Modal Convolution Kleene Algebras

• Modal convolution Kleene algebras are function spaces over structured categories or relational frames that lift both convolution and modal operators while ensuring controlled finite decomposability.
• They employ a Möbius-star recursive construction and relational ternary compositions to model sequential and concurrent operations, which supports robust program verification and weighted semantics.
• This framework generalizes classical convolution quantales and Kleene algebras by preserving induction principles and integrating modalities like Box and Diamond, impacting formal language and concurrency theories.

Searching arXiv for the specified papers and related work on convolution/modal Kleene algebras. Modal convolution Kleene algebras are convolution algebras of function spaces such as $K^C=\{f:C\to K\}$ or $Q^X=\{f:X\to Q\}$, where the underlying structure carries compositional data and the codomain is a Kleene algebra, a modal Kleene algebra, or a concurrent quantale. Their defining feature is that algebraic composition is lifted pointwise by convolution over factorizations in a category, catoid, or ternary relational frame, while modal operators are inherited from source–target maps or from binary relations extracted from the underlying ternary relations. In the setting of generalised Möbius categories, the key technical advance is that a recursive Möbius-style star construction yields bona fide convolution Kleene algebras, and in the modal case this produces $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$ satisfying the modal Kleene algebra axioms (Cranch et al., 29 Aug 2025). In the broader relational setting, the same perspective extends to concurrent composition, powerset liftings, shuffle languages, pomsets, and weighted graph models (Cranch et al., 2020).

1. Algebraic setting and basic construction

The starting point is a small category, or more generally a Möbius catoid, together with a semiring $S$. One forms the function space

$S^C=\{\,f:C\to S\,\},$

equipped with pointwise addition

$(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$

convolution

$$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$

and convolution unit

$id₀(x)=[x\text{ is an identity in }C].$

This makes $(S^C,+,*,0,id₀)$ into a semiring provided each $x\in C$ has finitely many factorizations $Q^X=\{f:X\to Q\}$0, a condition stated as finite $Q^X=\{f:X\to Q\}$1-decomposability (Cranch et al., 29 Aug 2025).

The categorical finiteness conditions are sharpened by the notion of a Möbius category, or Möbius catoid. In this formulation, arrows admit finite lengths $Q^X=\{f:X\to Q\}$2 and only finitely many decompositions of each length. Definition 3.1 states that a catoid $Q^X=\{f:X\to Q\}$3 is Möbius if every $Q^X=\{f:X\to Q\}$4 has a finite maximal decomposition-length $Q^X=\{f:X\to Q\}$5 and only finitely many ways to factor $Q^X=\{f:X\to Q\}$6 into $Q^X=\{f:X\to Q\}$7 at each step (Cranch et al., 29 Aug 2025). This finiteness is what permits the recursive definition of a star.

A parallel but more general formulation uses a relational bi-magma $Q^X=\{f:X\to Q\}$8, where $Q^X=\{f:X\to Q\}$9 are ternary relations encoding two forms of composition. Given a bi-prequantale $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$0, the function space $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$1 carries pointwise joins and two relational convolutions,

$(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$2

$(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$3

Under relational associativity and units, these convolutions lift the algebraic structure of $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$4 to $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$5 (Cranch et al., 2020).

These two presentations are closely aligned. The category/catoid formulation emphasizes Möbius finiteness and recursive star; the relational formulation emphasizes relational correspondence, multiple compositions, and modal operators. This suggests that modal convolution Kleene algebras are best understood not as a single isolated variety, but as a common lifting pattern from structured relational or categorical data into function spaces.

2. Möbius-star recursion and the Kleene algebra structure

The central obstacle in constructing convolution Kleene algebras on a wide class of structures is the definition of a suitable star. In the generalised Möbius-category setting, this is resolved by combining a generalisation of Möbius categories with a generalisation of a classical definition of a star for formal power series (Cranch et al., 29 Aug 2025).

If $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$6 is a Kleene algebra, meaning an additively idempotent semiring with a star $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$7 satisfying unfold and induction axioms, then for $(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$8 the star is defined recursively by

$(K^C,+,*,0,id₀,(–)^*,\Box,\Diamond)$9

for each identity $S$0, and

$S$1

where $S$2 denotes the source identity of $S$3 (Cranch et al., 29 Aug 2025).

By induction on $S$4, one obtains the star-unfold equations

$S$5

and both star-induction axioms. Lemma 3.4 gives the pointwise form

$S$6

while Lemma 3.5 states that if $S$7 then $S$8, and dually $S$9 (Cranch et al., 29 Aug 2025).

The resulting structural theorem is explicit. Theorem 3.2 states: let $S^C=\{\,f:C\to S\,\},$0 be a Möbius catoid and $S^C=\{\,f:C\to S\,\},$1 a Kleene algebra. Define convolution $S^C=\{\,f:C\to S\,\},$2 and star $S^C=\{\,f:C\to S\,\},$3 by the recursive clauses above. Then $S^C=\{\,f:C\to S\,\},$4 is a Kleene algebra (Cranch et al., 29 Aug 2025).

In the relational interchange setting, an analogous star construction is available when $S^C=\{\,f:C\to S\,\},$5 is graded and finitely decomposable. Theorem 2.4 defines stars recursively over the grade: $S^C=\{\,f:C\to S\,\},$6 and similarly for $S^C=\{\,f:C\to S\,\},$7. Under these hypotheses, $S^C=\{\,f:C\to S\,\},$8 becomes a concurrent Kleene algebra (Cranch et al., 2020).

A plausible implication is that the Möbius-star construction and the graded relational-star construction capture the same general phenomenon from two technical directions: one controlled by finite decomposition-length in catoids, the other by finite decomposability plus grading in relational frames.

3. Modal operators and modal Kleene algebra laws

A modal Kleene algebra is a Kleene algebra $S^C=\{\,f:C\to S\,\},$9 equipped with two operators

$(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$0

satisfying, for all $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$1,

$(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$2

$(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$3

together with the dual axioms for $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$4 by interchanging $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$5-arguments and $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$6, as well as the idempotence laws $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$7 and $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$8 (Cranch et al., 29 Aug 2025).

In the convolution setting, assume $(f+g)(x)=f(x)+g(x),\qquad 0(x)=0_S,$9 is a local Möbius catoid and $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$0 a modal Kleene algebra. For $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$1, the modal operators are defined by

$$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$2

$$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$3

Equivalently, in terms of source and target maps,

$$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$4

Theorem 3.3 states that if $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$5 is a modal Kleene algebra and $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$6 is local, then

$$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$7

is again a modal Kleene algebra (Cranch et al., 29 Aug 2025).

The paper specifies what is preserved under lifting: distribution of $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$8 and $$(f*g)(x)=\sum_{y,z\in C:\;y\circ z=x} f(y)\cdot g(z),$$9 over $id₀(x)=[x\text{ is an identity in }C].$0, compatibility with convolution through Box-Kleene and Diamond-Kleene interaction, the unfold and induction laws for $id₀(x)=[x\text{ is an identity in }C].$1, and domain/codomain closure laws (Cranch et al., 29 Aug 2025). This is the sense in which the modal structure is not appended externally, but induced from the source–target structure of the base catoid.

In the broader relational account, modalities arise by currying ternary relations into binary relations. For a binary relation $id₀(x)=[x\text{ is an identity in }C].$2, one has

$id₀(x)=[x\text{ is an identity in }C].$3

Applied to the ternary compositions, this yields diamond and box operators that “look left”, “look right”, or “look in the interior” of sequential or parallel composition (Cranch et al., 2020).

A recurrent point of comparison concerns whether modal structure must be introduced by an ad hoc domain operator. The comparison section of the Möbius-category work states the opposite for this construction: modal structure in $id₀(x)=[x\text{ is an identity in }C].$4 is induced by the source–target maps of $id₀(x)=[x\text{ is an identity in }C].$5, not by an ad hoc domain operator (Cranch et al., 29 Aug 2025).

4. Relational correspondence, concurrency, and higher-dimensional variants

The relational theory emphasizes a correspondence between properties of the underlying frame and laws of the lifted algebra. In the bi-magma setting, the underlying data are two ternary relations together with seven relational interchange laws $id₀(x)=[x\text{ is an identity in }C].$6–$id₀(x)=[x\text{ is an identity in }C].$7. Proposition 3.1 states that for each of the seven small interchange laws $id₀(x)=[x\text{ is an identity in }C].$8,

$id₀(x)=[x\text{ is an identity in }C].$9

Under mild non-degeneracy conditions and finiteness assumptions, the implication reverses, so interchange in $(S^C,+,*,0,id₀)$0 holds exactly when it holds in $(S^C,+,*,0,id₀)$1 and in $(S^C,+,*,0,id₀)$2 (Cranch et al., 2020).

This correspondence is made concrete by delta functions. Lemma 6.1 defines the point-mass $(S^C,+,*,0,id₀)$3 by

$(S^C,+,*,0,id₀)$4

Then

$(S^C,+,*,0,id₀)$5

so convolutions of deltas exactly shadow the ternary relational trees in the quantale $(S^C,+,*,0,id₀)$6. Proposition 6.2 shows that, under no-zero-divisor and non-empty-fiber assumptions,

$(S^C,+,*,0,id₀)$7

Proposition 6.3 then characterizes interchange through the interaction of relational and algebraic interchange laws (Cranch et al., 2020).

Concurrency enters by allowing two distinct lifted compositions, usually interpreted as sequential and parallel composition. The 2020 exposition states that concurrent quantales and concurrent Kleene algebras arise as convolution algebras $(S^C,+,*,0,id₀)$8 of functions from structures $(S^C,+,*,0,id₀)$9 with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras $x\in C$0 (Cranch et al., 2020). Its main examples include weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

The generalised Möbius-category work extends this line further by discussing concurrent convolution Kleene algebras and higher convolution Kleene algebras, including those on strict higher categories and higher relational monoids (Cranch et al., 29 Aug 2025). It also states that the $x\in C$1-dimensional convolution of a $x\in C$2-catoid $x\in C$3 yields a concurrent Kleene algebra $x\in C$4 supporting both interleaving (horizontal) and parallel (vertical) composition, with a two-star $x\in C$5 and $x\in C$6 for repeated interleaving or parallel repetition (Cranch et al., 29 Aug 2025).

This suggests that modal convolution Kleene algebras occupy a boundary zone between ordinary LLMs, algebraic concurrency, and higher-dimensional rewriting. The unifying mechanism is always convolution; what varies is the compositional arity and geometry present in the underlying frame.

5. Worked semantics and canonical examples

A worked example in the Möbius-category account uses weighted path semantics. Let $x\in C$7 be a finite directed graph with weights in the tropical semiring

$x\in C$8

Its path-category $x\in C$9 has objects $Q^X=\{f:X\to Q\}$00 and arrows all finite paths. Then $Q^X=\{f:X\to Q\}$01 is a convolution dioid; restricting to finitely supported $Q^X=\{f:X\to Q\}$02 yields a dioid with $Q^X=\{f:X\to Q\}$03. If $Q^X=\{f:X\to Q\}$04 is equipped with the trivial Kleene star $Q^X=\{f:X\to Q\}$05 for all $Q^X=\{f:X\to Q\}$06, then $Q^X=\{f:X\to Q\}$07 is a convolution Kleene algebra with

$Q^X=\{f:X\to Q\}$08

$Q^X=\{f:X\to Q\}$09

$Q^X=\{f:X\to Q\}$10

For $Q^X=\{f:X\to Q\}$11, $Q^X=\{f:X\to Q\}$12, $Q^X=\{f:X\to Q\}$13, and $Q^X=\{f:X\to Q\}$14, one obtains

$Q^X=\{f:X\to Q\}$15

$Q^X=\{f:X\to Q\}$16

and since $Q^X=\{f:X\to Q\}$17, $Q^X=\{f:X\to Q\}$18, one gets $Q^X=\{f:X\to Q\}$19; also

$Q^X=\{f:X\to Q\}$20

(Cranch et al., 29 Aug 2025).

The broader relational theory develops several canonical examples. For $Q^X=\{f:X\to Q\}$21-weighted words, one takes $Q^X=\{f:X\to Q\}$22 with

$Q^X=\{f:X\to Q\}$23

Then $Q^X=\{f:X\to Q\}$24 has sequential and parallel convolutions

$Q^X=\{f:X\to Q\}$25

recovering the complex algebra of weighted shuffle languages (Cranch et al., 2020).

For digraphs, one considers a class $Q^X=\{f:X\to Q\}$26 of finite directed graphs closed under serial composition $Q^X=\{f:X\to Q\}$27 and parallel composition $Q^X=\{f:X\to Q\}$28, together with the subsumption preorder $Q^X=\{f:X\to Q\}$29 iff there is a vertex-bijective graph morphism $Q^X=\{f:X\to Q\}$30. Then the antitone functions $Q^X=\{f:X\to Q\}$31 form a unital concurrent quantale under

$Q^X=\{f:X\to Q\}$32

(Cranch et al., 2020).

For pomsets and labeled partial orders, the same pattern uses series composition for $Q^X=\{f:X\to Q\}$33, parallel disjoint union for $Q^X=\{f:X\to Q\}$34, and subsumption via order-embeddings for $Q^X=\{f:X\to Q\}$35. The resulting convolution algebra $Q^X=\{f:X\to Q\}$36 recovers the classical true-concurrency concurrent Kleene algebra of pomset-languages (Cranch et al., 2020).

In the Boolean case $Q^X=\{f:X\to Q\}$37, convolution reduces to relational product: $Q^X=\{f:X\to Q\}$38 Hence $Q^X=\{f:X\to Q\}$39, and one recovers exactly the usual powerset lifts (Cranch et al., 2020). This gives the Boolean edge case in which modal convolution Kleene algebras become complex algebras of relational frames.

6. Comparisons, applications, and interpretive issues

The generalised Möbius-category account explicitly compares its construction with two earlier lines. First, Rosenthal’s convolution quantales allow an arbitrary complete-lattice $Q^X=\{f:X\to Q\}$40 and arbitrary catoid $Q^X=\{f:X\to Q\}$41, with convolution using $Q^X=\{f:X\to Q\}$42 and arbitrary joins. This yields $Q^X=\{f:X\to Q\}$43-ary sups and a star $Q^X=\{f:X\to Q\}$44, but loses finitary induction and in general does not land in a Kleene algebra (Cranch et al., 29 Aug 2025). Second, Kozen’s convolution Kleene algebras with tests arise as a special case $Q^X=\{f:X\to Q\}$45, where $Q^X=\{f:X\to Q\}$46 are tests and $Q^X=\{f:X\to Q\}$47 actions. The Möbius-star construction generalises this to arbitrary many-object categories $Q^X=\{f:X\to Q\}$48, embedding $Q^X=\{f:X\to Q\}$49-valued indicator functions as tests (Cranch et al., 29 Aug 2025).

These comparisons mark an important conceptual distinction. Quantales do not require Möbius conditions, but support only sup-based star. By contrast, the Möbius-star construction is designed precisely to recover the finitary unfold and induction behavior characteristic of Kleene algebra (Cranch et al., 29 Aug 2025). A common misunderstanding would therefore be to treat convolution quantales and convolution Kleene algebras as interchangeable. The comparison section makes clear that they differ exactly on the role of star and induction.

The applications identified in the source material are primarily semantic and verification-theoretic. In program verification, programs are interpreted as elements $Q^X=\{f:X\to Q\}$50, assertions as tests $Q^X=\{f:X\to Q\}$51, sequential composition $Q^X=\{f:X\to Q\}$52 represents next-program, and choice $Q^X=\{f:X\to Q\}$53 represents nondeterminism. Kleene star models while-loops as $Q^X=\{f:X\to Q\}$54, and the star-induction axiom corresponds to the Hoare-induction rule. The modal operators $Q^X=\{f:X\to Q\}$55 provide predicate transformers,

$Q^X=\{f:X\to Q\}$56

giving sound and complete axiomatizations of quantitative Hoare triples

$Q^X=\{f:X\to Q\}$57

(Cranch et al., 29 Aug 2025).

The same source identifies applications to the verification of weighted and probabilistic sequential and concurrent programs, using quantitative Hoare logics or predicate transformer algebras, and to algebraic reasoning in higher-dimensional rewriting (Cranch et al., 29 Aug 2025). The 2020 relational account similarly presents modal convolution Kleene algebras as natural models for reasoning about sequential and concurrent composition together with modalities, and lists classical languages over words, pomsets, weighted digraph or graph-type languages, incidence algebras in combinatorics, chop-modalities in interval logic, and separating conjunction in separation logic among the structures unified by the framework (Cranch et al., 2020).

A plausible implication is that the significance of modal convolution Kleene algebras lies less in a single signature than in a transferable construction principle. Whenever the underlying relational or categorical frame supports controlled factorization, convolution lifts composition, star lifts finite iteration, and modalities lift observational structure. Under that reading, the theory supplies a common algebraic interface for weighted semantics, concurrency, and modal reasoning.