Algebraic Investigation of Connected Components
This paper characterizes connected components of both directed and undirected graphs as atomic fixpoints. As algebraic structure for our investigations we combine complete Boolean algebras with the well-known theory of Kleene Algebra with domain. Using diamond operators as an algebraic generalization of relational image and preimage we show how connected components can be modeled as atomic fixpoints of functions operating on tests and prove some advanced theorems concerning connected components.
KeywordsDirected Acyclic Graph Undirected Graph Relation Algebra Diamond Operator Complete Boolean Algebra
The author is grateful to Bernhard Möller and the anonymous reviewers for thorough proofreading and valuable hints and remarks which helped to improve the paper.
- [Coq]The Coq proof assistant. https://coq.inria.fr/
- [DM11]Dang, H.-H., Möller, B.: Simplifying pointer Kleene algebra. In: Höfner, P., McIver, A., Struth, G. (eds.) Proceedings of 1st Workshop on Automated Theory Engineering, CEUR Workshop Proceedings, Wrocław, vol. 760, pp. 20–29. CEUR-WS.org (2011)Google Scholar
- [Glüa]Glück, R.: Atomic lattices. http://www.rolandglueck.de/Downloads/Atomiclattices.pdf
- [Glüb]Glück, R.: Tarksi rule vs. all-or-nothing property. http://www.rolandglueck.de/Downloads/Tarski_all_or_nothing.in
- [Isa]Isabelle. https://isabelle.in.tum.de/
- [Kah14]Kahl, W.: Graph transformation with symbolic attributes via monadic coalgebra homomorphisms. ECEASST 71 (2014)Google Scholar
- [KL95]King, D.J., Launchbury, J.: Structuring depth-first search algorithms in Haskell. In: Cytron, R.K., Lee, P. (eds.) Conference Record of POPL 1995, pp. 344–354. ACM Press (1995)Google Scholar
- [McC]McCune, W.: Prover9 and Mace4. https://www.cs.unm.edu/mccune/mace4/