Convolution Kleene Algebras

• Convolution Kleene Algebras are function-space liftings of Kleene algebra structures, defined via convolution over compositional domains like monoids and categories.
• They support recursive definitions of the Kleene star in both complete and finitary settings by employing techniques such as Möbius finiteness and graded decomposition.
• These algebras model applications from formal power series to weighted languages and concurrent systems, unifying sequential and parallel composition under a common framework.

Convolution Kleene algebras are function-space liftings of Kleene-algebraic structure from a value algebra $K$ to a domain $X$ that carries a notion of composition, such as a monoid, category, catoid, higher catoid, or relational interchange monoid. Their characteristic multiplication is convolution: for functions $f,g : X \to K$, the value at $x\in X$ is obtained by summing products $f(y)\cdot g(z)$ over decompositions of $x$ into $y$ and $z$. In Boolean instances this is a powerset lifting; in weighted instances the codomain elements are interpreted as weights; and in concurrent instances two composition relations induce sequential and parallel convolutions. The central technical issue is the construction of a suitable Kleene star on $K^X$: complete-lattice settings obtain it from joins of powers, whereas finitary settings require decomposition conditions such as grading or Möbius finiteness and a recursive star definition (Cranch et al., 2020, Cranch et al., 29 Aug 2025, Ésik et al., 2015).

1. Algebraic core and convolution construction

The background structure is a semiring or dioid $K$, often a Kleene algebra $X$0. In the formulations used for convolution, addition is idempotent, hence determines the natural order $X$1, and multiplication distributes over addition. A Kleene algebra then equips this dioid with a star satisfying unfold and induction laws, such as $X$2 together with the usual left and right induction principles (Cranch et al., 29 Aug 2025).

Given a structured domain $X$3, the function space $X$4 inherits pointwise addition,

$X$5

and convolution multiplication. In the monoidal case,

$X$6

In the relational presentation used for quantales and concurrent algebras, a ternary relation $X$7 encodes composition, and convolution is

$X$8

Relational associativity of $X$9 yields associativity of convolution, relational units yield a unit in $f,g : X \to K$0, and relational commutativity yields commutativity in the abelian case. When $f,g : X \to K$1, functions $f,g : X \to K$2 are characteristic functions of subsets of $f,g : X \to K$3, so convolution becomes the usual powerset lifting of a ternary relation to subsets (Cranch et al., 2020).

This construction subsumes standard algebras of formal power series, weighted languages, incidence algebras, path algebras, and weighted relations. In the 2025 generalization, the domain $f,g : X \to K$4 is allowed to be a catoid or higher catoid with set-valued composition, so convolution is no longer restricted to single-sorted monoids or ordinary categories. The same framework supports local functional cases such as categories and strict $f,g : X \to K$5-categories, as well as relational cases such as shuffle catoids and higher relational monoids (Cranch et al., 29 Aug 2025).

2. Defining the Kleene star on convolution algebras

The major obstruction in the subject is not convolution itself but the star. The 2025 work identifies this explicitly: convolution algebras on maps from monoids, groups, categories, and related structures are common, but a suitable star on $f,g : X \to K$6 is nontrivial outside complete settings. The proposed solution combines generalized Möbius categories or catoids with a generalized version of the classical recursive star for formal power series (Cranch et al., 29 Aug 2025).

A catoid $f,g : X \to K$7 carries a set-valued composition and source/target maps. An element $f,g : X \to K$8 has finite length when there is a finite bound on the degree of its decompositions into non-identity arrows, and $f,g : X \to K$9 is Möbius when every element is finitely decomposable. For a Möbius catoid $x\in X$0 and a Kleene algebra $x\in X$1, the star on $x\in X$2 is defined recursively. For identities $x\in X$3,

$x\in X$4

and for non-identities $x\in X$5,

$x\in X$6

This recursion is well founded because Möbius finiteness guarantees only finitely many relevant decompositions at each stage. The result is that $x\in X$7 becomes a Kleene algebra, and the same pattern extends to Conway semirings (Cranch et al., 29 Aug 2025).

A related finitary construction appears in the graded-relational-monoid setting. If $x\in X$8 is graded and finitely decomposable, and $x\in X$9 is a Kleene algebra, then $f(y)\cdot g(z)$0 admits a recursively defined star with

$f(y)\cdot g(z)$1

and for $f(y)\cdot g(z)$2,

$f(y)\cdot g(z)$3

This is the star construction used to obtain interchange Kleene algebras and concurrent Kleene algebras from graded relational interchange monoids (Cranch et al., 2020).

Complete or continuous settings use a different mechanism. In a $f(y)\cdot g(z)$4-continuous Kleene algebra, $f(y)\cdot g(z)$5, and multiplication preserves these power-suprema in both arguments. A $f(y)\cdot g(z)$6-continuous Kleene $f(y)\cdot g(z)$7-algebra further equips a semimodule with an infinite product $f(y)\cdot g(z)$8 satisfying axioms Ax1–Ax4, from which an $f(y)\cdot g(z)$9-operation $x$0 is derived. This route is canonical for complete LLMs, function semirings, and Büchi-style infinite behavior, but it relies on continuity or $x$1-continuity rather than Möbius finiteness (Ésik et al., 2015).

The contrast between these constructions is structurally significant. Möbius and graded approaches provide finitary recursive stars, whereas quantalic and $x$2-continuous approaches define star through joins of powers or least fixpoints. This distinction separates finitary regular-expression-style models from models that depend on arbitrary suprema (Cranch et al., 29 Aug 2025).

3. Domain, tests, and modal structure

Convolution Kleene algebras are closely related to domain-sensitive variants of Kleene algebra. In the relational setting of McLean, a binary relation algebra in signature $x$3 augments relational composition, union, and reflexive transitive closure with a domain operation

$x$4

which acts as a test marking states from which $x$5 has an outgoing transition. McLean identifies the free $x$6-algebra on generators with reduced pointed labelled finite rooted trees, and the free $x$7-algebra with regular subsets of those trees. Under relational semantics, the equational validities of Kleene algebras with domain form a decidable set, and for the fragment $x$8 the axioms of domain semirings provide a finite quasiequational axiomatization (McLean, 2019).

These results are relevant to convolution because the free models are not merely word languages but structured sets of trees equipped with composition, union, star, and domain. Composition on reduced trees is defined by gluing the point of one tree to the root of another and then reducing; on regular sets it is lifted pointwise and maximalized. This yields a tree-language semantics in which domain behaves as a unary test operation interacting algebraically with composition and union, a pattern directly aligned with modal and test-based convolution constructions (McLean, 2019).

The modal version of this pattern appears in both modal semirings and concurrent dynamic algebra. In the modal setting, domain and codomain operators on convolution algebras over catoids are defined by finite sums over source and target identities when the catoid has finite valency. If $x$9 is a modal semiring and $y$0 is a finitely 2-decomposable local catoid of finite valency, then $y$1 becomes a modal semiring; if $y$2 is Möbius and $y$3 a modal Kleene algebra, then $y$4 becomes a modal Kleene algebra (Cranch et al., 29 Aug 2025).

In the multirelational semantics of concurrent dynamic algebra, domain and antidomain are primitive and induce modal operators by

$y$5

That setting is not a standard convolution Kleene algebra, because sequential composition of multirelations is not globally associative, yet it provides a closely related design pattern: a weak semiring or trioid with tests, modalities, and a star as least fixpoint. The paper reconstructs Peleg’s concurrent dynamic logic in this framework and formalizes the main algebraic results in Isabelle/HOL (Furusawa et al., 2014).

4. Concurrency, interchange, and nonclassical composition

The concurrent branch of the theory begins with two ternary relations on the same carrier $y$6: one for sequential composition and one for parallel composition. Convolution then produces two products on $y$7, and relational interchange laws on $y$8 lift to algebraic interchange laws on the function space. The central inequality is the interchange law

$y$9

whose relational counterpart is encoded as RI7. If $z$0 is a relational interchange monoid and $z$1 an interchange quantale, then $z$2 is an interchange quantale; under the commutative-parallel and shared-unit assumptions, this specializes to a concurrent quantale. With grading and finite decomposability, the same lifting yields interchange Kleene algebras and concurrent Kleene algebras (Cranch et al., 2020).

Concrete domains include $z$3-weighted words with concatenation and shuffle, digraphs with series and parallel composition, posets, isomorphism classes of finite digraphs, and pomsets. In each case the codomain algebra provides the weights, the domain structure provides the decompositions, and convolution yields the sequential and concurrent composition of weighted predicates or weighted languages. The Boolean case again reduces to powerset semantics (Cranch et al., 2020).

The 2025 generalization extends this from 2-dimensional interchange to higher dimensions. A 2-catoid carries two catoid structures subject to interchange and globularity axioms; an $z$4-catoid carries one such structure in each dimension. Correspondingly, interchange semirings become $z$5-semirings, and interchange Kleene algebras become $z$6-Kleene algebras. If $z$7 is a local Möbius $z$8-catoid of finite valency and $z$9 an $K^X$0-Kleene algebra, then $K^X$1 is again an $K^X$2-Kleene algebra. The intended applications include concurrent convolution Kleene algebras on strict higher categories and higher relational monoids, as well as algebraic reasoning in higher-dimensional rewriting (Cranch et al., 29 Aug 2025).

A different, but closely related, account of concurrency is provided by multirelations $K^X$3. There the carrier admits both sequential composition and concurrent composition,

$K^X$4

while sequential composition is defined by quantifying over intermediate output sets and functions $K^X$5. The resulting algebras validate algebraic variants of Peleg’s axioms, but sequential composition is not associative in general and only satisfies restricted associativity when one argument is a domain or antidomain element. This demonstrates that convolution-flavored program algebras need not inherit the full semiring laws globally, even though they still support modalities and iteration (Furusawa et al., 2014).

5. Free models, continuous semantics, and representative examples

Several strands of the theory identify canonical free or free-like models. For ordinary $K^X$6-behavior, the free continuous Kleene $K^X$7-algebra on an alphabet $K^X$8 is $K^X$9, with union as addition, concatenation as multiplication, star as Kleene star, and infinite product

$K$0

For finitary $K$1-continuous Kleene $K$2-algebras, the free objects are $K$3 and $K$4, where $K$5 is the regular-language algebra and $K$6 consists of finite unions of infinite products of finitely many regular languages (Ésik et al., 2015).

McLean’s tree semantics provides a different free construction, centered on domain. In the reduced signature $K$7, generators produce reduced pointed labelled finite rooted trees; in the full signature $K$8, elements become regular subsets of these reduced trees. This gives a free relational Kleene algebra with domain whose elements behave like regular tree languages rather than word languages, with tree gluing as composition and point relocation as domain (McLean, 2019).

The path-oriented and categorical instances are equally central. For a finite graph $K$9, the path category $X$00 yields a convolution Kleene algebra $X$01; for interval categories over locally finite posets, one obtains incidence Kleene algebras in which convolution models the chop operator on intervals; for guarded-string categories, one obtains convolution Kleene algebras with tests. The 2025 paper also identifies weighted path algebras, weighted interval temporal logics with chop-star, and modal convolution Kleene algebras as direct instances of the general construction (Cranch et al., 29 Aug 2025).

Function semirings furnish a continuous, non-language-based class of examples. In the $X$02-continuous $X$03-setting, finitely additive, locally finite, $X$04-continuous functions on complete lattices form $X$05-continuous Kleene algebras under pointwise supremum and composition. For energy problems, the set $X$06 of energy functions on $X$07 is a $X$08-continuous Kleene algebra, and with an appropriate semimodule $X$09 of $X$10-continuous predicates into $X$11, the pair $X$12 is a $X$13-continuous Kleene $X$14-algebra and a symmetric bi-inductive semiring–semimodule pair. This semantics is used to analyze reachability and Büchi acceptance in energy automata (Ésik et al., 2015).

6. Decidability, limitations, and open directions

The theory is accompanied by strong positive results, but also by precise limits. On the positive side, McLean proves that under relational semantics the equational theory of Kleene algebras with domain is decidable. The proof builds condition automata for terms, constructs automata for differences, translates them back to terms using antidomain, and then reduces satisfiability to propositional dynamic logic without atomic propositions. Since PDL satisfiability is EXPTIME-complete, this yields a decision procedure for equational validity over relations, with an upper bound described as roughly $X$15EXPTIME through the construction (McLean, 2019).

The limitations are equally instructive. McLean’s regular sets of trees are closed under union and intersection but not under complement, and the paper leaves open whether they are closed under Heyting implication or the residuals of composition. These questions point toward residuated and Heyting-style enrichments of convolution tree semantics, but no closure theorem is established there (McLean, 2019).

In the concurrent dynamic algebra setting, several standard algebraic expectations fail. Sequential composition of multirelations is not associative in general, left distributivity over addition fails globally, diamonds are not additive over unions of tests, boxes are not multiplicative over meets of tests, and variants of Segerberg’s axiom are refuted in the multirelational model. The star remains available as a least fixpoint, but the ambient algebra is weaker than a classical Kleene algebra (Furusawa et al., 2014).

The Möbius construction also has a sharp boundary. Pair groupoids and certain infinite path categories are not Möbius, so the recursive star on convolution algebras does not apply there; for weighted relations and matrices, one must instead use Conway’s block-matrix star rather than the length-recursive Möbius star. This explains why convolution Kleene algebras and convolution quantales are complementary rather than interchangeable: quantales admit arbitrary joins and therefore a star defined by $X$16, while finitary Kleene-algebraic constructions require finite decomposability and a well-founded notion of length (Cranch et al., 29 Aug 2025).

The current research directions therefore follow two axes. One axis extends the algebraic envelope: modal convolution Kleene algebras, concurrent convolution Kleene algebras, and higher convolution Kleene algebras on strict higher categories and higher relational monoids. The other axis concerns metatheory and applications: quantitative Hoare logics, predicate transformer algebras, weighted interval temporal logics with chop-star, verification of weighted and probabilistic sequential and concurrent programs, and the completeness, decidability, and complexity of equational theories relative to convolution quantales and other complete models (Cranch et al., 29 Aug 2025).