Quantalic lambda-calculus and additive disjunction

Abstract

Motivated by the need to reason about case statements quantitatively, we extend quantalic linear lambda-calculus with additive disjunction. We show that the resulting equational system is sound. We also show that when certain continuity properties (of the underlying quantale) are adopted, it is additionally (approximately) complete. We present several models of the extended calculus, involving for example meta-theoretical properties in categorical logic (gluing), probabilistic, and quantum computation. As a concrete application, we illustrate how a probabilistic model, based on Banach spaces, can be synergistically used with the calculus' equational system to reason about Cauchy sequences of random walks. This highlights the emergent shift from "program semantics as the science of program equivalence" to flexible, quantitative perspectives, involving functional analysis and beyond.