# Quantitative Logics for Equivalence of Effectful Programs

## TL;DR

This paper introduces a framework of quantitative logics to analyze the behavioral equivalence of effectful programs, extending traditional notions to include probabilistic and nondeterministic effects.

## Contribution

It develops a new approach using quantitative modalities to characterize program equivalence, connecting it to Abramsky's applicative bisimilarity and ensuring congruence.

## Key findings

- Quantitative modalities can characterize program equivalence.
- Equivalence aligns with Abramsky's applicative bisimilarity.
- Results apply to nondeterministic, probabilistic, and store effects.

## Abstract

In order to reason about effects, we can define quantitative formulas to describe behavioural aspects of effectful programs. These formulas can for example express probabilities that (or sets of correct starting states for which) a program satisfies a property. Fundamental to this approach is the notion of quantitative modality, which is used to lift a property on values to a property on computations. Taking all formulas together, we say that two terms are equivalent if they satisfy all formulas to the same quantitative degree. Under sufficient conditions on the quantitative modalities, this equivalence is equal to a notion of Abramsky's applicative bisimilarity, and is moreover a congruence. We investigate these results in the context of Levy's call-by-push-value with general recursion and algebraic effects. In particular, the results apply to (combinations of) nondeterministic choice, probabilistic choice, and global store.