ABSTRACT
We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon M-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.
► BibTeX data
► References
[1] Wassim Abou-Jaoudé, Denis Thieffry, and Jérôme Feret. Formal derivation of qualitative dynamical models from biochemical networks. Biosystems, 149: 70-112, 2016. https://doi.org/10.1016/j.biosystems.2016.09.001.
https://doi.org/https://doi.org/10.1016/j.biosystems.2016.09.001
[2] Jiří Adámek, Horst Herrlich, and George Strecker. Abstract and concrete categories, 1990.
[3] Jakob L. Andersen, Christoph Flamm, Daniel Merkle, and Peter F. Stadler. A Software Package for Chemically Inspired Graph Transformation. In R. Echahed and M. Minas, editors, Graph Transformation (ICGT 2016), volume 9761 of Lecture Notes in Computer Science,, pages 73-88, Cham, 2016. Springer International Publishing. https://doi.org/10.1007/978-3-319-40530-8_5.
https://doi.org/https://doi.org/10.1007/978-3-319-40530-8_5
[4] Jakob L. Andersen, Christoph Flamm, Daniel Merkle, and Peter F. Stadler. Chemical Transformation Motifs - Modelling Pathways as Integer Hyperflows. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 16 (2): 510-523, 2019. https://doi.org/10.1109/tcbb.2017.2781724.
https://doi.org/https://doi.org/10.1109/tcbb.2017.2781724
[5] Jakob Lykke Andersen, Rolf Fagerberg, Christoph Flamm, Rojin Kianian, Daniel Merkle, and Peter F Stadler. Towards mechanistic prediction of mass spectra using graph transformation. MATCH Commun. Math. Comput. Chem., 80: 705-731, 2018a.
[6] Jakob Lykke Andersen, Christoph Flamm, Daniel Merkle, and Peter F Stadler. Rule composition in graph transformation models of chemical reactions. MATCH Commun. Math. Comput. Chem., 80 (661-704): 45, 2018b.
[7] Wolfgang Banzhaf, Christoph Flamm, Daniel Merkle, and Peter F. Stadler. Algorithmic Cheminformatics (Dagstuhl Seminar 14452). Dagstuhl Reports, 4 (11): 22-39, 2015. https://doi.org/10.4230/DagRep.4.11.22.
https://doi.org/https://doi.org/10.4230/DagRep.4.11.22
[8] Nicolas Behr. Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework. In Rachid Echahed and Detlef Plump, editors, Proceedings of theTenth International Workshop on Graph Computation Models (GCM 2019) in Eindhoven, The Netherlands, volume 309 of Electronic Proceedings in Theoretical Computer Science, pages 23-52. Open Publishing Association, 2019. https://doi.org/10.4204/eptcs.309.2.
https://doi.org/https://doi.org/10.4204/eptcs.309.2
[9] Nicolas Behr. Tracelets and tracelet analysis of compositional rewriting systems. In John Baez and Bob Coecke, editors, \rm Proceedings Applied Category Theory 2019, \rm University of Oxford, UK, 15-19 July 2019, volume 323 of Electronic Proceedings in Theoretical Computer Science, pages 44-71. Open Publishing Association, 2020. https://doi.org/10.4204/EPTCS.323.4.
https://doi.org/https://doi.org/10.4204/EPTCS.323.4
[10] Nicolas Behr and Jean Krivine. Rewriting theory for the life sciences: A unifying framework for CTMC semantics. In Fabio Gadducci and Timo Kehrer, editors, Graph Transformation, 13th International Conference, ICGT 2020, Held as Part of STAF 2020, Bergen, Norway, June 25-26, 2020, Proceedings, volume 12150 of Theoretical Computer Science and General Issues. Springer International Publishing, 2020. https://doi.org/10.1007/978-3-030-51372-6.
https://doi.org/https://doi.org/10.1007/978-3-030-51372-6
[11] Nicolas Behr and Pawel Sobocinski. Rule Algebras for Adhesive Categories. In Dan Ghica and Achim Jung, editors, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), volume 119 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:21, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.CSL.2018.11.
https://doi.org/https://doi.org/10.4230/LIPIcs.CSL.2018.11
[12] Nicolas Behr and Pawel Sobocinski. Rule Algebras for Adhesive Categories (extended journal version). Logical Methods in Computer Science, Volume 16, Issue 3, July 2020. URL https://lmcs.episciences.org/6615.
https://lmcs.episciences.org/6615
[13] Nicolas Behr, Vincent Danos, and Ilias Garnier. Stochastic mechanics of graph rewriting. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16. ACM Press, 2016a. https://doi.org/10.1145/2933575.2934537.
https://doi.org/https://doi.org/10.1145/2933575.2934537
[14] Nicolas Behr, Vincent Danos, Ilias Garnier, and Tobias Heindel. The algebras of graph rewriting. arXiv preprint arXiv:1612.06240, 2016b.
arXiv:1612.06240
[15] Nicolas Behr, Vincent Danos, and Ilias Garnier. Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems. Logical Methods in Computer Science, Volume 16, Issue 3, July 2020a. URL https://lmcs.episciences.org/6628.
https://lmcs.episciences.org/6628
[16] Nicolas Behr, Reiko Heckel, and Maryam Ghaffari Saadat. Efficient Computation of Graph Overlaps for Rule Composition: Theory and Z3 Prototyping. In Berthold Hoffmann and Mark Minas, editors, \rm Proceedings of the Eleventh International Workshop on Graph Computation Models, \rm Online-Workshop, 24th June 2020, volume 330 of Electronic Proceedings in Theoretical Computer Science, pages 126-144. Open Publishing Association, 2020b. https://doi.org/10.4204/EPTCS.330.8.
https://doi.org/https://doi.org/10.4204/EPTCS.330.8
[17] Gil Benkö, Christoph Flamm, and Peter F. Stadler. A Graph-Based Toy Model of Chemistry. Journal of Chemical Information and Computer Sciences, 43 (4): 1085-1093, 2003. https://doi.org/10.1021/ci0200570.
https://doi.org/https://doi.org/10.1021/ci0200570
[18] M. L. Blinov, J. R. Faeder, B. Goldstein, and W. S. Hlavacek. BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains. Bioinformatics, 20 (17): 3289-3291, 2004. https://doi.org/10.1093/bioinformatics/bth378.
https://doi.org/https://doi.org/10.1093/bioinformatics/bth378
[19] Paul Boehm, Harald-Reto Fonio, and Annegret Habel. Amalgamation of graph transformations: A synchronization mechanism. Journal of Computer and System Sciences, 34 (2-3): 377-408, 1987. https://doi.org/10.1016/0022-0000(87)90030-4.
https://doi.org/https://doi.org/10.1016/0022-0000(87)90030-4
[20] Pierre Boutillier, Mutaamba Maasha, Xing Li, Héctor F Medina-Abarca, Jean Krivine, Jérôme Feret, Ioana Cristescu, Angus G Forbes, and Walter Fontana. The Kappa platform for rule-based modeling. Bioinformatics, 34 (13): i583-i592, 2018. https://doi.org/10.1093/bioinformatics/bty272.
https://doi.org/https://doi.org/10.1093/bioinformatics/bty272
[21] Benjamin Braatz and Christoph Brandt. Graph transformations for the resource description framework. Electronic Communications of the EASST, 10, 2008. https://doi.org/10.14279/tuj.eceasst.10.158.
https://doi.org/https://doi.org/10.14279/tuj.eceasst.10.158
[22] Benjamin Braatz, Hartmut Ehrig, Karsten Gabriel, and Ulrike Golas. Finitary $\mathcal{M}$ -adhesive categories. Mathematical Structures in Computer Science, 24 (4): 240403-240443, 2014. https://doi.org/10.1017/S0960129512000321.
https://doi.org/https://doi.org/10.1017/S0960129512000321
[23] Ferdinanda Camporesi, Jérôme Feret, and Kim Quyên Lý. KaDE: A Tool to Compile Kappa Rules into (Reduced) ODE Models. In Computational Methods in Systems Biology, pages 291-299. Springer International Publishing, 2017. https://doi.org/10.1007/978-3-319-67471-1_18.
https://doi.org/https://doi.org/10.1007/978-3-319-67471-1_18
[24] J.R.B. Cockett and Stephen Lack. Restriction categories II: partial map classification. Theoretical Computer Science, 294 (1-2): 61-102, 2003. https://doi.org/10.1016/s0304-3975(01)00245-6.
https://doi.org/https://doi.org/10.1016/s0304-3975(01)00245-6
[25] Andrea Corradini, Ugo Montanari, Francesca Rossi, Hartmut Ehrig, Reiko Heckel, and Michael Löwe. Algebraic Approaches to Graph Transformation - Part I: Basic Concepts and Double Pushout Approach. In Handbook of Graph Grammars and Computing by Graph Transformations, Volume 1: Foundations, pages 163-246, 1997.
[26] Andrea Corradini, Tobias Heindel, Frank Hermann, and Barbara König. Sesqui-Pushout Rewriting. In A. Corradini, H. Ehrig, U. Montanari, L. Ribeiro, and G. Rozenberg, editors, Graph Transformations (ICGT 2006), volume 4178 of Lecture Notes in Computer Science, pages 30-45. Springer Berlin Heidelberg, 2006. https://doi.org/10.1007/11841883_4.
https://doi.org/https://doi.org/10.1007/11841883_4
[27] Andrea Corradini, Dominique Duval, Rachid Echahed, Frederic Prost, and Leila Ribeiro. AGREE – Algebraic Graph Rewriting with Controlled Embedding. In F. Parisi-Presicce and B. Westfechtel, editors, Graph Transformation (ICGT 2015), volume 9151 of Lecture Notes in Computer Science, pages 35-51, Cham, 2015. Springer International Publishing. https://doi.org/10.1007/978-3-319-21145-9_3.
https://doi.org/https://doi.org/10.1007/978-3-319-21145-9_3
[28] Vincent Danos and Cosimo Laneve. Graphs for Core Molecular Biology. In C. Priami, editor, Computational Methods in Systems Biology (CMSB 2003), volume 2602 of Lecture Notes in Computer Science, pages 34-46. Springer Berlin Heidelberg, 2003a. https://doi.org/10.1007/3-540-36481-1_4.
https://doi.org/https://doi.org/10.1007/3-540-36481-1_4
[29] Vincent Danos and Cosimo Laneve. Core Formal Molecular Biology. In P. Degano, editor, Programming Languages and Systems (ESOP 2003), volume 2618 of Lecture Notes in Computer Science, pages 302-318. Springer Berlin Heidelberg, 2003b. https://doi.org/10.1007/3-540-36575-3_21.
https://doi.org/https://doi.org/10.1007/3-540-36575-3_21
[30] Vincent Danos and Cosimo Laneve. Formal molecular biology. Theoretical Computer Science, 325 (1): 69 - 110, 2004. https://doi.org/10.1016/j.tcs.2004.03.065.
https://doi.org/https://doi.org/10.1016/j.tcs.2004.03.065
[31] Vincent Danos, Jérôme Feret, Walter Fontana, Russell Harmer, and Jean Krivine. Rule-Based Modelling of Cellular Signalling. In L. Caires and V.T. Vasconcelos, editors, Concurrency Theory (CONCUR 2007), volume 4703 of Lecture Notes in Computer Science, pages 17-41. Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-74407-8_3.
https://doi.org/https://doi.org/10.1007/978-3-540-74407-8_3
[32] Vincent Danos, Jérôme Feret, Walter Fontana, Russell Harmer, and Jean Krivine. Rule-based modelling, symmetries, refinements. In Jasmin Fisher, editor, Formal Methods in Systems Biology (FMSB 2008), volume 5054 of Lecture Notes in Computer Science, pages 103-122. Springer Berlin Heidelberg, 2008. https://doi.org/10.1007/978-3-540-68413-8_8.
https://doi.org/https://doi.org/10.1007/978-3-540-68413-8_8
[33] Vincent Danos, Jerome Feret, Walter Fontana, Russell Harmer, Jonathan Hayman, Jean Krivine, Chris Thompson-Walsh, and Glynn Winskel. Graphs, Rewriting and Pathway Reconstruction for Rule-Based Models. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012), volume 18 of Leibniz International Proceedings in Informatics (LIPIcs), pages 276-288, Dagstuhl, Germany, 2012. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.FSTTCS.2012.276.
https://doi.org/https://doi.org/10.4230/LIPIcs.FSTTCS.2012.276
[34] Vincent Danos, Tobias Heindel, Ricardo Honorato-Zimmer, and Sandro Stucki. Reversible Sesqui-Pushout Rewriting. In Holger Giese and Barbara König, editors, Graph Transformation (ICGT 2014), volume 8571 of Lecture Notes in Computer Science, pages 161-176, Cham, 2014. Springer International Publishing. https://doi.org/10.1007/978-3-319-09108-2_11.
https://doi.org/https://doi.org/10.1007/978-3-319-09108-2_11
[35] H. Ehrig and A. Habel. Graph Grammars with Application Conditions. Springer Berlin Heidelberg, Berlin, Heidelberg, 1986. https://doi.org/10.1007/978-3-642-95486-3_7.
https://doi.org/https://doi.org/10.1007/978-3-642-95486-3_7
[36] H. Ehrig, M. Pfender, and H. J. Schneider. Graph-grammars: An algebraic approach. In 14th Annual Symposium on Switching and Automata Theory (SWAT 1973), pages 167-180, 1973. https://doi.org/10.1109/SWAT.1973.11.
https://doi.org/https://doi.org/10.1109/SWAT.1973.11
[37] H. Ehrig, K. Ehrig, U. Prange, and G. Taentzer. Fundamentals of algebraic graph transformation. Monographs in Theoretical Computer Science. An EATCS Series, 2006a. https://doi.org/10.1007/3-540-31188-2.
https://doi.org/https://doi.org/10.1007/3-540-31188-2
[38] Hartmut Ehrig, Annegret Habel, Hans-Jörg Kreowski, and Francesco Parisi-Presicce. Parallelism and concurrency in high-level replacement systems. Mathematical Structures in Computer Science, 1 (03): 361, 1991. https://doi.org/10.1017/s0960129500001353.
https://doi.org/https://doi.org/10.1017/s0960129500001353
[39] Hartmut Ehrig, Julia Padberg, Ulrike Prange, and Annegret Habel. Adhesive High-Level Replacement Systems: A New Categorical Framework for Graph Transformation. Fundamenta Informaticae, 74 (1): 1-29, 2006b. https://doi.org/10.5555/1231199.1231201.
https://doi.org/https://doi.org/10.5555/1231199.1231201
[40] Hartmut Ehrig, Reiko Heckel, Grzegorz Rozenberg, and Gabriele Taentzer, editors. Graph Transformations (ICGT 2008), volume 5214 of Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2008. https://doi.org/10.1007/978-3-540-87405-8.
https://doi.org/https://doi.org/10.1007/978-3-540-87405-8
[41] Hartmut Ehrig, Ulrike Golas, Frank Hermann, et al. Categorical frameworks for graph transformation and HLR systems based on the DPO approach. Bulletin of the EATCS, (102): 111-121, 2010.
[42] Hartmut Ehrig, Ulrike Golas, Annegret Habel, Leen Lambers, and Fernando Orejas. $\mathcal{M}$-adhesive transformation systems with nested application conditions. Part 1: parallelism, concurrency and amalgamation. Mathematical Structures in Computer Science, 24 (04), 2014. https://doi.org/10.1017/s0960129512000357.
https://doi.org/https://doi.org/10.1017/s0960129512000357
[43] Rolf Fagerberg, Christoph Flamm, Rojin Kianian, Daniel Merkle, and Peter F. Stadler. Finding the K best synthesis plans. Journal of Cheminformatics, 10 (1), 2018. https://doi.org/10.1186/s13321-018-0273-z.
https://doi.org/https://doi.org/10.1186/s13321-018-0273-z
[44] Jerome Feret, Thomas Henzinger, Heinz Koeppl, and Tatjana Petrov. Lumpability abstractions of rule-based systems. Theoretical Computer Science, 431 (0): 137 - 164, 2012. https://doi.org/10.1016/j.tcs.2011.12.059.
https://doi.org/https://doi.org/10.1016/j.tcs.2011.12.059
[45] Ulrike Golas, Annegret Habel, and Hartmut Ehrig. Multi-amalgamation of rules with application conditions in $\mathcal{M}$-adhesive categories. Mathematical Structures in Computer Science, 24 (04), 2014. https://doi.org/10.1017/s0960129512000345.
https://doi.org/https://doi.org/10.1017/s0960129512000345
[46] Annegret Habel and Karl-Heinz Pennemann. Correctness of high-level transformation systems relative to nested conditions. Mathematical Structures in Computer Science, 19 (02): 245, 2009. https://doi.org/10.1017/s0960129508007202.
https://doi.org/https://doi.org/10.1017/s0960129508007202
[47] Annegret Habel and Detlef Plump. $\mathcal{M}, \mathcal{N}$ -Adhesive Transformation Systems. In H. Ehrig, G. Engels, H.J. Kreowski, and G. Rozenberg, editors, Graph Transformations (ICGT 2012), volume 7562 of Lecture Notes in Computer Science, pages 218-233. Springer Berlin Heidelberg, 2012a. https://doi.org/10.1007/978-3-642-33654-6_15.
https://doi.org/https://doi.org/10.1007/978-3-642-33654-6_15
[48] Annegret Habel and Detlef Plump. $\mathcal M, \mathcal N$ -Adhesive Transformation Systems. (Long Version), 2012b. URL http://formale-sprachen.informatik.uni-oldenburg.de/ skript/fs-pub/HaPl12b.pdf.
http://formale-sprachen.informatik.uni-oldenburg.de/~skript/fs-pub/HaPl12b.pdf
[49] Annegret Habel, Reiko Heckel, and Gabriele Taentzer. Graph grammars with negative application conditions. Fundam. Inf., 26 (3,4): 287-313, 1996. https://doi.org/10.3233/FI-1996-263404.
https://doi.org/https://doi.org/10.3233/FI-1996-263404
[50] Leonard A. Harris, Justin S. Hogg, José-Juan Tapia, John A. P. Sekar, Sanjana Gupta, Ilya Korsunsky, Arshi Arora, Dipak Barua, Robert P. Sheehan, and James R. Faeder. BioNetGen 2.2: advances in rule-based modeling. Bioinformatics, 32 (21): 3366-3368, 2016. https://doi.org/10.1093/bioinformatics/btw469.
https://doi.org/https://doi.org/10.1093/bioinformatics/btw469
[51] Frank Hermann, Andrea Corradini, and Hartmut Ehrig. Analysis of permutation equivalence in $\mathcal{M}$-adhesive transformation systems with negative application conditions. Mathematical Structures in Computer Science, 24 (4), 2014.
[52] Stephen Lack and Paweł Sobociński. Adhesive Categories. In Igor Walukiewicz, editor, Foundations of Software Science and Computation Structures (FoSSaCS 2004), volume 2987 of Lecture Notes in Computer Science, pages 273-288. Springer Berlin Heidelberg, 2004. https://doi.org/10.1007/978-3-540-24727-2_20.
https://doi.org/https://doi.org/10.1007/978-3-540-24727-2_20
[53] Stephen Lack and Paweł Sobociński. Adhesive and quasiadhesive categories. RAIRO - Theoretical Informatics and Applications, 39 (3): 511-545, 2005. https://doi.org/10.1051/ita:2005028.
https://doi.org/https://doi.org/10.1051/ita:2005028
[54] Michael Löwe. Algebraic Approach to Single-Pushout Graph Transformation. Theor. Comput. Sci., 109 (1-2): 181-224, March 1993. ISSN 0304-3975. https://doi.org/10.1016/0304-3975(93)90068-5.
https://doi.org/https://doi.org/10.1016/0304-3975(93)90068-5
[55] Michael Löwe. Polymorphic Sesqui-Pushout Graph Rewriting. In F. Parisi-Presicce and B. Westfechtel, editors, Graph Transformation, volume 9151, pages 3-18, Cham, 2015. Springer International Publishing. https://doi.org/10.1007/978-3-319-21145-9_1.
https://doi.org/https://doi.org/10.1007/978-3-319-21145-9_1
[56] Elaine Murphy, Vincent Danos, Jérôme Féret, Jean Krivine, and Russell Harmer. Rule-based modeling and model refinement. In Elements of Computational Systems Biology, pages 83-114. John Wiley & Sons, Inc., 2010. https://doi.org/10.1002/9780470556757.ch4.
https://doi.org/https://doi.org/10.1002/9780470556757.ch4
[57] Marisa Navarro Gomez, Fernando Orejas Valdés, Elvira Pino Blanco, and Leen Lambers. A logic of graph conditions extended with paths. In Actas de las XVI Jornadas de Programación y Lenguajes (PROLE 2016): Salamanca, septiembre de 2016, pages 1-15, 2016.
[58] James R. Norris. Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 1998.
[59] Julia Padberg. Towards M-Adhesive Categories based on Coalgebras and Comma Categories. arXiv:1702.04650, 2017.
arXiv:1702.04650
[60] Karl-Heinz Pennemann. Resolution-Like Theorem Proving for High-Level Conditions. In H. Ehrig, R. Heckel, G. Rozenberg, and G. Taentzer, editors, Graph Transformations (ICGT 2008), volume 5214 of Lecture Notes in Computer Science, pages 289-304. Springer Berlin Heidelberg, 2008. https://doi.org/10.1007/978-3-540-87405-8_20.
https://doi.org/https://doi.org/10.1007/978-3-540-87405-8_20
[61] Tatjana Petrov, Jerome Feret, and Heinz Koeppl. Reconstructing species-based dynamics from reduced stochastic rule-based models. In Proceedings Title: Proceedings of the 2012 Winter Simulation Conference (WSC). IEEE, 2012. https://doi.org/10.1109/wsc.2012.6465241.
https://doi.org/https://doi.org/10.1109/wsc.2012.6465241
[62] Hendrik Radke. A Theory of HR* Graph Conditions and their Application to Meta-Modeling. PhD thesis, Carl von Ossietzky Universität Oldenburg, Fakultät II, Department für Informatik, 2016.
[63] Grzegorz Rozenberg, editor. Handbook of Graph Grammars and Computing by Graph Transformations, Volume 1: Foundations. World Scientific, 1997. https://doi.org/10.1142/3303.
https://doi.org/https://doi.org/10.1142/3303
[64] Sven Schneider, Leen Lambers, and Fernando Orejas. Symbolic Model Generation for Graph Properties. In M. Huisman and J. Rubin, editors, Fundamental Approaches to Software Engineering (FASE 2017), volume 10202 of Lecture Notes in Computer Science, pages 226-243. Springer Berlin Heidelberg, 2017. https://doi.org/10.1007/978-3-662-54494-5_13.
https://doi.org/https://doi.org/10.1007/978-3-662-54494-5_13
Cited by
[1] Nicolas Behr, Jean Krivine, Jakob L. Andersen, and Daniel Merkle, "Rewriting theory for the life sciences: A unifying theory of CTMC semantics", Theoretical Computer Science 884, 68 (2021).
[2] Lars Fritsche, Jens Kosiol, Adrian Möller, and Andy Schürr, Lecture Notes in Computer Science 13961, 184 (2023) ISBN:978-3-031-36708-3.
[3] Nicolas Behr and Joachim Kock, "Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)", Electronic Proceedings in Theoretical Computer Science 372, 323 (2022).
[4] Nicolas Behr, Russ Harmer, and Jean Krivine, Lecture Notes in Computer Science 12741, 3 (2021) ISBN:978-3-030-78945-9.
[5] Nicolas Behr, Bello Shehu Bello, Sebastian Ehmes, and Reiko Heckel, "Stochastic Graph Transformation For Social Network Modeling", Electronic Proceedings in Theoretical Computer Science 350, 35 (2021).
[6] Xerxes D. Arsiwalla and Jonathan Gorard, "Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types", International Journal of Theoretical Physics 63 4, 83 (2024).
The above citations are from Crossref's cited-by service (last updated successfully 2024-04-18 12:27:58). The list may be incomplete as not all publishers provide suitable and complete citation data.
On SAO/NASA ADS no data on citing works was found (last attempt 2024-04-18 12:27:58).
This Paper is published in Compositionality under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.