Graded modal logic and counting message passing automata (2401.06519v2)
Abstract: We examine the relationship of graded (multi)modal logic to counting (multichannel) message passing automata with applications to the Weisfeiler-Leman algorithm. We introduce the notion of graded multimodal types, which are formulae of graded multimodal logic that encode the local information of a pointed Kripke-model. We also introduce message passing automata that carry out a generalization of the Weisfeiler-Leman algorithm for distinguishing non-isomorphic graph nodes. We show that the classes of pointed Kripke-models recognizable by these automata are definable by a countable (possibly infinite) disjunction of graded multimodal formulae and vice versa. In particular, this equivalence also holds between recursively enumerable disjunctions and recursively enumerable automata. We also show a way of carrying out the Weisfeiler-Leman algorithm with a formula of first order logic that has been augmented with H\"artig's quantifier and greatest fixed points.
- Descriptive complexity for distributed computing with circuits. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, August 28 to September 1, 2023, Bordeaux, France, volume 272 of LIPIcs, pages 9:1–9:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.9, doi:10.4230/LIPICS.MFCS.2023.9.
- Descriptive complexity for neural networks via boolean networks. CoRR, abs/2308.06277, 2023. URL: https://doi.org/10.48550/arXiv.2308.06277, arXiv:2308.06277, doi:10.48550/ARXIV.2308.06277.
- The logical expressiveness of graph neural networks. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020.
- An optimal lower bound on the number of variables for graph identification. Combinatorica, 12(4):389–410, 1992.
- Martin Grohe. The descriptive complexity of graph neural networks, 2023. arXiv:2303.04613.
- Weak models of distributed computing, with connections to modal logic. In Darek Kowalski and Alessandro Panconesi, editors, ACM Symposium on Principles of Distributed Computing, PODC ’12, pages 185–194. ACM, 2012.
- Weak models of distributed computing, with connections to modal logic. Distributed Comput., 28(1):31–53, 2015.
- Antti Kuusisto. Modal Logic and Distributed Message Passing Automata. In Computer Science Logic 2013 (CSL 2013), volume 23 of Leibniz International Proceedings in Informatics (LIPIcs), pages 452–468, 2013.
- Emptiness problems for distributed automata. Inf. Comput., 272:1–17, 2020. doi:10.1016/j.ic.2019.104503.
- Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI conference on artificial intelligence, number 01 in 33, pages 4602–4609, 2019.
- Fabian Reiter. Asynchronous distributed automata: A characterization of the modal mu-fragment. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 100:1–100:14, 2017.
- The reduction of a graph to canonical form and the algebra which appears therein. nti, Series, 2(9):12–16, 1968.
- How powerful are graph neural networks? In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL: https://openreview.net/forum?id=ryGs6iA5Km.