On automata networks dynamics: an approach based on computational complexity theory.

Alejandro Maass - Eric Goles (Chile) / Sylvain Sené - Guillaume Theyssier (France)

Abstract 

An automata network (AN) is a network of entities, each holding a state from a finite set and related by a graph structure called an interaction graph. Each node evolves according to the states of its neighbors in the interaction graph, defining a discrete dynamical system. This thesis work explores two main questions: a) what is the link between dynamical and computational properties of an AN? and b) what is the impact of the interaction graph topology on the global dynamics of an AN?. In order to tackle the first question a notion of computational complexity of an AN family is defined in terms of the computational complexity of decision problems related to the dynamics of the network. On the other hand, the dynamical complexity of a particular AN family is defined in terms of the existence of attractors of exponential period. A strong link between these two last definitions is presented in terms of the notion of simulation between AN families. In this context, complexity is characterized from a localized standpoint by studying the existence of structures called coherent gadgets which satisfy two properties: i) they can locally interact in a coherent way as dynamical systems and ii) they are capable of simulating a finite set of functions defined over a fixed finite set. Finally, the second question is addressed in the context of a well-known family called freezing automata networks. An AN is freezing if there is an order on states such that the state evolution of any node is non-decreasing in any orbit. A general model checking problem capturing many classical decision problems is presented. In addition, when three graph parameters, the maximum degree, the treewidth and the alphabet size are bounded, a fast-parallel algorithm that solves general model checking problem is presented. Moreover, it is shown that the latter problem is unlikely to be fixed-parameter tractable on the treewidth parameter as well as on the alphabet size when considered as single parameters.

Keywords: discrete math - dynamical systems - computational complexity - complex systems - automata networks - systems biology.

• 2022 Exploring the Dynamics of Fungal Cellular Automata, Carlos S. Sepúlveda, Eric Goles, Martín Ríos-Wilson, Andrew Adamatzky. International Journal of Unconventional Computing (TO APPEAR, see https://arxiv.org/abs/2210.14615 for the pre-print)

• 2022 Introducing the activity parameter for elementary cellular automata, Pablo Concha-Vega, Eric Goles, Pedro Montealegre, Martín Ríos-Wilson, Julio Santivañez, International Journal of Modern Physics C, 2250121, 2022.

• 2022 On the complexity of generalized Q2R automaton, Eric Goles, Pedro Monteale- gre, Marco Montalva-Medel, Martín Ríos-Wilson, Advances in Applied Mathematics 138, 102355, Academic Press, 2022/‌7/‌1.

• 2021 On the Complexity of Asynchronous Freezing Cellular Automata, Eric Goles, Diego Maldonado, Pedro Montealegre, Martín Ríos-Wilson, Information and Computation, Information and Computation, 2021, 104764, ISSN 0890-5401. In Press. [Online version] [Preprint] 

• 2021 Generating Boolean functions on totalistic automata networks, Andrew Adamatzky, Eric Goles, Pedro Montealegre, Martín Ríos-Wilson, Int. J. Unconv. Comput. 16(4): 343-391 [Online version] [Preprint] 

• 2020 On the effects of firing memory in the dynamics of conjunctive networks, Eric Goles, Pedro Montealegre, Martín Ríos-Wilson, Discrete and Continuous Dynamical Systems - A, 40(10): 5765-5793. [Online version] [Preprint]  

• 2022 Computing Power of Hybrid Models in Synchronous Networks., Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Iván Rapaport, Martín Ríos-Wilson, Ioan Todinca, In: Conference On Principles Of Distributed Systems (OPODIS 2022). [Proceedings version] [Preprint]

• 2022 Brief Announcement: Computing Power of Hybrid Models in Synchronous Networks., Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Iván Rapaport, Martín Ríos-Wilson, Ioan Todinca, In: 36th International Symposium on Distributed Computing (DISC 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik. [Proceedings version] [Preprint]

• 2021 On the impact of treewidth in the computational complexity of freezing dynamics, Eric Goles, Pedro Montealegre, Martín Ríos-Wilson, Guillaume Theyssier, In: De Mol L., Weiermann A., Manea F., Fernández-Duque D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science, vol 12813. Springer, Cham. [Proceedings version] [Preprint]

• 2019 On the effects of firing memory in the dynamics of conjunctive networks, Eric Goles, Pedro Montealegre, Martín-Ríos Wilson, In: Castillo-Ramirez A., de Oliveira P. (eds) Cellular Automata and Discrete Complex Systems. AUTOMATA 2019. Lecture Notes in Computer Science, vol 11525. Springer, Cham. [Proceedings version] [Preprint] 

• 2021 Computing the Probability of Getting Infected: On the Counting Complexity of Bootstrap Percolation, Pedro Montealegre and Martín Ríos-Wilson, In: Automata and Complexity. Essays presented to Eric Goles on the occasion of his 70th birthday, Emergence, Complexity and Computation, Springer. To appear [Preprint]

• 2022  Exploring the Dynamics of Fungal Cellular Automata, Carlos Sepúlveda, Eric Goles,Martín Ríos-Wilson, Andrew Adamatzky, arXiv:2210.14615v1. [Preprint] 

• 2022 Computing Power of Hybrid Models in Synchronous Networks., Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Iván Rapaport, Martín Ríos-Wilson, Ioan Todinca, arXiv:2208.02640v1, [Preprint] 

• 2022 Intrinsic Simulations and Universality in Automata Networks, Martín Ríos- Wilson and Guillaume Theyssier, arXiv:2209.09527v2. [Preprint]

• 2021 On the impact of treewidth in the computational complexity of freezing dynamics, Eric Goles, Pedro Montealegre, Martín Ríos-Wilson, Guillaume Theyssier, hal-02665147, [Preprint]

• 2021 On Symmetry versus Asynchronism: at the Edge of Universality in Automata Networks, Martín Ríos-Wilson and Guillaume Theyssier, Submitted to Journal of Computer and System Sciences, arXiv:2105.08356. [Preprint]

Organizing XVIII Escuela de Verano en Matemáticas Discretas

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Organizing  the Workshop Escuela Predoctoral DISC 2021

Organizing Committee of the workshop The Automata Factory 3 3 (2020)