Towards a Semantic Measure of the Execution Time in Call-by-Value lambda-Calculus (Long Version)

TL;DR

This paper explores a semantic approach to measure execution time in call-by-value lambda calculus using a linear logic-based model, revealing limitations and proposing future refinements for accurate timing analysis.

Contribution

It introduces a semantic framework for estimating execution time in call-by-value lambda calculus and highlights the need for syntax and semantics refinement for precise measurement.

Findings

Interpretation non-emptiness indicates normalizability.

Type derivation size correlates with execution time.

Quantitative info in derivations does not extend to types.

Abstract

We investigate the possibility of a semantic account of the execution time (i.e. the number of \beta_v-steps leading to the normal form, if any) for the shuffling calculus, an extension of Plotkin's call-by-value {\lambda}-calculus. For this purpose, we use a linear logic based denotational model that can be seen as a non-idempotent intersection type system: relational semantics. Our investigation is inspired by similar ones for linear logic proof-nets and untyped call-by-name {\lambda}-calculus. We first prove a qualitative result: a (possibly open) term is normalizable for weak reduction (which does not reduce under abstractions) if and only if its interpretation is not empty. We then show that the size of type derivations can be used to measure the execution time. Finally, we show that, differently from the case of linear logic and call-by-name {\lambda}-calculus, the quantitative information enclosed in type derivations does not lift to types (i.e. to the interpretation of terms). To get a truly semantic measure of execution time in a call-by-value setting, we conjecture that a refinement of its syntax and operational semantics is needed.