We investigate the computational efficiency of the sharing graphs of Lamping [Lam90], Gonthier, Abadi, and L'evy [GAL92], and Asperti [Asp94], designed to effect so-called optimal evaluation, with the goal of reconciling optimality, efficiency, and the clarification of reasonable cost models for the -calculus. Do these graphs suggest reasonable cost models for the -calculus? If they are optimal, are they efficient? We present a brief survey of these optimal evaluators, identifying their common characteristics, as well as their shared failures. We give a lower bound on the efficiency of sharing graphs by identifying a class of -terms that are normalizable in \Theta(n) time, and require \Theta(n) "fan interactions, " but require\Omega\Gammaq n ) bookkeeping steps. For [GAL92], we analyze this anomaly in terms of the dynamic maintenance of deBruijn indices for intermediate terms. We give another lower bound showing that sharing graphs can do \Omega\Gammao n ) work (via fan interactio...

We propose a new technique for hardware synthesis from higherorder functional languages with imperative features based on Reynolds’s Syntactic Control of Interference. The restriction on contraction in the type system is useful for managing the thorny issue of sharing of physical circuits. We use a semantic model inspired by game semantics and the geometry of interaction, and express it directly as a certain class of digital circuits that form a

Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reduction process is local and asynchronous, and all the operations are made explicit, including discarding and copying of data. Our aim is to bridge the gap between the above formalisms by showing how to understand interaction nets in a term rewriting framework. This allows us to transfer results from one paradigm to the other, deriving syntactical properties of interaction nets from the (well-studied) properties of term rewriting systems; in particular concerning termination and modularity. Keywords: term rewriting, interaction nets, termination, modularity. 1 Introduction Term rewriting systems provide a general framework for specifying and reasoning about computation. They can be regarde...

The oe-calculus adds explicit substitutions to the -calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the oe-calculus to provide a linear calculus of explicit substitutions, called xDILL, which analogously describes the implementation of linear functional programming languages. Our main observation is that there are non-trivial interactions between linearity and explicit substitutions and that xDILL is therefore best understood as a synthesis of its underlying logical structure and the technology of explicit substitutions. This is in contrast to the oe-calculus where the explicit substitutions are independent of the underlying logical structure. Keywords: -calculus, explicit substitutions, linear logic 1 Introduction This paper combines the technologies of explicit substitutions and linearity in a mathematically consistent way. We start by describing these technologies and the...

We reveal a symmetric structure in the ho/n games model of innocent strategies, introducing rigid strategies, a concept dual to bracketed strategies. We prove a direct definability theorem of general innocent strategies with respect to a simply typed language of extended Bohm trees, which gives an operational meaning to rigidity in call-byname. A corresponding factorization of innocent strategies into rigid ones with some form of conditional as an oracle is constructed. 1

Optimal graph reduction technology for the -calculus, as developed by Lamping, with modifications by Asperti, Gonthier, Abadi, and L'evy, has a well-understood local dynamics based on a standard menagerie of reduction rules, as well as a global context semantics based on Girard's geometry of interaction. However, the global dynamics of graph reduction has not been subject to careful investigation. In particular, graphs lose their structural resemblence to -terms after only a few graph reduction steps, and little is known about graph reduction strategies that maintain efficiency or structure. While the context semantics provides global information about the computation, its use as part of a reduction strategy seems computationally infeasible. We propose a tractable graph reduction strategy that preserves computationally relevant global structure, and allows us to efficiently bound the computational resources needed to implement optimal reduction. A simple canonical representation for gr...

An examination of Girard's execution formula suggests implementations of the Geometry of Interaction at the syntactic level. In this paper we limit our scope to groundtype terms and study the parallel aspects of such implementations, by introducing a family of abstract machines which can be directly implemented. These machines address all the important implementation issues such as the choice of an inter-thread communication model, and allow to incorporate specic strategies for dividing the computation of the execution path into smaller tasks. 1

This paper describes a framework for flow analysis of programs with higher-order functions with normal-order reduction. The framework is based on an abstract machine derived from the Geometry of Interaction semantics for reduction in linear logic proof nets. By standard methods from abstract interpretation the transition system defined by the machine induces a set of equations defining the flow between the program points. This set of equations defines a collecting semantics for the program and is amenable to further analysis by abstraction-based approximation. As examples of its application we show how to obtain information about strictness, control-flow and usage of data.

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the reduction strategy) and a lower bound to the number of steps to normal form (for certain reduction strategies). Weights are then exploited in proving strong soundness theorems for various subsystems of linear logic, namely elementary linear logic, soft linear logic and light linear logic. 1

Abstract. Two common misbeliefs about encodings of the λ-calculus in interaction nets (INs) are that they are good only for strategies that are not very well understood (e.g. optimal reduction) and that they always have to deal in a complex way with boxes. In brief, the theory of interaction nets is more or less disconnected from the standard theory: we can do things in INs that we cannot do with terms, which is true [5, 10]; and we cannot do in INs things that can easily be done with terms. This paper contributes to fighting this misbelief by showing that the standard call-by-name and call-by-value strategies of the λ-calculus are encoded in interaction nets in a very simple and extensible way, and in particular that these encodings do not need any notion of box. This work can also be seen as a first step towards a generic approach to derive graph-based abstract machines. 1