A semantic measure of the execution time in linear logic

Abstract

International audienceWe give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped MELL net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut-link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus

Towards a Semantic Measure of the Execution Time in Call-by-Value lambda-Calculus

“…The approach of [11,12] relies on a crucial lemma to find an upper bound (and hence the exact length) of the execution time: it relates the size of a type derivation to the size of its conclusion, for a normal term/proof-net. In λ sh this lemma should claim that "For every sh-normal form t, if π x 1 : P 1 , .…”

“…We conjecture that in order to overcome this counterexample (and to successfully follow the method of [11,12] to get a purely semantic measure of the execution time) we should change the syntax and the operational semantics of our calculus, always remaining in a call-by-value setting equivalent (from the termination point of view) to λ sh and the other ones studied in [2]. Intuitively, in Ex.…”

“…Starting from [11], research on relational semantics/non-idempotent intersection types has proliferated: various works in the literature explore their power in bounding the execution time or in characterizing normalization [12,8,6,27,5,13,34,28,9,31]. All these works study relational semantics/non-idempotent intersection types either in LL proof-nets (the graphical representation of proofs in LL), or in some variant of ordinary (i.e.…”

“…Following the same approach used in [11] for the call-by-name λ -calculus and in [12] for LL proofnets, we prove that in the shuffling calculus λ sh : Finally, we show that, differently from the case of LL and call-by-name λ -calculus, we are not able to lift the quantitative information enclosed in type derivations to types (i.e. to the interpretation of terms) following the same technique used in [11,12], as our Ex.…”

Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction

“…His analysis uses very basic information on the terms (their length, or height, and order), but gives bounds that are in general very rough. On the other hand other groups, including Dal Lago and Laurent [13], De Carvalho [9], or Bernardet and Lengrand [3], use semantic structures (respectively, game semantics, relational semantics, or non-idempotent intersection types) to capture abstractly the precise complexity of particular terms. Their bounds are much more precise on particular terms, but require information on the terms whose extraction is in general as long to obtain as actual execution.…”