[Paper Review] Convolution, Separation and Concurrency

Brijesh Dongol, Ian J. Hayes+1 more|arXiv (Cornell University)|Oct 15, 2014

Logic, programming, and type systems21 references3 citations

TL;DR

This paper introduces a generic algebraic framework based on formal power series and convolution to unify and generalize key constructs in program verification, including separation logic, interval logics, and real-time system analysis. By lifting partial semigroups into quantales via convolution, it enables a uniform derivation of Hoare logics and concurrency operations, with the frame rule of separation logic derived via equational reasoning on convolution-based structures.

ABSTRACT

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.

Motivation & Objective

• To unify diverse computational models—such as separation logic, interval logics, and stream interval functions—under a common algebraic framework based on formal power series and convolution.
• To establish a generic lifting theorem that constructs quantales of power series from partial semigroups and target quantales, enabling algebraic reasoning across diverse domains.
• To derive propositional Hoare logics in a uniform way for all instances of the lifting construction, including state, predicate, and resource transformers.
• To demonstrate how convolution serves as a natural concurrency operation in commutative settings, enabling new spatial concurrency models in dynamic and real-time systems.
• To provide a foundation for formalizing and implementing verification tools, including support for concurrent systems with infinite behaviors.

Proposed method

• Defining convolution as a Cauchy product over formal power series: $(f times g)ig(xig) = extstyleigoplus_{x = y ullet z} f(y) ullet g(z)$, where $\bullet$ is a semigroup operation.
• Lifting partial semigroups $S$ and target quantales $Q$ into a power series quantale $\mathcal{P}(S, Q)$, where convolution becomes the multiplication operation.
• Using bi-semigroups and bi-quantales to model dual operations such as temporal chop and spatial separation, enabling combined temporal and spatial reasoning.
• Deriving propositional Hoare logics generically within the power series quantale by encoding preconditions and postconditions as elements of the assertion quantale.
• Applying the lifting construction to concrete instances: languages, binary relations, matrices, traces, intervals, and stream interval functions.
• Establishing locality of predicate transformers via a pointwise convolution inequality: $(f\,p) \ast q \leq f(p \ast q)$, which enables equational derivation of the frame rule.

Experimental results

Research questions

• RQ1 Can convolution in formal power series serve as a unifying algebraic foundation for diverse program verification logics, including separation logic and interval logics?
• RQ2 How can the lifting of semigroups into power series quantales be generalized to support both finite and infinite system behaviors?
• RQ3 To what extent can concurrency be modeled algebraically through commutative convolution, and how does this relate to spatial and temporal composition in real-time systems?
• RQ4 Can Hoare-style inference rules be derived generically across all instances of the power series construction, including for predicate and state transformers?
• RQ5 How can the frame rule of separation logic be derived purely through equational reasoning in the power series framework?

Key findings

• The power series construction lifts any partial semigroup $S$ and target quantale $Q$ into a new quantale $\mathcal{P}(S, Q)$, where convolution becomes the multiplication operation.
• The frame rule of separation logic can be derived via equational reasoning using a pointwise locality condition on predicate transformers: $(f\,p) \ast q \leq f(p \ast q)$.
• Convolution-based separation logic arises naturally from resource monoids, with the assertion quantale formed as a power series over the monoid, and convolution corresponding to separating conjunction.
• A new concurrency operation emerges in the commutative case, enabling spatial concurrency in stream interval functions for dynamic and real-time systems.
• The framework supports both finite and infinite system behaviors by using appropriate source semigroups, such as those modeling infinite traces or streams.
• The entire lifting construction and predicate transformer approach have been formalized in Isabelle/HOL, enabling future tool development for concurrent and separation logic verification.