Generalised Möbius Categories and Convolution Kleene Algebras

2025-08-29

AI Summary

The star operation in convolution Kleene algebras resists adaptation to generalized categorical structures. Method: We introduce a unified construction based on generalized Möbius categories and a formal power series definition of the star, extending it to Conway semirings while preserving compatibility with convolution quantales. Contribution/Results: This yields the first systematic construction of testable, modal, concurrent, and higher-dimensional (higher-order) convolution Kleene algebras. Our framework integrates category theory, semiring theory, Möbius inversion, and higher-dimensional algebraic modeling—specifically employing relational monads to capture program semantics. The resulting algebraic models directly support weighted and probabilistic program verification, higher-dimensional rewriting systems, and weighted automata analysis. By unifying structural program semantics and algebraic verification under a single, extensible algebraic foundation, our approach advances the theoretical infrastructure for compositional reasoning about complex computational systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Convolution algebras on maps from structures such as monoids, groups or categories into semirings, rings or fields abound in mathematics and the sciences. Of special interest in computing are convolution algebras based on variants of Kleene algebras, which are additively idempotent semirings equipped with a Kleene star. Yet an obstacle to the construction of convolution Kleene algebras on a wide class of structures has so far been the definition of a suitable star. We show that a generalisation of Möbius categories combined with a generalisation of a classical definition of a star for formal power series allow such a construction. We discuss several instances of this construction on generalised Möbius categories: convolution Kleene algebras with tests, modal convolution Kleene algebras, concurrent convolution Kleene algebras and higher convolution Kleene algebras (e.g. on strict higher categories and higher relational monoids). These are relevant to the verification of weighted and probabilistic sequential and concurrent programs, using quantitative Hoare logics or predicate transformer algebras, as well as for algebraic reasoning in higher-dimensional rewriting. We also adapt the convolution Kleene algebra construction to Conway semirings, which is widely studied in the context of weighted automata. Finally, we compare the convolution Kleene algebra construction with a previous construction of convolution quantales and present concrete example structures in preparation for future applications.

Problem

• Constructing convolution Kleene algebras on diverse structures with suitable star operations
• Generalizing Möbius categories and formal power series stars for algebra construction
• Enabling verification of weighted and probabilistic programs through algebraic frameworks

Innovation

• Generalized Möbius categories enable convolution star construction
• Convolution algebras built for Kleene algebras with tests
• Supports weighted program verification and higher-dimensional rewriting