Synchronism vs Asynchronism in Boolean networks
SSynchronism vs Asynchronismin Boolean automata networks
Mathilde Noual , Laboratoire I3S, UMR 7271 - UNS CNRS, Universit´e de Nice-Sophia Antipolis,06900 Sophia Antipolis, France; [email protected] IXXI, Institut rhˆone-alpin des syst`emes complexes, Lyon, France
Abstract.
We show that synchronism can significantly impact on net-work behaviours, in particular by filtering unstable attractors inducedby a constraint of asynchronism. We investigate and classify the differ-ent possible impacts that an addition of synchronism may have on thebehaviour of a Boolean automata network. We show how these relateto some strong specific structural properties, thus supporting the ideathat for most networks, synchronism only shortcuts asynchronous tra-jectories. We end with a discussion on the close relation that apparentlyexists between sensitivity to synchronism and non-monotony.
Keywords:
Automata network, synchronism, asynchronism, attractor, updat-ing mode, elementary transition, atomic transition.
Introduction
In works involving automata networks, synchronism has often either been con-sidered as a founding hypothesis, as in [10] and the many studies that followedin its lead, or, on the contrary, in lines with [18], it has been disregarded alto-gether to the benefit of pure asynchrony. In some applied contexts, theoreticalsynchronism is sometimes understood as simultaneity although this restrictiveinterpretation relies on a formalisation of duration which conflicts with the dis-crete nature of automata networks. More simply, the synchronous occurrence oftwo changes in a network can be regarded as occurrences that are close enoughin time to disallow any other significant event in between them. This naturallydefines a much more general notion of time flow that is strongly relative to theset of events underwent by the network. And thus it yields substantial represen-tational capacity to the notion of synchronism, justifying the attention that wepropose to give to it in this paper.Comparisons have been made between different kinds of ways of updating au-tomata states, involving variable degrees of synchronism in both probabilistic[2,6,7,12,17] (with cellular automata) and deterministic frameworks [1,4,5,9,15,16].In particular, for the algorithmic purpose of finding the shortest path to a stableconfiguration, Robert[16] compared Boolean automata network behaviours un-der the parallel and sequential update schedules. In this context, he noted three a r X i v : . [ c s . D M ] D ec frequent (but not systematic) phenomena” that can be observed through theeffect of parallelisation: the “ bursting ”, the “ aggregation ” and the “ implosion ”of attraction basins. Here, we focus on attractors, both stable and unstable. Andconsidering more generally state transition systems rather than just determinis-tic dynamical systems, we propose to investigate synchronism per se , and analyseits input to the design of Boolean automata network behaviours. More precisely,we propose to consider asynchronous transition graphs representing the set of allpunctual and atomic events of a network and we propose to explore the conse-quences of adding to it a synchronous transition, representing a new possibilityto perform a punctual but non-atomic change. We propose to identify the caseswhere such an addition of synchronism changes substantially a network’s possibleasymptotic behaviours or its evolutions towards them. Thus, we are looking fornetworks for which synchronism does not just shortcut asynchronous trajectoriesbut rather also adds some new ones that can not be mimicked asynchronously.This leads to classify the possible impacts of non-sequentialisable transitions andthen the sensitivity of networks to synchronism. Preliminaries
Notations –
By default, V = { , . . . , n − } denotes a set of n ∈ N automatanumbered from 0 to n −
1. We let B = { , } . Any x ∈ B n is called a configu-ration and its component x i ∈ B is regarded as the state of automaton i ∈ V .In this paper, special attention is paid to switches of automata states startingin a given configuration. For this reason, we introduce the following notations: ∀ x = x . . . x n − ∈ B n , ∀ i ∈ V , x i = x . . . x i − ¬ x i x i +1 . . . x n − and ∀ W = W (cid:48) (cid:93) { i } ⊆ V , x W = ( x i ) W (cid:48) = ( x W (cid:48) ) i .Also, to compare two configurations x, y ∈ B n , we use: D( x, y ) = { i ∈ V ; x i (cid:54) = y i } and the Hamming distance d( x, y ) = | D( x, y ) | . Finally, to switch values from B to { − , } , we let s : b ∈ B (cid:55)→ b − ¬ b ∈ { − , } . Networks – A Boolean automata network ( ban ) of size n is comprised of n interacting automata. Formally, it is a set N = { f i ; i ∈ V } of n Booleanfunctions specifying “how the net of automata works”. Function f i : B n → B specifies the behaviour of automaton i ∈ V in any configuration x ∈ B n . Itis called the transition function of i . We focus on functions that are locallymonotone w.r.t. all their components, i.e. ∀ i ∈ V , ∀ j ∈ V we assume:that either ∀ x ∈ B n , s ( x j ) · (f i ( x ) − f i ( x j )) ≥ ∀ x ∈ B n , s ( x j ) · (f i ( x ) − f i ( x j )) ≤
0. (2)At the end of this paper, non-monotony is discussed. Until then, we assume all ban s to be monotone , that is, to involve transition functions that are locallymonotone w.r.t. all their components. etwork structures –
The structure of N is the digraph G = ( V , A ) whosenode set is V (thus, automata are also called nodes) and whose arc set is: A = { ( j, i ) ∈ V ; ∃ x ∈ B n , f i ( x ) (cid:54) = f i ( x j ) } . V − ( i ) denotes the in-neighbourhoodof i ∈ V in G . The local monotony of transition functions allows us to sign the arcs of G . ∀ ( j, i ) ∈ A and ∀ x ∈ B n s.t. f i ( x ) (cid:54) = f i ( x j ), we let sign( j, i ) = s ( x j ) · (f i ( x ) − f i ( x j )) = s ( x j ) · s (f i ( x )) which equals +1 if (1) is satisfied and − j, i ) = 0 when ( j, i ) / ∈ A . Naturally, we define the signof a path or cycle in G as the product of the signs of the arcs it involves. Thus,a positive path globally transmits “information” directly whereas a negative onetransmits its negation. Instabilities and frustrations –
For every x ∈ B n , we define the set: U ( x ) = { i ∈ V ; f i ( x ) (cid:54) = x i } . Automata in U ( x ) are said to be unstable (or “callingfor a change or updating”[13]) in x and those in U ( x ) = V \ U ( x ) are said tobe stable in x . Informally, the number u( x ) = | U ( x ) | of instabilities in x canbe understood as the velocity or momentum of N in x . Configurations x suchthat u( x ) = 0 are called stable . Our first lemma which will be very useful in thesequel, relates instabilities to ban structures. Its proof is simple so we skip it. Lemma 1 (loops). ∀ i ∈ V , ∀ x ∈ B n , i ∈ U ( x ) ∩ U ( x i ) ⇒ sign( i, i ) =+1 and i ∈ U ( x ) ∩ U ( x i ) ⇒ sign( i, i ) = − . ∀ x ∈ B n , we introduce the set of arcs that are frustrated[3,8,19,20] in x : FRUS ( x ) = { ( j, i ) ∈ A ; s ( x j ) · s ( x i ) = − sign( j, i ) } (3)Our second preliminary lemma states that adding frustrated arcs incoming anunstable automaton cannot make it stable. Again, we skip its proof which mainlyrelies on the local monotony of transition functions. Lemma 2 (frustrations & instabilities). ∀ i ∈ V , ∀ x, y ∈ B n : (cid:0) i ∈ U ( x ) ∧ FRUS ( x ) ∩ V − ( i ) ⊂ FRUS ( y ) ∩ V − ( i ) (cid:1) ⇒ i ∈ U ( y ) .Transitions and and transition graphs – An elementary transition of a ban N represents an effective , punctual and possible change in N . It is any coupleof configurations ( x, y ) ∈ B n × B n , noted x y , which satisfies: ∅ (cid:54) = D( x, y ) ⊆ U ( x ). The size of an elementary transition x y equals d( x, y ). Digraph( B n , { x y ; x, y ∈ B n } ) is called the elementary transition graph ( etg )of N . It represents all punctual/elementary events that N can undergo. Thereare two main types of elementary transitions x y . Those of size d( x, y ) > non-atomic or synchronous and are written x y . Those of sized( x, y ) = 1 are called asynchronous or atomic and are written x y (theyare s.t. ∃ i ∈ V , y = x i ). Digraph T a = ( B n , { x y ; x, y ∈ B n } ) is calledthe asynchronous transition graph or atg . It represents only those eventsthat N can undergo which involve only one local automaton state change. Thetransitive closure of (resp. ) is denoted by (resp. ). Derivations are ordered lists of these non necessarily elementary transitionswritten x x . . . x (cid:96) − x (cid:96) but in the sequel, we abuse lan-guage and also speak of a derivation x y . on-sequentialisable transitions and critical cycles A cycle of a ban N is a sub-graph of its structure G corresponding to a closeddirected walk, with possibly repeated nodes but no repeated edges. ∀ x ∈ B n , wesay that a cycle C = ( V C , A C ) of N is x -critical if: V C ⊂ U ( x ) ∧ A C ⊂ FRUS ( x ). Note that for an isolated cycle, since | V − ( i ) | = 1 , ∀ i ∈ V C , a node isunstable if and only if its sole incoming arc is frustrated. A critical cycle of N isone that is x -critical for some x ∈ B n . All main results of this paper mention thesetypes of cycles. This yields some importance to Proposition 1 below which derivesfrom the following which, by (3), holds for any x -critical cycle C = ( V C , A C ) oflength (cid:96) and sign s : (cid:89) ( j,i ) ∈ A C − sign( j, i ) = ( − (cid:96) × s = (cid:89) ( j,i ) ∈ A C s ( x j ) · s ( x i ) = 1. ++ − + + + Fig. 1.
Signed ban struc-ture whose Hamiltoniancycle C = ( V C , A C ) is asin Proposition 1. If f : x (cid:55)→ x ∧ ( x ∨ x ), then C cannot be critical be-cause 2 ∈ U ( x ) and A C ⊂ FRUS ( x ) cannot be satis-fied at once. Proposition 1.
A cycle that is critical is either posi-tive with an even length or negative with an odd length.
Let us emphasise that although a positive (resp. neg-ative) cycle with an even (resp. odd) length is criticalwhen it is isolated, when embedded in larger struc-tures, it may loose this property ( cf.
Figure 1).Now, the next result sets the backbone of the article:it shows how critical cycles are the main structuralaspects of a ban underlying its possibility to performsynchronous changes that cannot be mimicked asyn-chronously. First, let us say that x y is sequen-tialisable if it is asynchronous or if it can be brokeninto an derivation x y involving smaller transi-tions. A synchronous transition x y which is notsequentialisable is called a normal transition and israther written x y . Proposition 2 (sequentialisable transitions and critical cycles).
Let x y be a synchronous transition of an arbitrary ban N . There is a deriva-tion x x D x D (cid:93) D . . . x D (cid:93) D ... (cid:93) D m − = y of N such that D( x, y ) = (cid:85) t
Consider the digraph H = (D( x, y ) , FRUS ( x )) and let δ : D( x, y ) →{ , , . . . , m − } be a topological ordering of the nodes of H : ∀ j, i ∈ D( x, y ) , ( j, i ) ∈ RUS ( x ) ⇒ δ ( i ) ≤ δ ( j ) s.t. if j and i do not belong to the same cycle in H (andthus do not belong to the same x -critical cycle of G ), then ( j, i ) ∈ FRUS ( x ) ⇒ δ ( i ) < δ ( j ). Now, let D t = { i ∈ D( x, y ) , δ ( i ) = t } and x (0) = x . Based onLemma 2, an induction on t < m proves that ∀ t < m, x ( t ) x ( t +1) = x ( t ) D t is a transition of N . (cid:117)(cid:116) The next lemma considers the case where the only critical cycles of N are Hamil-tonian cycles. Lemma 3.
Let N be a ban whose critical cycles all have node set V . Then,either N has a unique transition x y , or it has two x y and y x .In the first case, every i ∈ U ( y ) bears a positive loop ( i, i ) ∈ A . In both cases theendpoints of these transitions can be reached by no asynchronous derivation.Proof. Suppose that x y and x (cid:48) y (cid:48) are two normal transitions. UsingProposition 2, if x (cid:48) (cid:54) = y , then W = D( x, x (cid:48) ) (cid:40) U ( x ) = V and D( x (cid:48) , y ) = V \ W (cid:40) U ( x (cid:48) ) = V . In this case, x x (cid:48) y is a derivation of N involving smallertransitions than x y , in contradiction with x y being normal. Thus,if x y is not the unique normal transition of N , then the only other oneis y x . For any normal transition z z (cid:48) = z V , and ∀ i ∈ V , z z (cid:48) i is a transition of N . By hypothesis and by Proposition 2, it is sequentialisable: z z (cid:48) i . Since z z (cid:48) is not however, this implies i ∈ U ( z (cid:48) i ) , ∀ i ∈ V . Thus,the endpoint of any normal transition of N can be reached by no asynchronousderivation. And since ∀ i ∈ V , i ∈ U ( y i ), any i ∈ U ( y ) is such that sign( i, i ) =+1 by Lemma 1. (cid:117)(cid:116) Impact of synchronous transitions
Let us introduce some new vocabulary to describe the transition graphs of N . Stable configurations and terminal strongly connected components of thesegraphs are called attractors (abusing language because it may be that an attrac-tor doesn’t attract anything). Attractors that are not stable configurations aresaid to be unstable. Now, while presenting notations relative to the atg (under-scripted by ’a’), let us continue introducing terminology relative to any transitiongraph of N , in particular its etg . ∀ x ∈ B n , we let O a ( x ) = { y ∈ B n ; x y } and B a ( x ) = { y ; y x } . Also, we let A a ( x ) denote the set of attractors that x can reach in T a . We say that a configuration x is recurrent when it belongs toan attractor and we denote this attractor by [ x ] a (then, A a ( x ) = { [ x ] a } ). Thebasin of an attractor [ x ] a is B a ([ x ] a ) = B a ( x ) \ [ x ] a . Non-recurrent configurationsare called transient .Let us consider an arbitrary synchronous transition x y of N and let T a (cid:48) =( B n , T a ∪{ ( x, y ) } ) denote the transition graph obtained by adding this transitionto the atg T a . We introduce notations A ( z ), B ( z ), O ( z ) and [ x ] relative to T a (cid:48) naturally as we did above for T a . In the sequel, we say that an attractor A of T a is destroyed by x y if all its configurations are transient in T a (cid:48) . Generally,since ∀ z ∈ B a ( x ) ∪{ x } , A ( z ) = A a ( z ) ∪A a ( y ), the addition of x y to T a canave several possible consequences on the asymptotic evolution of N starting ina configuration z ∈ B a ( x ) ∪ { x } . We list them now exhaustively.1. We say that it has no impact when x is transient in T a and A a ( y ) ⊂A a ( x ) = A ( x ). In this case, x y ‘only’ adds to T a some new derivationsfrom x to the configurations of the orbit O a ( x ) of x . It does not change theresult of any network evolution. In particular, if x y is sequentialisable,then it shortcuts some derivations starting in x . But on the contrary, it canalso deviate some derivations (when ∃ z ∈ O a ( x ) ∩ O a ( y ) s.t. y z is noshorter than x z ).Obviously, all synchronous transitions x y that do have an impact on theasymptotic evolution of N are normal.2. We say that transition x y has little or F-impact ( cf. Figure 2) if x istransient in T a and A ( x ) = A a ( x ) ∪ A a ( y ) (cid:54) = A a ( x ). Here, the addition of x y adds some new degrees of f reedom to the asymptotic outcomes ofthe evolutions of N from any configuration z ∈ B a ( x ) ∪ { x } .Thus, it causesthe growth of the basins B a ( A ) , A ∈ A a ( y ). ∀ x ∈ B n , (cid:26) f ( x ) = x ∧ ¬ x f ( x ) = ¬ x ∧ x + −− + Fig. 2.
Transition functions, structure and modified atg T a (cid:48) of a ban N whose normaltransition 11 00 has F-impact. This is a special case of F-impact induced by onecritical, Hamiltonian cycle with no automata outside of it. Its impact, precisely, consistsin making reachable an reachable attractor ( cf. Lemmas 3 and 4).
Note that with addition of synchronous transitions that have no or F-impact,the set of recurrent configurations of T a equals that of T a (cid:48) .3. We say that transition x y has G-impact ( cf. the end of Example 1)on the asymptotic evolution of N when, in T a , x is recurrent, y is transientand A a ( y ) = A a ( x ) = { [ x ] a } . In this case, y is connected to x and recurrentin T a (cid:48) . The addition of x y to T a causes attractor [ x ] a to absorb allderivations from y to [ x ] a and g row into [ x ] = [ y ] without being destroyed.4. We say that transition x y has D-impact ( cf. Example 1) if x and y are both recurrent in T a and A a ( y ) \ A a ( x ) (cid:54) = ∅ . In this case, the addition of x y d estroys the unstable attractor [ x ] a by emptying it into (the basinsof) the attractors A ∈ A a ( y ) \ A a ( x ). scc { x ∈ B | x ∨ x = 1 } stable configuration0000 Fig. 3.
Schematic representation of T a (cid:48) , the atg of the ban of Example 1 augmentedwith normal transition 1100 0000 which updates both automata 0 and 1 simul-taneously. The shaded ellipse corresponds a strongly connected component which isterminal in T a but not in T a (cid:48) nor the etg with the added possibility of 1100 0000. It can be checked that the four types of impact listed above are disjoint andcover all possible cases. It follows as a particular consequence that a uniquenormal transition is not enough to merge attractors. Let us emphasise that aconfiguration that is recurrent in the atg can become transient with the addi-tion of synchronism (if N has a normal transition with D-impact), in particular,it can become transient in the etg . Conversely, synchronism can turn a tran-sient configuration into a recurrent one (if N has a transition with G-impact).Synchronism can however not create new attractors from scratch. Indeed, if allconfigurations of a set X ⊂ B n are transient in the atg , then in the etg as wellas in its sub-graph the atg , there necessarily exists a derivation outgoing X .The addition of x y to the atg has no or little ( i.e. F-) impact when x istransient in the atg . To change the asymptotics of N (rather than just some ofits evolutions towards it), x y needs to have G- or D-impact. And for this, inthe atg , an unstable attractor [ x ] a is needed. It can only be induced by a negativecycle in the structure G of N [14]. Further, considering Hamiltonian criticalcycles again as in Lemma 3, the last point of this section evidences the need toembed critical cycles in a larger, structural ’environment’ to obtain G- and D-impact transitions. In other terms, N must have a critical cycle C = ( V C , A C )as well as nodes i ∈ V \ V C (cid:54) = ∅ outside of it if the addition of synchronism isto significantly impact on its behaviour and change its asymptotics. Lemma 4.
Let N be a ban with no normal transitions of size smaller than itssize n . Then, any transition x y either has no impact on the asymptoticsof N or it has F-impact. In the latter case, y is a stable with an empty basin B a ( y ) = ∅ in the atg and all nodes of the structure G of N have a positive loop.Proof. Let x y = x V be a normal transition of N . Since ∀ i ∈ V , V \ { i } ⊂ U ( x ) = V , Proposition 2 implies x y i , ∀ i ∈ V . Thus, ∀ z ∈ B n , y z ⇒ x z . And either U ( y ) (cid:54) = ∅ in which case A a ( y ) ⊂ A a ( x ) (and x y hasno impact), or y is stable in which case, by Lemma 3, its basin is empty and allautomata bear a positive loop. (cid:117)(cid:116) xample 1 (D- and G-impact). Let N = { f i ; i < } be the ban of size 4 whosetransition functions and signed structure are given below: ∀ x ∈ B n , h ( x ) = x ∨ ( x ∧ ¬ x )h ( x ) = x ∨ ( ¬ x ∧ x )h ( x ) = ¬ x ∧ x h ( x ) = x ∧ ¬ x + − + + +++ − − − The etg of this ban has two attractors: one unstable and one stable (configu-ration 0000). When x = x = 1 and x = x = 0, the simultaneous update ofautomata 0 and 1 has an effect that cannot be mimicked by a series of atomicupdates ( cf. Figure 3). If it could, strongly connected component [1100] a wouldnot be terminal in the atg . From Propositions 1 and 2 we know that this isessentially due to the positive cycle of length 2 induced by automata 0 and 1.Building on this example, we can derive an example of a ban with a G-impacttransition. Indeed, consider a fifth automaton i = 4 ∈ V s.t. f ( x ) = ¬ ( x ∨ x ∨ x ∨ x ) ∧ ¬ x and let f ( x ) = h ( x ) ∨ ( x ∧ ¬ x ∧ ¬ x ) and ∀ i ∈ { , , } ,let f i ( x ) = h i ( x ). It can be checked that h ( x ) (cid:54) = f ( x ) ⇒ x = 00001 and asa consequence, adding the same normal transition as before to the atg of thisnew ban yields the transition graph below:0011 scc { x ∈ B | x ∨ x = 1 ∧ x = } scc { x ∈ B | x ∨ x = 1 ∧ x = } On the basis of the previous classification of the impact of synchronous transi-tions, we now take a more abstract point of view to propose a list of the differenttypes of sensitivity that a ban may have to (the addition of) synchronism. Nat-urally, we say that N has no sensitivity to synchronism if none of its normalsynchronous transitions has any impact ( cf. Point 1 in the previous section). Wesay that it has
F- and G-sensitivity to synchronism when, respectively, ithas normal transitions with F- and G-impact ( cf.
Point 2 and 3). When N hasormal transitions with D-impact ( cf. Point 4), two cases may occur. Indeed, let x y be a normal transition of N that has D-impact. Then, there may be an-other D-impact normal transition y (cid:48) x (cid:48) such that [ x ] a = [ x (cid:48) ] a (cid:54) = [ y ] a = [ y (cid:48) ] a , i.e. x and x (cid:48) on the one hand, and y and y (cid:48) on the other belong to the same un-stable attractors. In this case, the two normal transitions x y and y (cid:48) x (cid:48) cause attractors [ x ] a and [ y ] a of the atg to m erge ([ x ] = [ y ] = [ x ] a (cid:93) [ y ] a ). Hence, N is said to have M-sensitivity to synchronism . If there is no other normaltransition connecting [ y ] a to [ x ] a , then attractor [ x ] a is effectively d estroyed bythe addition of x y and N is said to have D-sensitivity to synchronism .This and the results presented above as well as, notably, the series of remarksmade at the end of the previous section yield Proposition 3 below.
Proposition 3.
1. Sensitivity to synchronism requires the existence of a crit-ical cycle, and thus of an positive cycle with an even length or a negativecycle with an odd length.2. G-, D- and M-sensitivity require the existence of a critical cycle of lengthstrictly smaller than the ban size as well as of a negative cycle.3. Unless N has a Hamiltonian critical cycle and positive loops on all of itsautomata, to have F-sensitivity, N also needs to have a critical cycle oflength strictly smaller than the ban size. Sensitivity to synchronism & non-monotony
Obviously, to be sensitive to synchronism, a ban must involve at least two au-tomata. It can be checked that there are no monotone ban s of size 2 that areD- or M-sensitive (we say very sensitive ) to synchronism, but there are somenon-monotone ones ( cf.
Figure 4). xx = y x = y x { , } = y Fig. 4. atg of a strongly con-nected ban N of size 2 s.t.f , f ∈ { x (cid:55)→ x ⊕ x , x (cid:55)→¬ ( x ⊕ x ) } , augmented withnormal transition x y . N is D-sensitive to synchronism. However, interestingly, the monotone, D-sensitive ban of Example 1 actually also involves non-monotone actions. Indeed, it only involves a fewmonotone individual interactions between fourautomata but these are architectured into a wid-get that can globally mimic a punctual non-monotone action in the right configuration andwith the right synchronous update of automatastates. More precisely but informally, in this wid-get, a non-monotone action is structurally splitinto two parts. These two parts consist in thetwo halves of a xor : ( x x ) (cid:55)→ x ∧ ¬ x and( x x ) (cid:55)→ ¬ x ∧ x . They are encoded separatelyin the transition functions f and f of two differ-ent automata connected by what can be a crit-ical cycle by Proposition 1. When the controlson these two parts are lifted ( i.e. when x = x = 0 so that we do indeed havef ( x ) = x ∧ ¬ x and f ( x ) = ¬ x ∧ x ), the synchronous update of automata and 1 simultaneously applies f and f . Instantly, this amounts to combin-ing influences underwent by 0 and 1 by “simulating” a or connector betweentheir transition functions, thereby outputting the global action f ( x ) ∨ f ( x ).Precisely, this puts together the two halves of a xor with a ∨ and produces aglobal non-monotone action. Examining the widget of Example 1, one can noticethat the automata that it involves have different roles. Roughly, automata 0 and1 encode the non-monotone action mentioned above. The role of automata 2and 3 is to make “use” of it and ensure the necessary unstable attractor. Thisattractor is made dependent on automata 0 and 1 by requiring x ∨ x = 1. Moreprecisely, the widget is designed so that the unstable attractor is characterisedby this condition. In the atg , if the condition becomes true, it remains true.Every configuration x such that x = x = 0 reaches the stable configuration. + s + − s s s s s ab ¬ c ¬ ab ¬ ca ¬ b ¬ c a ¬ bc ¬ abc x = ¬ a ¬ bc ¬ a ¬ b ¬ c y = abc Fig. 5.
Generic signed structure ( ∀ i, j ∈ V , s ji = sign( j, i ) = sign( i, j )) and modified atg T a (cid:48) of all monotone ban s of size 3 that are very sensitive (necessarily D-sensitive)to synchronism, e.g. N = { f : x (cid:55)→ x ∨ ( x ∧ ¬ x ) , f : x (cid:55)→ x ∨ ( ¬ x ∧ x ) , f : x (cid:55)→¬ x ∧ ( x ∨ x ) } . For all instances of these ban s, in the starting point x of the normaltransition, f (cid:0) f ( x ) f ( x ) x (cid:1) ∈ { x ⊕ x , ¬ ( x ⊕ x ) } . These remarks suggest that there is a tight relationship between significant sen-sitivity to synchronism and non-monotony . Let us add that the smallest mono-tone ban s that are sensitive to synchronism have size 3. They are monotoneencodings of the non-monotone sensitive ban s of size 2 ( cf. Figure 4). This isproven by building around a normal transition x ∈ B y ∈ B (that mustsatisfy d( x, y ) = 2 < ban s have an atg and a signed structure of the form of thoserepresented in Figure 5, and they have D-sensitivity. Conclusion and perspectives