Work

Quantitative Equational Logic

Year: 2026

Type: article

Abstract: We develop a quantitative analogue of equational reasoning, which we call quantitative equational logic. The quantitative equations use, instead of classical equality, quantitative equalities, which are equalities indexed with nonnegative reals. Thus, s = ɛ t means that “ s and t are points in a metric space and their distance is less than ɛ”. Quantitative equalities will be used to encode behavioural distances, with ɛ being an upper bound on the measure of dissimilarity between two terms. We develop the metatheory of this subject. We define a notion of quantitative algebra, which is the quantitative analogue of universal algebra. We prove a completeness theorem for quantitative equational logic, and we show that we obtain monads on suitable categories of metric spaces. We present a set of examples where the free algebra of a quantitative equational theory corresponds to some well-known structure. These examples are: Hausdorff metrics from quantitative semilattices; p -Wasserstein metrics (hence also the Kantorovich metric), and the total variation metric.

Source: Journal of the ACM

DOI: 10.1145/3818603

Authors: Giorgio Bacci, Radu Mardare, Prakash Panangaden, Gordon Plotkin

Institutions: Allied American University, Heriot-Watt University, McGill University

Language: English

Topic: Advanced Algebra and Logic

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: Natural Sciences and Engineering Research Council of Canada, Engineering and Physical Sciences Research Council

Award: A correct-by-construction approach to approximate computation