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

1

Preliminaries

2

Some known results and open questions

Sylvain Sené Natural computation: Automata networks 2/12

Preliminaries

Plan

1

Preliminaries

2

Some known results and open questions

Sylvain Sené Natural computation: Automata networks 3/12

Preliminaries

Definitions

A Boolean automata network (BAN) of size n is a function f : Bn Ñ Bn x “ px0,x1,...,xn´1q ÞÑ fpxq “ pf0pxq,f1pxq,...,fn´1pxqq , where @i P t0,...,n´1u, xi P B is the state of automaton i, and Bn is the set of configurations. The interaction graph of f is the signed digraph Gpfq : pV,E Ď V ˆVq where: V “ t0,...,n´1u; pj,iq P E is positive if Dx P X “ t0,1un s.t. fipx0,...,xj´1,0,xj`1,...,xn´1q “ 0 et fipx0,...,xj´1,1,xj`1,...,xn´1q “ 1; pj,iq P E is negative if Dx P X “ t0,1un s.t. fipx0,...,xj´1,0,xj`1,...,xn´1q “ 1 et fipx0,...,xj´1,1,xj`1,...,xn´1q “ 0.

Sylvain Sené Natural computation: Automata networks 4/12

Preliminaries

Definitions

A Boolean automata network (BAN) of size n is a function f : Bn Ñ Bn x “ px0,x1,...,xn´1q ÞÑ fpxq “ pf0pxq,f1pxq,...,fn´1pxqq , where @i P t0,...,n´1u, xi P B is the state of automaton i, and Bn is the set of configurations. f : t0,1u4 Ñ t0,1u4 f “ ¨ ˚ ˚ ˝ f0pxq “ x0 _x1 ^x3 f1pxq “ x0 ^px1 _x2q f2pxq “ x3 f3pxq “ x0 _x1 ˛ ‹ ‹ ‚ 3 1 2

Sylvain Sené Natural computation: Automata networks 4/12

Preliminaries

Automata updates

f0pxq “ x0 _x1 ^x3 f3pxq “ x0 _x1 f1pxq “ x0 ^px1 _x2q f2pxq “ x3

0101 1 2 3

Sylvain Sené Natural computation: Automata networks 5/12

Preliminaries

Automata updates

f0pxq “ x0 _x1 ^x3 f3pxq “ x0 _x1 f1pxq “ x0 ^px1 _x2q f2pxq “ x3

0101 1 2 3

t2u “ 0

Sylvain Sené Natural computation: Automata networks 5/12

Preliminaries

Automata updates

0001

t3u t1u

f0pxq “ x0 _x1 ^x3

Asynchronous transitions

f3pxq “ x0 _x1 f1pxq “ x0 ^px1 _x2q f2pxq “ x3

0101 1 2 3 0000 0100 1000 1001 1100 1101

t2u t0u

Sylvain Sené Natural computation: Automata networks 5/12

Preliminaries

Automata updates

1100 1101

t2u t0u

0001

t3u t1u t0,1u t0,1u

f0pxq “ x0 _x1 ^x3 f3pxq “ x0 _x1 f1pxq “ x0 ^px1 _x2q f2pxq “ x3

0101 1 3 0000 0100 2 1000 1001

“ 0 1 “

Sylvain Sené Natural computation: Automata networks 5/12

Preliminaries

Automata updates

3 0000 0100 1000 1001 1100 1101

t2u t0u

0001

t3u t1u t1,3u t1,2,3u t1,2u t2,3u t0,1,3u t0,2u t0,1,2,3u t0,1u t0,1,2u t0,3u t0,2,3u

f0pxq “ x0 _x1 ^x3

Synchronous transitions

f3pxq “ x0 _x1 f1pxq “ x0 ^px1 _x2q f2pxq “ x3

0101 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, 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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

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˛pfq “ pt0,1un,T Ď t0,1un ˆpPpVqzHqˆt0,1unq, where ˛ represents a given “fair” update mode. f : t0,1u3 Ñ t0,1u3 f “ $ & % f0pxq “ x1 _x2 f1pxq “ x0 ^x2 f2pxq “ x2 ^px0 _x1q 1 2

Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries

BAN behaviour

Parallel evolution

The parallel transition graph of f is the digraph Gπpfq “ pt0,1un,Tq, with T “ tpx,V,fpxqq | x P t0,1unu.

000 001 010 011 100 101 110 111

toto

Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries

BAN behaviour

Parallel evolution

The parallel transition graph of f is the digraph Gπpfq “ pt0,1un,Tq, with T “ tpx,V,fpxqq | x P t0,1unu.

000 001 010 011 100 101 110 111

An attractor of pf,˛q is a terminal SCC of de G˛pfq. A fixed point (stable configuration) is a trivial attractor. A limit cycle (stable oscillation) is a non trivial attractor.

Sylvain Sené Natural computation: Automata networks 6/12

Preliminaries

BAN behaviour

Asynchronous update mode

The asynchronous transition graph of f is the digraph Gαpfq “ pt0,1un,Tq, with: T “ tpx,i,yq | x,y P t0,1un et y “ px0,...,xi´1,fipxq,xi`1,...,xn´1qu.

000 001 010 011 100 101 110 111 1 1 1 1 1 1 1 1 2 2 2 2 2

Sylvain Sené Natural computation: Automata networks 6/12

Some known results and open questions

Plan

1

Preliminaries

2

Some known results and open questions

Sylvain Sené Natural computation: Automata networks 7/12

Some known results and open questions

About computability

˛ Turing universality Theorem (McCulloch & Pitts, 1943) Threshold BANs are Turing-complete. Idea of a neat proof Ñ Any Turing machine can be simulated by a cellular automaton (Smith, 1971). Ñ Any cellular automaton can be simulated by a threshold BAN (Goles & Martínez, 1990).

Sylvain Sené Natural computation: Automata networks 8/12

Some known results and open questions

About computability

˛ Intrinsic simulation Problem (Demongeot, Noual & S., 2012) Cycles, double-cycles... What else? ? Problem (Bridoux, Guillon, Perrot, S. & Theyssier, 2017) Simulating pf,˛1q by pg,˛2q. Ñ Possible? How? At which cost?...

Sylvain Sené Natural computation: Automata networks 8/12

Some known results and open questions

Interaction cycles

2 kinds of cycles: the positive and the negative ones. 1 4 5 3 2

C `

6 1

C ´

6 4 5 3 2 an even number of negative arcs an odd number of negative arcs

Sylvain Sené Natural computation: Automata networks 9/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 1 2 3 4 5 6 7 $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 4 6 3 2 7 1 5 $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 7 1 5 4 6 3 2 7 1 5 $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6 4 6 3 2

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 4 6 3 2 7 1 5 3 2 7 1 5 $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

• 4

6 3 2 7 1 5

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 3 2 7 1 5 3 2 7 1 5 $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

• 4

6 3 2 7 1 5

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 3 2 7 1 5 3 1 5

• $

’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

• 4

6 3 2 7 1 5

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 3 1 5 3 1 5

• $

’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

• 4

6 3 2 7 1 5

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Robert, 1986) No cycles in Gpfq ù ñ G˛pfq admits a unique attractor, a fixed point. Idea of the proof Ñ Remark that @i P V,δ ´

i

“ 0, fipxq “ t0,1u. Ñ Take a DAG and use an induction on automata depth.

t´1 t 3 1 5

• $

’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % f0pxq “ 1 f1pxq “ x2 f2pxq “ x0 ^x4 f3pxq “ x2 _px6 ^x7q f4pxq “ 0 f5pxq “ x2 ^x7 f6pxq “ 1 f7pxq “ x4 _x6

• 4

6 3 2 7 1 5

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Thomas, 1981; Richard & Comet, 2007; Noual & S., 2012) G˛pfq admits several fixed points ù ñ G contains a positive cycle. History of this result Ñ Proposed in 1981 as a conjecture. Ñ Proven in 2007 for ˛ “ α. Ñ Generalised to any ˛ in 2012. Idea of the proof Ñ Admit the result of 2007. Ñ Consider the elementary update mode. Ñ For any BAN with no positive cycles:

Either it doesn’t contain any cycle and Robert’s theorem holds. Or it does contain a negative cycle, which prevents removing all the local unstabilities.

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Theorem (Thomas, 1981; Richard & Comet, 2007; Noual & S., 2012) G˛pfq admits several fixed points ù ñ G contains a positive cycle. Example Elementary dynamics of a negative 6-cycle 2 6 2 `6

2

˘ “ 12 2 `6

3

˘ “ 40 2 `6

5

˘ “ 12

111101 100111

...

110011 111110 111100

... ... ...

111111

...

100000 000000 010101 111000

automata number of unstable in the layer number of config.

101010

011101

...

101110 001101

...

111010

...

101000 011101 110101 000000 111100 111111 000111 000011 111000 111110 011111 110000 100000 000001 001111

3 1 5 4

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

About counting and characterising

Problem Given an interaction graph G “ pV,Eq, determine φpGq “ maxptcardpFPpG,fqq | f : Bn Ñ Bnuq. Ñ τ`pGq: positive feedback vertex set of G, i.e. min. number of automata meeting all the positive cycles. Ñ νpGq: packing number of G, i.e. mac number of disjoint cycles in G. Theorem (Aracena, 2008; Aracena, Richard & Salinas, 2017) νpGq`1 ď φpGq ď 2τ`pGq ď 2hpνpGqq. Ñ Refine the bounds. Ñ Show that φpGq ď 2hpνpGqq without using the theorem of Reed, Robertson, Seymour & Thomas, 1996 stating that @G, τpGq ď hpνpGqq. Ñ What about limit cycles?

Sylvain Sené Natural computation: Automata networks 10/12

Some known results and open questions

Last but not least: Structural robustness

Class 70 Class 71

Perfect robustness Mean robustness

Class 72 Class 73

Weak robustness Null robustness

Noual, Regnault & S., 2012; Noual & S., 2017

Sylvain Sené Natural computation: Automata networks 11/12