Complexity of limit-cycle problems in Boolean networks
Florian Bridoux, Caroline Gaze-Maillot, Kévin Perrot, Sylvain Sené
CComplexity of limit-cycle problems in Boolean networks
Florian Bridoux , Caroline Gaze-Maillot , K´evin Perrot , and Sylvain Sen´e Aix Marseille Univ., Univ. Toulon, CNRS, LIS, UMR 7020, Marseille, France. Univ. Cˆote d’Azur, CNRS, I3S, UMR 7271, Sophia Antipolis, France.
Abstract
Boolean networks are a general model of interacting entities, with applicationsto biological phenomena such as gene regulation. Attractors play a central role, andthe schedule of entities update is a priori unknown. This article presents resultson the computational complexity of problems related to the existence of updateschedules such that some limit-cycle lengths are possible or not. We first provethat given a Boolean network updated in parallel, knowing whether it has at leastone limit-cycle of length k is NP -complete. Adding an existential quantification onthe block-sequential update schedule does not change the complexity class of theproblem, but the following alternation brings us one level above in the polynomialhierarchy: given a Boolean network, knowing whether there exists a block-sequentialupdate schedule such that it has no limit-cycle of length k is NP NP -complete. Boolean networks (BNs) were introduced by McCulloch and Pitts in the 1940s throughthe well known formal neural networks [17] that are specific BNs governed by a multi-dimensional threshold function. Informally, BNs are finite dynamical systems in whichentities having Boolean states may interact with each other over discrete time. Af-ter their introduction, neural networks were studied in depth from the mathematicalstandpoint. Among the main works on them are the introduction by Kleene of finiteautomata and regular expression [16], first results on the dynamical behaviors of linearfeedback shift register [13] and linear networks [8]. These researches led Kauffman andThomas (independently) from the end of the 1960s to develop the use of BNs in thecontext of biological networks modeling [15, 27], which has paved the way to numerousapplied works at the interface between molecular biology, computer science and discretemathematics. In parallel, theoretical developments were done in the framework of linearalgebra and numerical analysis by Robert [24], and in that of dynamical system theoryand computational models, which constitutes the lens through which we look at BNs inthis paper.In this context, numerous studies have already been led and have brought very im-portant results. Considering that a BN can be defined as a collection of local Boolean1 a r X i v : . [ c s . D M ] J a n unctions (each of these defining the discrete evolution of one entity over time giventhe states of the entities that influence it), it can be represented by a directed graphat the static level, classically called the interaction digraph. Moreover, as a BN is bydefinition of finite size here, it is trivial to see that the trajectory of any of its configu-rations (or global state) ends into a cycle that can be a fixed point or a limit-cycle. Themain theoretical objective in the domain is twofold: obtaining (combinatorial or alge-braic) characterizations of the dynamics of such objects, through either their definitionas collections of Boolean functions or their interaction graphs, and understanding thecomplexity of finding such characterizations.In these lines, Robert showed that retroaction cycles between entities in the in-teraction graph are necessary for a BN to have a non-trivial dynamical behavior [25]and Thomas conjectured strong relations between these retroaction cycles (well knownas positive and negative cycles) and the existence of multi-stationarity (several fixedpoints) or limit-cycles [28] which were proven later [21, 23, 22]. A notable fact aboutthese seminal works is that they underline clearly that retroaction cycles are the en-gines of behavioral complexity (or dynamical richness). More recently, a real effort hasbeen impulsed on the understanding of retroaction cycles. In particular, Demongeotet al. characterized exhaustively the behaviors of retroaction cycles and some of theirintersections [7]. Furthermore, the problem of counting the number of fixed points andlimit-cycles has mushroomed. Advances have been done concerning fixed points [3, 5].Nevertheless, due to the high dependence of limit-cycle appearance according to theupdate schedule ( i.e. the way / order under which entities are updated over time), nogeneral combinatorial results have been obtained, except for retroaction cycles [7]. Inrelation to complexity theory, the main known results based on BNs are: determining ifa BN admits fixed points is NP-complete, counting fixed points is k is NP-complete whatever the update schedule (in the class ofblock-sequential updating modes, that is updating modes defined as ordered partitionsof the set of entities). Furthermore, we show that determining if there exists a block-sequential update schedule such that a given BN admits no limit-cycles of length k isNP NP -complete.In what follows, Section 2 presents the main definitions that are used in the paper.Section 3 gives a brief state of the art of the problematic addressed. The main resultsof the paper are given in Section 4 and are followed by a conclusion developing someperspectives of this work. 2 Definitions
We denote N + the set of strictly positive integers, and [ n ] = { , . . . , n } for some n ∈ N + .For x ∈ { , } n and i ∈ [ n ], we denote x i the component i of x , and x + e i the vectorof { , } n obtained by flipping component i of x (addition is performed modulo 2). Thesymbol ⊕ is used for the binary operator exclusive or ( xor ). A Boolean network (BN) is a function f : { , } n → { , } n , that we see as n localfunctions f , . . . , f n with f i : { , } n → { , } for each i ∈ [ n ]. The interaction digraph of a BN f captures the actual dependencies among its components, and is defined as G f = ( V, A ), with V = [ n ] and( i, j ) ∈ A ⇐⇒ ∃ x ∈ { , } n : f j ( x ) (cid:54) = f j ( x + e i ) . The arcs of the interaction digraph may be assigned signs σ : A → { + , − , ±} as follows: • σ ( i, j ) = + when ∃ x ∈ { , } n : x i = 0 ∧ f j ( x ) > f j ( x + e i ), • σ ( i, j ) = − when ∃ x ∈ { , } n : x i = 0 ∧ f j ( x ) < f j ( x + e i ), • σ ( i, j ) = ± when both conditions above hold.For convenience, we may use various symbols to denote the components of the net-work, but as it will always be a finite set a bijection with [ n ] is straightforward. The size of a BN is its number of components. The configuration space is { , } n , and it remains to explain how components are up-dated. Given a BN f , a configuration x and a subset I ⊆ [ n ], we denote f ( I ) ( x ) theconfiguration obtained by updating components of I only, i.e. for any i ∈ [ n ] , f ( I ) ( x ) i = (cid:26) f i ( x ) if i ∈ Ix i otherwise.Remark that f ([ n ]) = f . A block-sequential update schedule is an ordered partition of [ n ],denoted W = ( W , . . . , W t ), and a BN f updated according to W gives the deterministicdiscrete dynamical system on { , } n defined as f ( W ) = f ( W t ) ◦ · · · ◦ f ( W ) ◦ f ( W ) . The update schedule ([ n ]) is called parallel (or synchronous ). Parenthesis are used to differentiate update schedules from iterations of a function. f ( x ) = x f ( x ) = x f ( x ) = x f ( x ) = x f ( x ) = x Figure 1: Two BNs and their respective interaction digraphs (all arcs are positive).Left: for W = ( { } , { } ) we have φ ( f ( W ) ) = 0, whereas for the parallel mode we have φ ( f ) = 1. Right: for W (cid:48) = ( { } , { , } ) we have φ ( f (cid:48) ( W (cid:48) ) ) = 1 with ↔ ,whereas for the parallel mode we have φ ( f (cid:48) ) = 0. Given that the configuration space is finite and the dynamics is deterministic, the orbit ofany configuration convergences to a fixed point (a configuration x such that f ( W ) ( x ) = x )or to a limit-cycle (a configuration x such that ( f ( W ) ) k ( x ) = x for some length k ∈ N + ,and such that ( f ( W ) ) (cid:96) ( x ) (cid:54) = x for any (cid:96) ∈ [ k − f , an update schedule W , and k ∈ N + , we denote Φ k ( f ( W ) ) the set ofconfigurations in limit-cycles of length k , i.e. Φ k ( f ( W ) ) = { x ∈ { , } n | ( f ( W ) ) k ( x ) = x and ∀ ≤ (cid:96) < k, ( f ( W ) ) (cid:96) ( x ) (cid:54) = x } and φ k ( f ( W ) ) = | Φ k ( f ( W ) ) | k the number of limit-cycles of length k . Remark that for afixed k , the quantity φ k ( f ( W ) ) may vary depending on W (see Figure 1).For retroaction cycles (such as those of Figure 1), the dynamical behavior in terms ofnumber of limit-cycles of size k , whatever the update schedule, is entirely characterizedin [7] on the basis of [11]. Remark 1.
Note that an input BN f is encoded with its local functions as propositionalformulas (see also Remark 3 at the end). We are interested in the following decision problems related to attractors in thedynamics of BNs, and especially limit-cycles. k -limit-cycle problem ( k -LC) Input: a BN f updated in parallel. Question: does φ k ( f ) ≥ Remark 2. f updated in parallel is not a limitation here, since one can transform inpolynomial time a BN f and an updated schedule W into a BN f (cid:48) updated in parallelsuch that f (cid:48) = f ( W ) (simply construct local functions of f (cid:48) from those of f and W ), aspresented in [26]. lock-sequential k -limit-cycle problem (BS k -LC) Input: a BN f . Question: does there exist W block-sequential such that φ k ( f ( W ) ) ≥ Block-sequential no k -limit-cycle problem (BS no k -LC) Input: a BN f . Question: does there exist W block-sequential such that φ k ( f ( W ) ) = 0?Fixed points are invariant over block-sequential update schedules [10], consequently 1 -LC and BS -LC are identical. However, the last two problems are not complementof each other, because there exist some instance positive in both (see Figure 1 for anexample).For the reductions giving complexity lower bounds, we need the following classicalproblems. For a formula ψ on { λ , . . . , λ n } and a partial assignment v : { λ , . . . , λ s } →{ , } for some s ∈ [ n ], we denote ψ [ v ] the substitution ψ [ λ ← v ( λ ) , . . . , λ s ← v ( λ s )]. Input: a 3-CNF formula ψ on { λ , . . . , λ n } . Question: is ψ satisfiable? ∃∀ -3-SAT Input: a 3-CNF formula ψ on { λ , . . . , λ n } and s ∈ [ n ]. Question: is there an assignment v of λ , . . . , λ s such thatall assignments of λ s +1 , . . . , λ n satisfy ψ [ v ]? is a well known NP -complete problem [14], and ∃∀ -3-SAT is NP NP -complete [20](one level above in the polynomial hierarchy). Also, note that NP NP = NP co - NP since anoracle language or its complement are equally useful. The k -limit-cycle problem is known to be NP -complete for k = 1 [9], and the fixedpoints of a BN are invariant for any block-sequential update schedule [10]. It has beenproven in [2] that given a BN f , it is NP -complete to know whether there exist two block-sequential update schedules W, W (cid:48) such that f ( W ) (cid:54) = f ( W (cid:48) ) (that is, they differ on at leastone configuration). This problem is indeed surprisingly difficult, but the proof relies ona basic construction similar to Theorem 1 for k = 1. More over, in [4], the authors studythe computational complexity of limit cycle problems. Given a BN f , an update schedule W and a limit-cycle C of f ( W ) , it is NP -complete to know whether there exists anotherupdate schedule W (cid:48) (not equivalent to W ) such that f ( W ) also has the limit-cycle C .Some variants of this problem are deduced to be NP -complete: knowing whether thesets of limit cycles are equal, and whether the sets of limit-cycles share at least oneelement. This work focuses on finding block-sequential update schedules sharing limitcycles. After writing this article, we learned that the PhD thesis of G´omez [12] containsresults of a very close flavor: given a BN f , determining whether it is possible to find a5lock-sequential update schedule W such that f ( W ) has at least one limit cycle (of anylength greater than two) is NP -complete, even when restricted to AND-OR networks.Moreover, the problem of finding a block-sequential W such that f ( W ) has only fixedpoints is NP -hard. In the sequel, we prove an analogous bound for the existence problem(Corollary 1. Our construction also has only AND-OR local functions), and a strongertight bound for the non-existence problem (Theorems 3, 4 and Corollary 2). As adifference, in our setting the length of the limit-cycle is fixed in the problem definition.It is also proven in [12] that given a BN f and two configurations x, y , is there a W suchthat f ( W ) ( x ) = y ? is an NP -complete problem.Eventually, questions on the maximum number of fixed points possible when onlythe interaction digraph of a BN is provided, have already let some complexity classeshigher than NP appear in problems related to the attractors of BNs [6]. The constructions presented in this section are gradually extended with more involvedarrangements of components, to prove complexity lower bounds from formula satisfactionproblems. The first result adapts a folklore proof for fixed points (case k = 1). Theorem 1. k -LC is NP -complete for any k ∈ N + .Proof. The problem belongs to NP because one can check in polynomial time a certificateconsisting of one configuration x ∈ { , } n of the limit-cycle of length k . Indeed, to checkthat x ∈ { , } n is in a limit-cycle of size k , it is sufficient to check that f ( x ) , . . . , f k − ( x )are different from x and that f k ( x ) equals x .To show that it is NP -hard, we present a reduction from . Given a 3-CNFformula ψ on { λ , . . . , λ n } with m clauses C , . . . , C m ∈ ( { λ , . . . , λ n }∪{¬ λ , . . . , ¬ λ n } ) ,we construct the following BN of size n + m + k . The components are { λ , . . . , λ n } ∪ { C , . . . , C m } ∪ { ψ , . . . , ψ k } and the local functions are • f λ i ( x ) = x λ i for i ∈ [ n ], • f C j ( x ) = (cid:87) λ i ∈ C j x λ i ∨ (cid:87) ¬ λ i ∈ C j ¬ x λ i for j ∈ [ m ], • f ψ ( x ) = ¬ x ψ ∧ x ψ k ∧ ( x C ∧ · · · ∧ x C m ), • f ψ i ( x ) = ¬ x ψ i ∧ x ψ i − for i ∈ { , . . . , k } .If k = 1, then we set f ψ ( x ) = ¬ x ψ ∨ ( x C ∧ · · · ∧ x C m ). An example signed interactiondigraph of this BN is presented on Figure 2.The idea is that to get a limit-cycle of length k , one is forced to find in x λ , . . . , x λ n an assignment satisfying ψ , in order to have x C j = for all j ∈ [ m ] and a configurationcycling through x φ , . . . , x φ k . Otherwise if x λ , . . . , x λ n does not satisfy ψ , then theattractor is a fixed point (except for the case k = 1). The articulation between theformula assignment and the limit-cycle of length k hinges upon f ψ .6 λ λ λ λ C C C ψ ψ ψ ψ Figure 2: Signed interaction digraph of the BN obtained for k = 4 and the 3-CNFformula ψ = ( λ ∨ λ ∨ ¬ λ ) ∧ ( ¬ λ ∨ λ ∨ ¬ λ ) ∧ ( ¬ λ ∨ ¬ λ ∨ λ ). Negative arcs ( − )are red with a flat head, positive arcs (+) are black (there are no ± arcs).Let us now prove that ψ is satisfiable if and only if the BN has a limit-cycle of length k . Suppose ψ is satisfied for v : { λ , . . . , λ n } → { , } , then the following configuration x ∈ { , } n + m + k is part of a limit-cycle of length k : • x λ i = v ( λ i ) for all i ∈ [ n ], • x C j = for all j ∈ [ m ], • x ψ = and x ψ = · · · = x ψ k = .Indeed, the state of components { λ , . . . , λ n } ∪ { C , . . . , C m } do not change, and theunique state in the cycle of components { ψ , . . . , ψ k } moves one component forwardat each step (all other components being in state ), and comes back to the initialconfiguration x in k steps, i.e. f k ( x ) = x .For the reverse direction, suppose there is a limit-cycle of length k , and let x be one ofits configurations. Remark that in any attractor, the states of components { λ , . . . , λ n } are fixed, and so are the states of components { C , . . . , C m } . As a consequence, in thelocal function f ψ , the evaluation of the part ( x C ∧ · · · ∧ x C m ) is fixed. For the sake ofcontradiction suppose that it is evaluated to , then so is x ψ , then so is x ψ , etc , and x is a fixed point (in the case k = 1 we have f ψ ( x ) (cid:54) = x ψ ). Therefore, components { C , . . . , C m } are all in state , which, according to their local functions, is possible if andonly if each of them has at least one of its predecessors in state if it appears positivelyin the corresponding clause, or in state if it appears negatively. As a conclusion thestates of components { λ , . . . , λ n } in x correspond to a valuation satisfying ψ .The second result initiates the consideration of update schedules in complexity stud-ies of the dynamics of BNs. However, with an additional existential quantification onthe update schedule the problem remains NP -complete (there was already an existentialquantification on configurations for the existence of a limit-cycle), and it turns out thatthe same construction proves it. The cycle in the interaction digraph. orollary 1. BS k -LC is NP -complete for any k ∈ N + .Proof. This problem still belongs to the class NP , as one can check in polynomial time acertificate consisting of a block-sequential W on [ n ] and one configuration x ∈ { , } n ofthe limit-cycle of length k . Indeed, it is sufficient to check that f ( W ) ( x ) , . . . , ( f ( W ) ) k − ( x )are different from x and that ( f ( W ) ) k ( x ) equals x .For the NP -hardness we use the same construction as in the proof of Theorem 1.Indeed, remark that the existential quantification on a block-sequential update schedulefits the reasoning. For the left to right direction of the if and only if we use the same x with W = [ n ]. And for the reverse direction, if ψ is not satisfiable then for any block-sequential update schedule any configuration converges to a fixed point (the upper partis always fixed, and x ψ = fixes the cycle).We have seen in Theorem 1 and Corollary 1 that with two consecutive existentialquantifications (one for a block-sequential update schedule and one for a configurationof a limit-cycle) the problem remains in NP . However, BS no k -LC corresponds to anexistential quantification (for a block-sequential update schedule) followed by a universalquantification (for the absence of a limit-cycle). The next results therefore jump onelevel above in the polynomial hierarchy. Theorem 2. BS no k -LC is in NP NP for any k ∈ N + .Proof. The problem belongs to the class NP co - NP = NP NP , as one can guess non-deterministically a block-sequential update schedule W and then check in polynomialtime (in NP ), using an oracle in co - NP , whether φ k ( f ( W ) ) = 0. Once W is fixed thislast question is indeed in co - NP , as it is the complement of k-LC , see Remark 2 andTheorem 1.The hardness proof is splitted into three results, developing some incremental mech-anisms and constructions. Theorem 3. BS no k -LC is NP NP -hard for all k even and strictly greater than .Proof. We present a reduction from ∃∀ -3-SAT . Given a 3-CNF formula ψ on { λ , . . . , λ n } with m clauses denoted as usual C , . . . , C m , and an integer s ∈ [ n ], we construct thefollowing BN of size 2 s + n + m + k + 2. The components are { Ω , ψ } ∪ { λ , . . . , λ n } ∪ { λ (cid:48) , . . . , λ (cid:48) s } ∪ { λ (cid:48)(cid:48) , . . . , λ (cid:48)(cid:48) s } ∪ { C , . . . , C m } ∪ { ψ , . . . , ψ k − } and the local functions are • f Ω ( x ) = ¬ x Ω , • f λ (cid:48) i ( x ) = f λ (cid:48)(cid:48) i ( x ) = x Ω for i ∈ [ s ], • f λ i ( x ) = x λ (cid:48) i ⊕ x λ (cid:48)(cid:48) i for i ∈ [ s ], and f λ i ( x ) = x λ i for i ∈ [ n ] \ [ s ], • f C j ( x ) = (cid:87) λ i ∈ C j x λ i ∨ (cid:87) ¬ λ i ∈ C j ¬ x λ i for j ∈ [ m ], • f ψ ( x ) = x C ∧ · · · ∧ x C m , • if i ∈ { , . . . , k − } is even then f ψ i ( x ) = (cid:26) x ψ i if x ψ = ∨ x Ω = x ψ i − k otherwise ,8 λ λ λ λ λ λ λ λ λ λ λ C C C ψ ψ ψ ψ ψ Figure 3: Signed interaction digraph of the BN obtained for k = 4, the 3-CNF formula ψ = ( λ ∨ λ ∨¬ λ ) ∧ ( ¬ λ ∨ λ ∨¬ λ ) ∧ ( ¬ λ ∨¬ λ ∨ λ ), and s = 3. Negative arcs ( − ) arered with a flat head, positive arcs (+) are black, positive-negative arcs ( ± ) are dashedwith both colors and heads. Components Ω and ψ are both connected to components ψ , ψ , ψ and ψ with arcs of sign ± . • if i ∈ { , . . . , k − } is odd then f ψ i ( x ) = (cid:26) x ψ i if x ψ = ∨ x Ω = x ψ i − otherwise .An example signed interaction digraph of this BN is presented on Figure 3.The idea is that to prevent a possible limit-cycle of length k to take place on com-ponents { ψ , . . . , ψ k − } , one is forced to solve the ∀∃ -3-SAT instance and let x ψ = inany configuration x that is part of an attractor. The existential variables are assigned inthe block-sequential update schedule (on the updates of λ i , λ (cid:48) i and λ (cid:48)(cid:48) i relative to the up-date of Ω, for i ∈ [ s ]), and the universal variables all appear in both states in attractors(thanks to the positive loops on components { λ s +1 , . . . , λ n } ).Let us now prove that there exists an assignment v : { λ , . . . , λ s } → { , } such thatall assignments v (cid:48) : { λ s +1 , . . . , λ n } → { , } verify ψ [ v ][ v (cid:48) ] ≡ , if and only if there existsa block-sequential update schedule W such that f ( W ) has a no limit-cycle of length k .Suppose there exists such an assignment v , then we define W = ( T (cid:48) , { Ω } , F (cid:48) ∪ { λ (cid:48)(cid:48) , . . . , λ (cid:48)(cid:48) s } , { λ , . . . , λ n } ∪ { C , . . . , C m } ∪ { ψ } ∪ { ψ , . . . , ψ k − } )with T (cid:48) = { λ (cid:48) i | v ( λ i ) = } and F (cid:48) = { λ (cid:48) i | v ( λ i ) = } (for i ∈ [ s ]). We claim that f ( W ) has no limit-cycle of length k . Indeed, the state of components { λ , . . . , λ s } correspondto the valuation v , because λ (cid:48) i and λ (cid:48)(cid:48) i for positive (resp. negative) variables are updatedbefore and strictly after (resp. both strictly after) component Ω flips his state when itis updated, therefore are equal (resp. not equal) when local functions f λ i compute theirxor. The states of components { Ω }∪{ λ (cid:48) , . . . , λ (cid:48) s }∪{ λ (cid:48)(cid:48) , . . . , λ (cid:48)(cid:48) s } all flip at each step (hencethe required conditions on k ), but the states of components { λ , . . . , λ s } are fixed. Thestates of components { λ s +1 , . . . , λ n } are also fixed in any attractor, to arbitrary values9mong { , } . As v satisfies ψ for any valuation v (cid:48) : { λ s +1 , . . . , λ n } → { , } , the statesof components { C , . . . , C m } and ψ are all fixed to in any attractor. Hence, in anyattractor we have: • Ω flips its state at each time step, • ψ is fixed to state .The local functions of components { ψ , . . . , ψ k − } are designed to prevent any limit-cycle of length k in this case: each of them is of the form f ψ i ( x ) = x ψ i , i.e. fixed. As aconclusion any attractor is in a limit-cycle of length 2 (cid:54) = k .For the reverse direction we consider the contrapositive, suppose that there is noassignment v : { λ , . . . , λ s } → { , } such that all assignments v (cid:48) : { λ s +1 , . . . , λ n } →{ , } verify ψ [ v ][ v (cid:48) ] ≡ . From what precedes, for any block-sequential update schedule W there exists a configuration x part of an attractor, with x λ s +1 , . . . , x λ n chosen suchthat the state of ψ is fixed to . Without loss of generality let use set x Ω = . Recallthat in any attractor the states of components { Ω } ∪ { λ (cid:48) , . . . , λ (cid:48) s } ∪ { λ (cid:48)(cid:48) , . . . , λ (cid:48)(cid:48) s } flip ateach time step, the states of components { λ , . . . , λ n } ∪ { C , . . . , C m } are fixed, and that k is even. Now if we let x ψ = x ψ = and x ψ = · · · = x ψ k − = , then we claim that x is in a limit-cycle of length k . We have to consider that in W , each ψ i may eitherbe updated before Ω, or strictly after Ω, and we also have to consider the parity of i .According to local functions f ψ i , and because the state of component ψ is fixed to , thefour cases are as follows (recall that initially x Ω = ): • if i is even and ψ i is updated before Ω, then component ψ i copies the state of ψ i − k at even time steps and is unchanged at odd time steps, • if i is even and ψ i is updated strictly after Ω, then component ψ i copies the stateof ψ i − k at odd time steps and is unchanged at even time steps, • if i is odd and ψ i is updated before Ω, then component ψ i copies the state of ψ i − at odd time steps and is unchanged at even time steps, • if i is odd and ψ i is updated strictly after Ω, then component ψ i copies the stateof ψ i − at even time steps and is unchanged at odd time steps.Now observe that in any case, thanks to the parity of i and the order of ψ i relative tocomponent Ω, when ψ i copies the state of ψ i − k , it is not possible that ψ i − k has already copied the state of ψ i − k . As a consequence, at each time step thecouple of states moves one component forward along the cycle { ψ , . . . , ψ k − } , andafter k time steps we have ( f ( W ) ( x )) k = x (and not before).In the construction above, the fact that f Ω ( x ) = ¬ x Ω imposes that any configurationconverges to a limit-cycle of even length. Component Ω acts as a clock. For k = 2 wecan adapt the construction by letting x ψ stop this clock when the formula is satisfied,then in this case any configuration converges to a fixed point. Corollary 2. BS no k -LC is NP NP -hard for k = 2 .Proof. We present again a reduction from ∃∀ -3-SAT , with a slightly modified construc-tion from Theorem 3. Given a 3-CNF formula ψ on { λ , . . . , λ n } with m clauses denoted10s usual C , . . . , C m , and an integer s ∈ [ n ], we construct the following BN of size2 s + n + m + 2. The components are { Ω , ψ } ∪ { λ , . . . , λ n } ∪ { λ (cid:48) , . . . , λ (cid:48) s } ∪ { λ (cid:48)(cid:48) , . . . , λ (cid:48)(cid:48) s } ∪ { C , . . . , C m } and the local functions are • f Ω ( x ) = ¬ x Ω ∧ ¬ x ψ , • f λ (cid:48) i ( x ) = f λ (cid:48)(cid:48) i ( x ) = x Ω for i ∈ [ s ], • f λ i ( x ) = x λ (cid:48) i ⊕ x λ (cid:48)(cid:48) i for i ∈ [ s ], and f λ i ( x ) = x λ i for i ∈ [ n ] \ [ s ], • f C j ( x ) = (cid:87) λ i ∈ C j x λ i ∨ (cid:87) ¬ λ i ∈ C j ¬ x λ i for j ∈ [ m ], • f ψ ( x ) = ( x C ∧ · · · ∧ x C m ) ∨ x ψ .In this construction, the valuation of existential variables is still encoded in theblock-sequential update schedule W , and all combinations of states on components cor-responding to universal variables still appear in attractors. Now if the formula ψ is anegative instance of ∃∀ -3-SAT , then for any W there exists a complete valuation (ex-istential and universal variables) not satisfying the formula, hence in some attractor wehave x ψ = , and component Ω flips at each step, giving a limit-cycle of length 2. Onthe contrary, if ψ is a positive instance of ∃∀ -3-SAT , then there exists a W such that allcomplete valuations satisfy the formula, hence in all attractors we have x ψ = (suppose x ψ = , then it will converge to state under update schedule W ). Finaly, if x ψ = then the attractor is a fixed point (it fixes component Ω, then λ (cid:48)(cid:48) i , λ (cid:48) i , λ i , then C j ), thusin this case there is no limit-cycle of length other than 1.The idea presented in Corollary 2 of stopping a clock when the formula is satisfied(the clock gives a limit-cycle of length k , and stopping it leads to a fixed point), can beextended to any k >
2. The challenge here is to design a clock giving a limit-cycle oflength k for any block-sequential update schedule. Theorem 4. BS no k -LC is NP NP -hard for any k > .Proof. The reduction is again from ∃∀ -3-SAT . Given a 3-CNF formula ψ on { λ , . . . , λ n } with m clauses denoted as usual C , . . . , C m , and an integer s ∈ [ n ], we construct a BNof size s + n + m + k + (cid:100) log ( BS k +1 ) (cid:101) + 3 with BS k the number of block-sequential updateschedules of size k , on the components { Ω , . . . , Ω k } ∪ { ω , . . . , ω (cid:100) log ( BS k +1 ) (cid:101) } ∪ { stop }∪ { λ , . . . , λ n } ∪ { λ (cid:48) , . . . , λ (cid:48) s } ∪ { C , . . . , C m } ∪ { ψ } . The number of block-sequential update schedules of size k equals the number of ordered partitionsof a set of k elements, also known as ordered Bell number (sequence A000670 in the
OEIS [1]). We have BS n = k (cid:88) i =0 i ! (cid:26) ki (cid:27) = k (cid:88) i =0 i (cid:88) j =0 ( − i − j (cid:32) ij (cid:33) j k using the Stirling numbers of the second kind (denoted with {} ) counting the number of surjective mapsfrom a set of i elements to a set of k elements [18]. k is fixed in the problem definition, hence we do not need to consider thegrowth of log ( BS k +1 ), which is a constant from the point of view of BS no k -LC .The idea is to build a clock of length k on the k + 1 components Ω, with some state moving forward at each step. However, it will not move forward from componentsΩ i to Ω i +1 , etc modulo k , but instead it will move forward according to the order ofcomponents Ω in the current update schedule, which is supposed to be encoded (inbinary ) on components ω (positive loops on components ω will let them take any fixedvalue in attractors). Similarly to the construction of Theorem 3, the update order of λ i , λ (cid:48) i compared to clock component Ω encodes existential variables in W , and positiveloops on universal variables let them take any fixed value in attractors. Finally, x ψ = will stop the clock. Regarding the logics of the proof, if ψ is a positive instancethen one can choose W with components Ω updated in parallel and λ (cid:48) i encoding theexistential variables to satisfy ψ , then component ψ will be in state in any attractor(thanks to the construction, regardless of the update schedule encoded on components ω ) hence leading to fixed points only. If ψ is a negative instance, then for any W we can set components ω accordingly to have a working clock of length k , and nomatter the encoding of existential variables there exists a choice of states on componentscorresponding to universal variables such that ψ is in state , letting the clock tick foreverand create a limit-cycle of length k .The local functions are • f stop ( x ) = x stop ∨ x ψ ∨ error ( x ω , . . . , x ω (cid:100) log2( BS k +1) (cid:101) ), where error ( ω ) equals when components ω do not encode a block-sequential update schedule, • f ω i ( x ) = x ω i for i ∈ [ (cid:100) log ( BS k +1 ) (cid:101) ], • for the definition of Ω i , let us consider the update schedule encoded on compo-nents ω in some configuration x , and denote j ( x ) , . . . , j k ( x ) the lexicographicallyminimal permutation of 0 , . . . , k such that Ω j ( x ) (cid:52) x ω Ω j ( x ) (cid:52) x ω · · · (cid:52) x ω Ω j k ( x ) ,where a (cid:52) x ω b means that component a is updated prior to or simultaneously withcomponent b in the update schedule encoded on components ω in configuration x ;for i ∈ { , . . . , k } , f Ω i ( x ) = ¬ x stop ∧ if i = j p ( x ) and x Ω jp − x ) = and (cid:16) x ω (cid:54) = ( { Ω , . . . , Ω k } ) or i (cid:54) = k (cid:17) otherwise,with x ω the block-sequential update schedule encoded on components ω , • f λ (cid:48) i ( x ) = x Ω for i ∈ [ s ], • for i ∈ [ s ], f λ i ( x ) = (cid:26) x Ω ⊕ x λ (cid:48) i if x Ω = x λ i otherwise,and for i ∈ [ n ] \ [ s ], f λ i ( x ) = x λ i , • f C j ( x ) = (cid:87) λ i ∈ C j x λ i ∨ (cid:87) ¬ λ i ∈ C j ¬ x λ i for j ∈ [ m ], • f ψ ( x ) = ( x C ∧ · · · ∧ x C m ). TODO:
Can we nevertheless consider it, just for fun, in this footnote? Since k is a constant we can consider any computable encoding of the block-sequential updateschedules, for example their numbering according to the lexicographic order (each subset of { Ω , . . . , Ω k } corresponds to a digit on k bits). ω ω ω ω ...Ω Ω Ω Ω Ω Ω stop λ λ λ λ λ λ λ λ C C C ψ Figure 4: Signed interaction digraph of the BN obtained for k = 5 ( BS = 4683), the3-CNF formula ψ = ( λ ∨ λ ∨ ¬ λ ) ∧ ( ¬ λ ∨ λ ∨ ¬ λ ) ∧ ( ¬ λ ∨ ¬ λ ∨ λ ), and s = 3.Negative arcs ( − ) are red with a flat head, positive arcs (+) are black, positive-negativearcs ( ± ) are dashed with both colors and heads. All components stop and ω , . . . , ω are connected to all components Ω , . . . , Ω with arcs of sign ± .An example signed interaction digraph of this BN is presented on Figure 4. First remarkthat if x stop = then x converges to a fixed point (the clock stops), hence we will considerthereafter only attractors from configurations with component stop in state .Suppose ψ is a negative instance of ∃∀ -3-SAT . For any block-sequential updateschedule W , consider a configuration x such that components ω encode the projectionof W on the clock components. If W is not the parallel update schedule, the clock hasthe following dynamics (time goes downward, one step per line):Ω j k ( x ) Ω j k − ( x ) Ω j k − ( x ) . . . Ω j ( x ) Ω j ( x ) Ω j ( x ) . . . . . . ... ... ... ... ... ... . . . . . . . . . . . . and if W is the parallel update schedule ( { Ω , . . . , Ω k } ), the clock has the following13ynamics (time goes downward, one step per line):Ω Ω Ω . . . Ω k − Ω k − Ω k . . . . . . ... ... ... ... ... ... . . . . . . . . . . . . thus we have a clock of length k in any case: • when W is not parallel the minimum component according to (cid:52) x ω and the lexico-graphical order is skipped (the state moves two components forward), • when W is parallel component Ω k is discarded (it remains in state and the clockticks on components Ω , . . . , Ω k − ).Furthermore, for i ∈ [ s ] component λ (cid:48) i goes to state exactly once every k steps, andthe relative positions of components λ (cid:48) i , λ i , Ω fixes the value of component λ i : • if ( λ (cid:48) i = W Ω ) or (Ω ≺ W λ i (cid:52) W λ (cid:48) i ) or ( λ i (cid:52) W λ (cid:48) i ≺ W Ω ) or ( λ (cid:48) i ≺ W Ω ≺ W λ i )then x λ i = , • otherwise x λ i = .Since the instance ψ is negative, for any assignment of states to components λ , . . . , λ s (corresponding to existential variables), we can set the states of components λ s +1 , . . . , λ n (corresponding to universal variables) so that at least one clause C j is not satisfied hence x C j = and x ψ = 0. As a consequence f stop ( x ) = , i.e. the clock is not stopped, andtherefore it creates a limit-cycle of length k .Suppose ψ is a positive instance of ∃∀ -3-SAT , with v : { λ , . . . , λ s } → { , } anassignment such that for all v (cid:48) : { λ s +1 , . . . , λ n } → { , } we have ψ [ v ][ v (cid:48) ] ≡ . We define W = ( T (cid:48) , { Ω , . . . , Ω k } , R ) , with T (cid:48) = { λ (cid:48) i | v ( λ i ) = } and R all the other components. We consider a casedisjunction on the starting configuration. • If components ω encode the parallel update schedule, then from what preceeds thestates of components λ i for i ∈ [ s ] encode v and component ψ will eventually bein state , so does component stop and the clock stops, leading to a fixed point. • If components ω do not encode the parallel update schedule, then from the defini-tion of local function f Ω i we will have a clock of length k + 1 (with state movingone component forward at each step, in an order given by the update scheduleencoded on components ω ). However, it does not alter the fact that the statesof components λ i for i ∈ [ s ] encode v , therefore the same deductions apply: theconfiguration converges to a fixed point.We can conclude that under update schedule W , any configuration converges to a fixedpoint hence there is no limit-cycle of length k .14 emark 3. Encoding local function as truth tables of the components it effectively de-pends on (its in-neighbors in the interaction digraph) would also lead to the same com-plexity results, because all the constructions presented for hardness results can be adaptedso that each component depends on a bounded number of components (the resulting in-teraction digraph has a bounded in-degree), given that k is a constant. We have characterized precisely the computational complexity of problems related to,given a BN, the existence or not of limit-cycles of some fixed length k , with the quantifieralternation of “does there exist an update schedule such that all configurations are notin a limit-cycle of size k ” bringing us to level Σ P of the polynomial hierarchy.Remark that all the constructions presented in our reductions (except for Theorem ?? which is subsumed by Theorem ?? ) are such that the resulting BN has either some limitcycles of size k , or only fixed points. Consequently, the same results directly hold forthe problem φ k ( f ( W ) ) is replaced by φ ≥ k ( f ( W ) ) = (cid:80) (cid:96) ≥ k φ (cid:96) ( f ( W ) ), i.e. we consider limit-cycles of length at least k instead of exactly k . With little additional work the proofsmay also be adapted to φ ≤ k ( f ( W ) ) = (cid:80) (cid:96) ≤ k φ (cid:96) ( f ( W ) ), i.e. if we consider limit-cycles oflength at most k (fixed points should be transformed into limit-cycles of length largerthat k ).Finally, if k is part of the input, is there a drastic complexity increase as observedfor problems related to the number of fixed points in [6]? The construction presented inthe proof of Theorem 4 makes heavy use of being a k constant.We hope that these first results on the complexity of deciding the existence of limit-cycles in Boolean networks opens a promising research direction, confronting the neces-sary difficulty of considering a diversity of update schedules. The lens of computationalcomplexity reveals, via the gadgets employed in lower bound constructions, mechanismsat the heart of Boolean network’s dynamical richness. The authors are thankful to project ANR-18-CE40-0002-01 “FANs”, project ECOS-CONICYT C16E01, project STIC AmSud CoDANet 19-STIC-03 (Campus France 43478PD),for their funding.
References [1] OEIS A000679. The online encyclopedia of integer sequences. https://oeis.org/A000670 .[2] J. Aracena, J. Demongeot, ´E. Fanchon, and M. Montalva. On the number of dif-ferent dynamics in Boolean networks with deterministic update schedules.
Math.Biosci. , 242:188–194, 2013. 153] J. Aracena, J. Demongeot, and E. Goles. Fixed points and maximal independentsets in AND–OR networks.
Discr. Appl. Math. , 138:277–288, 2004.[4] J. Aracena, L. G´omez, and L. Salinas. Limit cycles and update digraphs in Booleannetworks.
Disc. Appl. Math. , 161:1–12, 2013.[5] J. Aracena, A. Richard, and L. Salinas. Number of fixed points and disjoint cyclesin monotone Boolean networks.
SIAM J. Discr. Math. , 31:1702–1725, 2017.[6] F. Bridoux, N. Durbec, K. Perrot, and A. Richard. Complexity of maximum fixedpoint problem in Boolean networks. In
Proc. of CiE’2019 , volume 11558 of
LNCS ,pages 132–143, 2019.[7] J. Demongeot, M. Noual, and S. Sen´e. Combinatorics of Boolean automata circuitsdynamics.
Discr. Appl. Math. , 160:398–415, 2012.[8] B. Elspas. The theory of autonomous linear sequential networks.
IRE Trans. CircuitTheory , 6:45–60, 1959.[9] P. Floreen and P. Orponen. On the computational complexity of analyzing Hopfieldnets.
Complex Systems , 3:577–587, 1989.[10] E. Goles and S. Mart´ınez.
Neural and automata networks: dynamical behavior andapplications . Kluwer Academic Publishers, 1990.[11] E. Goles and M. Noual. Block-sequential update schedules and Boolean automatacircuits. In
Proc. of AUTOMATA’2010 , DMTCS, pages 41–50, 2010.[12] L. G´omez.
Dynamics of discrete networks with deterministic updates schedules.Application to genetic regulatory networks . PhD thesis, Univ. Concepci´on, 2015.[13] D. A. Huffman. Canonical forms for information-lossless finite-state logical ma-chines.
IRE Trans. Inform. Theory , 5:41–59, 1959.[14] R. M. Karp. Reducibility among combinatorial problems.
Complexity of ComputerComputations , pages 85–103, 1972.[15] S. A. Kauffman. Homeostasis and differentiation in random genetic control net-works.
Nature , 224:177–178, 1969.[16] S. C. Kleene. Representation of events in nerve nets and finite automata. ProjectRAND RM–704, US Air Force, 1951.[17] W. S. McCulloch and W. Pitts. A logical calculus of the ideas immanent in nervousactivity.
J. Math. Biophys. , 5:115–133, 1943.[18] M. Noual and S. Sen´e. Towards a theory of modelling with Boolean automatanetworks - I. Theorisation and observations. arXiv:1111.2077, 2011.1619] P. Orponen. Neural networks and complexity theory. In
Proc. of MFCS’1992 ,volume 629 of
LNCS , pages 50–61, 1992.[20] C. H. Papadimitriou.
Computational complexity . Addison-Wesley, 1994.[21] ´E. Remy, P. Ruet, and D. Thieffry. Graphic requirements for multistability andattractive cycles in a Boolean dynamical framework.
Adv. Appl. Math. , 41:335–350,2008.[22] A. Richard. Negative circuits and sustained oscillations in asynchronous automatanetworks.
Adv. Appl. Math. , 44:378–392, 2010.[23] A. Richard and J.-P. Comet. Necessary conditions for multistationarity in discretedynamical systems.
Discr. Appl. Math. , 155:2403–2413, 2007.[24] F. Robert. Blocs-H-matrices et convergence des m´ethodes it´eratives classiques parblocs.
Linear Algebra Appl. , 2:223–265, 1969.[25] F. Robert. It´erations sur des ensembles finis et automates cellulaires contractants.
Linear Algebra Appl. , 29:393–412, 1980.[26] F. Robert.
Discrete iterations: a metric study . Springer, 1986.[27] R. Thomas. Boolean formalization of genetic control circuits.
J. Theor. Biol. ,42:563–585, 1973.[28] R. Thomas. On the relation between the logical structure of systems and theirability to generate multiple steady states or sustained oscillations. In