Natural computation Automata networks Sylvain Sen ENS Cachan - PowerPoint PPT Presentation

Natural computation Automata networks Sylvain Sené ENS Cachan visits Marseille 23rd November 2017

Plan Preliminaries 1 Some known results and open questions 2 Sylvain Sené Natural computation: Automata networks 2/12

Preliminaries Plan Preliminaries 1 Some known results and open questions 2 Sylvain Sené Natural computation: Automata networks 3/12

Preliminaries Definitions 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 j , i q P E is positive if D x P X “ t 0 , 1 u n s.t. f i p x 0 ,..., x j ´ 1 , 0 , x j ` 1 ,..., x n ´ 1 q “ 0 et f i p x 0 ,..., x j ´ 1 , 1 , x j ` 1 ,..., x n ´ 1 q “ 1; p j , i q P E is negative if D x P X “ t 0 , 1 u n s.t. f i p x 0 ,..., x j ´ 1 , 0 , x j ` 1 ,..., x n ´ 1 q “ 1 et f i p x 0 ,..., x j ´ 1 , 1 , x j ` 1 ,..., x n ´ 1 q “ 0. Sylvain Sené Natural computation: Automata networks 4/12

Preliminaries Definitions 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 t 0 , 1 u 4 Ñ t 0 , 1 u 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 Natural computation: Automata networks 4/12

Preliminaries 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 Natural computation: Automata networks 5/12

Preliminaries 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 Natural computation: Automata networks 5/12

Preliminaries 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 Natural computation: Automata networks 5/12

Preliminaries 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 Natural computation: Automata networks 5/12

Preliminaries 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 Natural computation: Automata networks 5/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour Update modes The update mode defines the network behaviour Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. 0 1 t 0 , 1 u 3 Ñ t 0 , 1 u 3 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. 0 1 t 0 , 1 u 3 Ñ t 0 , 1 u 3 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries BAN behaviour 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 “ pt 0 , 1 u n , T Ď t 0 , 1 u n ˆp P p V qzHqˆt 0 , 1 u n q , where ˛ represents a given “fair” update mode. 0 1 t 0 , 1 u 3 Ñ t 0 , 1 u 3 f : $ f 0 p x q “ x 1 _ x 2 & f “ f 1 p x q “ � x 0 ^ x 2 2 f 2 p x q “ � x 2 ^p x 0 _ x 1 q % Sylvain Sené Natural computation: Automata networks 6/12