Synchronism vs asynchronism in Boolean automata networks Sylvain - PowerPoint PPT Presentation

Synchronism vs asynchronism in Boolean automata networks Sylvain Sené MOVE seminar 18th January 2018

Outline Introduction 1 Main definitions 2 3 Deterministic periodic updates 4 Non-deterministic updates Sylvain Sené Synchronism vs asynchronism in BANs 2/27

Introduction Outline Introduction 1 Main definitions 2 3 Deterministic periodic updates 4 Non-deterministic updates Sylvain Sené Synchronism vs asynchronism in BANs 3/27

Introduction BANs, non formally § A discrete computational model of interaction systems. § From a theoretical standpoint: § Simple setting and representation. § Able to capture dynamically a lot of behavioural intricacies and heterogeneities. § From a more practical/applied standpoint: § Originate from neural theoretical modelling (McCulloch, Pitts, 1943). § Developed in the context of genetics (Kauffman, 1969; Thomas, 1973). § The most used mathematical objects for genetic regulation qualitative modelling. Sylvain Sené Synchronism vs asynchronism in BANs 4/27

Introduction The (a-)synchronicity problematic(s) § The causality of events along time depends on the relation between automata updates and “time” but... § How to define this relation? § How to study the causal perturbations due to changes of this relation? § Mathematical pertinence: § Neat problematic at the frontier of dynamical systems, combinatorics, complexity and computability. § Biological pertinence: § Genetic expression and chromatin dynamics. § A remaining question: does model synchronicity stand for modelled system simultaneity? Sylvain Sené Synchronism vs asynchronism in BANs 5/27

Main definitions Outline Introduction 1 Main definitions 2 3 Deterministic periodic updates 4 Non-deterministic updates Sylvain Sené Synchronism vs asynchronism in BANs 6/27

Main definitions BANs and interaction graphs A Boolean automata network (BAN) of size n is a function B n B n f : Ñ f p x q “ p f 0 p x q , f 1 p x q ,..., f n ´ 1 p x qq , x “ p x 0 , x 1 ,..., x n ´ 1 q ÞÑ where @ i P t 0 ,..., n ´ 1 u , x i P B is the state of automaton i , and B n is the set of configurations. The interaction graph of f is the signed digraph G p f q : p V , E Ď V ˆ V q where: § V “ t 0 ,..., n ´ 1 u ; § p i , j q P E is positive if D x P B n s.t. f j p x 0 ,..., x i ´ 1 , 0 , x i ` 1 ,..., x n ´ 1 q “ 0 and f j p x 0 ,..., x i ´ 1 , 1 , x i ` 1 ,..., x n ´ 1 q “ 1; § p i , j q P E is negative if D x P B n s.t. f j p x 0 ,..., x i ´ 1 , 0 , x i ` 1 ,..., x n ´ 1 q “ 1 and f j p x 0 ,..., x i ´ 1 , 1 , x i ` 1 ,..., x n ´ 1 q “ 0. Sylvain Sené Synchronism vs asynchronism in BANs 7/27

Main definitions BANs and interaction graphs A Boolean automata network (BAN) of size n is a function B n B n f : Ñ f p x q “ p f 0 p x q , f 1 p x q ,..., f n ´ 1 p x qq , x “ p x 0 , x 1 ,..., x n ´ 1 q ÞÑ where @ i P t 0 ,..., n ´ 1 u , x i P B is the state of automaton i , and B n is the set of configurations. 0 1 B 4 Ñ B 4 f : $ f 0 p x q “ � x 0 _ x 1 ^ x 3 ’ ’ ’ f 1 p x q “ x 0 ^p x 1 _ x 2 q & “ f f 2 p x q “ � x 3 ’ ’ ’ f 3 p x q “ x 0 _� x 1 % 3 2 Sylvain Sené Synchronism vs asynchronism in BANs 7/27

Main definitions Automata updates 0101 0 1 f 0 p x q “ � x 0 _ x 1 ^ x 3 f 1 p x q “ x 0 ^p x 1 _ x 2 q f 3 p x q “ x 0 _� x 1 3 2 f 2 p x q “ � x 3 Sylvain Sené Synchronism vs asynchronism in BANs 8/27

Main definitions Automata updates t 2 u 0101 0 1 f 0 p x q “ � x 0 _ x 1 ^ x 3 f 1 p x q “ x 0 ^p x 1 _ x 2 q “ 0 f 3 p x q “ x 0 _� x 1 3 2 f 2 p x q “ � x 3 Sylvain Sené Synchronism vs asynchronism in BANs 8/27

Main definitions Automata updates t 2 u Asynchronous transitions 0101 t 0 u t 1 u t 3 u 0000 0001 0100 1000 1001 1100 1101 0 1 f 0 p x q “ � x 0 _ x 1 ^ x 3 f 1 p x q “ x 0 ^p x 1 _ x 2 q f 3 p x q “ x 0 _� x 1 3 2 f 2 p x q “ � x 3 Sylvain Sené Synchronism vs asynchronism in BANs 8/27

Main definitions Automata updates t 2 u 0101 t 0 u t 1 u t 0 , 1 u t 3 u t 0 , 1 u 0000 0001 0100 1000 1001 1100 1101 1 “ “ 0 0 1 f 0 p x q “ � x 0 _ x 1 ^ x 3 f 1 p x q “ x 0 ^p x 1 _ x 2 q f 3 p x q “ x 0 _� x 1 3 2 f 2 p x q “ � x 3 Sylvain Sené Synchronism vs asynchronism in BANs 8/27

Main definitions Automata updates t 2 u Synchronous transitions 0101 t 1 , 3 u t 0 u t 1 u t 0 , 3 u t 1 , 2 , 3 u t 0 , 2 u t 3 u t 0 , 1 u t 1 , 2 u t 0 , 2 , 3 u t 0 , 1 , 3 u t 2 , 3 u t 0 , 1 , 2 u t 0 , 1 , 2 , 3 u 0000 0001 0100 1000 1001 1100 1101 0 1 f 0 p x q “ � x 0 _ x 1 ^ x 3 f 1 p x q “ x 0 ^p x 1 _ x 2 q f 3 p x q “ x 0 _� x 1 3 2 f 2 p x q “ � x 3 Sylvain Sené Synchronism vs asynchronism in BANs 8/27

Main definitions Update modes and BAN behaviours § An update mode is a way of organising the automata updates along time. § It can be deterministic (periodic or not) or non-deterministic (stochastic or not). § There exists an infinite number of update modes. Sylvain Sené Synchronism vs asynchronism in BANs 9/27

Main definitions Update modes and BAN behaviours § An update mode is a way of organising the automata updates along time. § It can be deterministic (periodic or not) or non-deterministic (stochastic or not). § There exists an infinite number of update modes. § The update mode defines the network behaviour. § The behaviour of a BAN f is described by a transition graph G ˛ p f q “ p B n , T Ď B n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Synchronism vs asynchronism in BANs 9/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 Parallel evolution 000 011 110 toto 010 101 001 100 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 Parallel evolution 000 § An attractor of p f , ˛ q is a terminal SCC of G ˛ p f q . 011 110 § A fixed point (stable configuration) is a trivial attractor. 010 101 001 § A limit cycle (stable oscillation) is a non-trivial attractor. 100 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 pt 0 u , t 1 u , t 2 uq -sequential evolution 001 010 110 100 101 101 101 100 000 000 000 110 011 111 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 pt 0 u , t 1 u , t 2 uq -sequential evolution 001 010 101 100 000 110 011 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 pt 0 , 2 u , t 1 uq -block-sequential evolution Number of ordered 000 000 partitions: n ´ 1 010 001 001 ˆ n ˙ B ord ÿ B ord “ , n k k k “ 0 110 111 101 100 100 011 with B ord “ 1. 0 111 110 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 pt 0 , 2 u , t 1 uq -block-sequential evolution Number of ordered 000 partitions: n ´ 1 010 001 ˆ n ˙ B ord ÿ B ord “ , n k k k “ 0 110 101 100 011 with B ord “ 1. 0 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 pt 0 , 2 u , t 1 uq -block-sequential evolution Number of ordered 000 partitions: n ´ 1 010 001 ˆ n ˙ B ord ÿ B ord “ , n k k k “ 0 110 101 100 011 with B ord “ 1. 0 111 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 Asynchronous evolution ` t 0 , 2 u -synchronous transitions 0 0 2 1 1 0 0 000 2 001 100 2 101 1 1 1 1 1 1 0 010 011 110 111 2 2 0 0 0 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Main definitions Some examples B 3 Ñ B 3 0 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % 2 1 Asynchronous evolution ` t 0 , 2 u -synchronous transitions 0 0 2 1 1 0 0 000 2 001 100 2 101 1 1 1 1 1 1 0 010 011 110 111 2 2 0 0 0 Sylvain Sené Synchronism vs asynchronism in BANs 10/27

Deterministic periodic updates Outline Introduction 1 Main definitions 2 3 Deterministic periodic updates 4 Non-deterministic updates Sylvain Sené Synchronism vs asynchronism in BANs 11/27