Convolution and concurrency

Abstract

We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Catoids and modal convolution algebras

“…Remark 4.2. If X is merely an st-multimagma, then PX forms a prequantale instead of a quantale [12,5]. This weaker result is needed in Section 6.…”

“…Remark 4.6. If X is an st-multimagma and Q a prequantale, then the convolution algebra Q X is a prequantale [12,5]. This result is needed in Section 7.…”

“…Other relational or partial monoids and convolution algebras are discussed in [5,11,12]. For examples based on categories and non-local catoids see Section 9.…”

“…Such correspondence triangles between X, Q and Q X have already been studied, and adapted to cases where Q is merely a semiring [5]. For Q = 2, they are known from substructural logics.…”

“…For Q = 2, they are known from substructural logics. They have also been extended to correspondences between concurrent relational monoids and concurrent semirings and quantales [5], where two quantalic compositions interact via a weak interchange law akin to that of higher categories [27].…”