suggest that there is no decrease in performance of humans for some complex tasks that ideally require such nonlinear operations. However, there are other such tasks where human performance is degraded and this appears to be due more to the complexity of the task and the nature of the correlations in the image than the order of the task. This suggests applications in diagnostic imaging where it might be advantageous for a machine viewer to substitute or work in conjunction with the human observer.