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

## 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 challenges in transferring quantitative info from derivations to types.

## Key findings

- Interpretation non-emptiness characterizes normalizability.
- Type derivation size correlates with execution time.
- Quantitative info does not naturally lift to types.

## Abstract

We investigate the possibility of a semantic account of the execution time (i.e. the number of beta-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.