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 analyze the inherent complexity of implementing L'evy's notion of optimal evaluation for the -calculus, where similar redexes are contracted in one step via so-called parallel fi-reduction. Optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction. We prove that the cost of parallel fi-reduction is not bounded by any Kalm'ar-elementary recursive function. Not merely do we establish that the parallel fi-step cannot be a unit-cost operation, we demonstrate that the time complexity of implementing a sequence of n parallel fi-steps is not bounded as O(2 n ), O(2 2 n ), O(2 2 2 n ), or in general, O(K ` (n)) where K ` (n) is a fixed stack of ` 2s with an n on top. A key insight, essential to the establishment of this nonelementary lower bound, is that any simply-...

In the last ten years there has been a steady interest in optimal reduction of -terms (or, more generally, of functional programs). The very story started, in fact, more than twenty

Sharing graphs are an implementation of linear logic proof-nets in such a way that their reduction never duplicate a redex. In their usual formulations, proof-nets present a problem of coherence: if the proof-net N reduces by standard cut-elimination to N 0, then, by reducing the sharing graph of N we donot obtain the sharing graph of N 0.Wesolve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is con uent and terminating. The proof of this fact exploits an algebraic semantics for sharing graphs. 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...

Sharing graphs are the structures introduced by Lamping to implement optimal reductions of lambda calculus. Gonthier's reformulation of Lamping's technique inside Geometry of Interaction, and Asperti and Laneve's work on Interaction Systems have shown that sharing graphs can be used to implement a wide class of calculi. Here, we give a general characterization of sharing graphs independent from the calculus to be implemented. Such a characterization rests on an algebraic semantics of sharing graphs exploiting the methods of Geometry of Interaction. By this semantics we can de ne an unfolding partial order between proper sharing graphs, whose minimal elements are unshared graphs. The least-shared instance of a sharing graph is the unique unshared graph that the unfolding partial order associates to it. The algebraic semantics allows to prove that we can associate a semantical read-back to each unshared graph and that such a read-back can be computed

We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).