On the complexity of acyclic modules in automata networks
aa r X i v : . [ c s . D M ] F e b On the complexity of acyclic modules inautomata networks
K´evin Perrot , Pacˆome Perrotin , and Sylvain Sen´e Universit´e publique Aix-Marseille Univ., Univ. de Toulon, CNRS, LIS, UMR 7020,Marseille, France
Abstract
Modules were introduced as an extension of Boolean automata net-works. They have inputs which are used in the computation said modulesperform, and can be used to wire modules with each other. In the presentpaper we extend this new formalism and study the specific case of acyclicmodules. These modules prove to be well described in their limit behav-ior by functions called output functions. We provide other results thatoffer an upper bound on the number of attractors in an acyclic modulewhen wired recursively into an automata network, alongside a diversity ofcomplexity results around the difficulty of deciding the existence of cyclesdepending on the number of inputs and the size of said cycle.
Automata networks (ANs) are a generalisation of Cellular automata (CAs).While classical CAs require a n -dimensional lattice with uniform local func-tions, ANs can be built on any graph structure, and with any function at eachvertex of the graph. They have been applied to the study of genetic regulationnetworks [16, 25, 17, 8, 10] where the influence of different genes (inhibition,activation) are represented by automata whose functions mirror together theglobal dynamics of the network. This application in particular motivates thedevelopment of tools to understand, predict and describe the dynamics of ANsin an efficient way. In the worst case, studying the dynamics of an AN ( i.e. analysing the behavior of all possible configurations of the system) will alwaystake an exponential amount of time in the size of the network. Attempts usingmainly combinatorics have been made to predict and count specific limit behav-ior of the system without enumerating the entire network’s dynamics [3, 11, 5].Other studies focused on understanding the dynamics of such complex systemsby considering them as compositions of bricks simpler to analyse [6, 23, 9] andpropose to study manners of controlling these bricks and/or systems [7, 20]. In1ine with such approaches and [13] the authors developed in [21] the formalismof modules. They are ANs with inputs, and operators called wirings that allowmodules to be composed into larger modules, and eventually into ANs. In thispaper we propose an exploration of a specific type of modules, namely acyclicmodules, which do not include cycles in their interaction graph. The presentpaper also introduces output functions, which characterise the behavior of anacyclic module as a function of the inputs of the network over time. Outputfunctions allow us to characterise the dynamics of a network while forgettingits inner structure, illustrated by Theorem 1, which shows that if two acyclicmodules have equivalent output functions, they also have isomorphic attractors.In Section 2 we propose definitions of ANs, modules and wirings. Section 3presents definitions of acyclicity in modules and related concepts and results.Finally in Section 4 we explore complexity results around acyclic modules andtheir inputs. General notations.
We denote B the set of Booleans B = { , } . For Λ analphabet, we denote Λ n the set of vectors of size n with values in Λ. For x ∈ Λ n ,we might denote x by x x . . . x n . For example, a vector x ∈ B defined suchthat x = 1, x = 0, x = 1 can alternatively be denoted by x = 101. For S an ordered set of labels, x ∈ Λ S , s in S , and f a function which takes x as an input, we might denote f ( x ) = s as a simplification of f ( x ) = x s . For G a digraph, we denote by V ( G ) the set of its vertices and by A ( G ) the setof its arcs. Let G, G ′ be two digraphs, we denote G ⊆ G ′ if and only if G isan induced subdigraph of G ′ , that is V ( G ) ⊆ V ( G ′ ) and u, v ∈ V ( G ) implies( u, v ) ∈ A ( G ) ⇔ ( u, v ) ∈ A ( G ′ ). For f : A → B , and C ⊆ A , we denote f | C the function defined over f | C : C → B such that f | C ( x ) = f ( x ) for all x ∈ C .For x ∈ Λ S , for any function f : R → S (for some set R ), we define x ◦ f as( x ◦ f ) r = x f ( r ) , for all r ∈ R . For X = ( x , x , . . . , x k ) a sequence of x i ∈ Λ S ,we define X ◦ f as the sequence ( x ◦ f, x ◦ f, . . . , x k ◦ f ). In most of ourexamples, the alphabet Λ will be B and the set S finite, hence x ∈ Λ S will beconsidered as a Boolean vector (according to some order on S ). ANs are composed by a set S of automata. Each automaton in S , or node, isat any time in a state in Λ. Gathering those isolated states into a vector ofdimension | S | provides us with a configuration of the network. More formally,a configuration of S over Λ is a vector in Λ S . The state of every automatonis bound to evolve as a function of the configuration of the entire network.Each node has a unique function, called a local function that is predefined anddoes not change over time. A local function is thus a function f defined over f : Λ S → Λ. An AN is described as a set which provides a local function to2 cb 000 001100 010 101 011110 111
Figure 1: (Left) Interaction digraph and of (right) dynamics of the network ofExample 2.1.every node in the network. Formally, an automata network F is a set of localfunctions f s over S and Λ for every s ∈ S . Example 2.1.
For
Λ = B , and S = { a, b, c } , let F be a Boolean AN with localfunctions f a ( x ) = ¬ a , f b ( x ) = a ∨ ¬ c , and f c ( x ) = ¬ c ∧ ¬ a . The configuration of an AN is updated using the local functions. The proto-col by which the local functions are applied is called its update schedule. Manydifferent update schedules exist (actually, there are an infinite number of these),and it is well known that changing the update schedule of ANs can change theobtained dynamics [22, 15, 4, 18]. The update schedule used in this paper isthe parallel update schedule, in which every node udpates its value according toits local function at each time step. Thus, considering a configuration x of anAN F , the update F ( x ) of F over x is the configuration such that for all s ∈ S , F ( x ) s = f s ( x ), where f s is the local function assigned to s in F . Example 2.2.
Following the previous example, we can see that F (000) = 111 , F (010) = 111 and that F (111) = 010 . ANs are usually represented by the influence that automata hold on eachother. As such the visual representation of an AN is a directed graph, called aninteraction digraph, whose nodes are the automata of the network, and arcs arethe influences that link the different automata. Formally, s influences s ′ if andonly if there exist two configurations x, x ′ such that ∀ r ∈ S, r = s ⇐⇒ x r = x ′ r ,and F ( x ) s ′ = F ( x ′ ) s ′ . From this, we define the interaction digraph of F as thedirected graph with nodes S such that ( s, s ′ ) is an arc of the digraph if and onlyif s influences s ′ . For instance the interaction digraph of the network developedin Example 2.1 is depicted in Figure 1.To encapsulate the entire behavior of the network, one needs to enumerateall the possible configurations the network, namely the elements of Λ S , anddescribe the global update function upon this set. This is often done via anothergraphical representation, which is another digraph, called the dynamics of thenetwork. Intuitively, this graph defines an arc from x to x ′ if and only if theupdate of the network over the configuration x results in the configuration x ′ .Formally, the dynamics of F can be represented as the digraph G with vertexset Λ S , such that ( x, x ′ ) is an arc in G if and only if F ( x ) = x ′ . The dynamicsof the network developed in Example 2.1 is presented in Figure 1.3he dynamics of a network is a large object. A commonly studied part ofthis object is called the attractors of the networks. An attractor is a sequenceof configurations which constitutes a cycle in the dynamics of the network.Alternatively, the attractors of a network can be defined as the set of non trivialstrongly connected components of its dynamics. Formally, an attractor of F isa connected component of the subdigraph G L ⊆ G , such that x is a node in G L if and only if there exists k ∈ N \ { } such that F k ( x ) = x . Notice that,classically in the domain of ANs, An attractor of size one is called a fixed point ,whereas an attractor of size greater than one is called a limit cycle . Example 2.3.
In our example, the attractors of F are the configurations and since they verify F (010) = 010 and F (111) = 111 . For any otherconfiguration, updating the network more than two times changes the state ofthe network to or . Alternatively, the configuration and form theonly non trivial strongly connected component of the dynamics of this network. Informally, modules can be described as ANs with inputs. More formally, fora given module, we introduce a new set of labels, usually denoted I , whichcontains the inputs of the module. By convention, inputs will be denoted withGreek letters. A local function of a module does not only depend on the states ofthe automata of the network, but also on the evaluations of the inputs. Inputsare not automata, and do not have a state; but it is interesting to suggestthat inputs are added nodes of the network that do not admit local functions.Formally, by considering S and I as sets of labels, and Λ as an alphabet, a module is a set which, for every s ∈ S , defines a local function f s : Λ S ∪ I → Λ. Example 2.4.
For
Λ = B , S = { a, b, c } and I = { α, β, γ } let M be a modulewith local functions f a ( x, i ) = ¬ b ∨ α , f b ( x, i ) = a ∨ ¬ c ∨ β ∨ ¬ α , and f c ( x, i ) = ¬ c ∧ ¬ γ . The digraph representation of a module is similar to that of an AN; theinputs are added for clarity as incident arrows to the nodes they influence. Forinstance, the module of Example 2.4 is illustrated in Figure 3. As well, updatinga module over the parallel update schedule is similar to updating an AN. Theinputs are introduced with specific notations which are detailed below. Let x and i be configurations over S and I respectively. The update of a module M over x and i , denoted M ( x, i ), is defined as a configuration over S such that, forall s ∈ S , M ( x, i ) s = f s ( x, i ), where f s is the local function assigned to s in M . Example 2.5.
Let us update the module M over the node configuration x =011 and the input configuration i = 000 . We compute f a ( x, i ) = ¬ ∨ , f b ( x, i ) = 0 ∨ ¬ ∨ ∨ ¬ and f c ( x, i ) = ¬ ∧ ¬ , thus giving M (011 , . Since it will be convenient to update a module over multiple iterations atonce, we will generally consider a sequence of input configurations of the form4 cb ααβ γ d e δ a cb ααβ γ d e Figure 2: Illustration of the wiring of Example 2.7. Interaction digraphs of themodules (left) M , (center) M ′ and (right) M ′′ .( i , i , . . . , i m ). For α, β, . . . the inputs of the considered module, we will denotefor convenience α , β , . . . the evaluation of those inputs in the configuration i , and so on, denoting α k , β k , . . . the evaluation of the respective inputs inthe configuration i k . We will denote by M ( x, ( i , i , . . . , i m )) the execution of m updates of the module M starting with configuration x , taking the inputconfiguration i k at update number k . Formally, it is defined recursively as: M ( x, ( i , i , . . . i m )) = M ( M ( x, i ) , ( i , . . . , i m )), with M ( x, ∅ ) = x. Modules are a formalism of composition and decomposition of ANs. As such,we define the process of composing modules together as wiring. Wirings exist intwo forms. One is recursive, and proposes the rearrangement of a single moduleby connecting inputs of the module to itself. The second type of wiring isnon-recursive, and defines the combination of two modules into one, connectinginputs of one module to the nodes of the other. When an input is connected, anyfunction depending on the value of that input relies on the state of the connectednode instead. Those two sorts of wirings were proven to be universal to composeany network from elementary parts [21]. Wiring operations are defined uponan object that specifies the operated connections, usually denoted ω which is apartial function defined from a subset of inputs of the second module to nodesof the first. Recursive wiring.
Let M be a module with label sets S and I which, for every s ∈ S , defines the local function f s . For ω : I S a partial function, we define (cid:8) ω M the module which, for every s ∈ S , defines the local function f ′ s such that: ∀ x ∈ Λ S ∪ I \ dom( ω ) , f ′ s ( x ) = f s ( x ◦ ˆ ω ) , with ˆ ω ( k ) = (cid:26) ω ( k ) if k ∈ dom( ω ) k if k ∈ S ∪ I \ dom( ω ) . Example 2.6.
For
Λ = B , S = { a, b, c } and I = { α, β, γ } let M be a modulewith local functions f a ( x, i ) = ¬ b ∨ α , f b ( x, i ) = a ∨ ¬ c ∨ β ∨ ¬ α , and f c ( x, i ) = ¬ c ∧ ¬ γ . Let us define a partial function ω : I → S such that dom( ω ) = { α, γ } ,and ω ( α ) = c and ω ( γ ) = a . The result of the recursive wiring (cid:8) ω M is a odule with label sets S ′ = S and I ′ = { β } with local functions f ′ a ( x, i ) = ¬ b ∨ c , f ′ b ( x, i ) = a ∨ ¬ c ∨ β ∨ ¬ c , and f ′ c ( x, i ) = ¬ c ∧ ¬ a . Non-recursive wiring.
Let M and M ′ be two modules with respective labelsets S, I , and S ′ , I ′ . We denote f s and f ′ s ′ the local functions defined respectivelyin M and M ′ for every s ∈ S and s ′ ∈ S ′ . For ω : I ′ S a partial function, wedefine M ω M ′ the module with label sets S ∪ S ′ and I ∪ I ′ \ dom ( ω ) which,for every s ∈ S ∪ S ′ , defines the local function f ′′ s such that: ∀ x ∈ Λ S , f ′′ s ( x ) = (cid:26) f s ( x | S ∪ I ) if s ∈ Sf ′ s ( x ◦ ˆ ω ) if s ∈ S ′ with ˆ ω ( k ) = (cid:26) ω ( k ) if k ∈ dom( ω ) k if k ∈ S ,for S = S ∪ S ′ ∪ I ∪ I ′ \ dom( ω ) . Example 2.7.
For
Λ = B , S = { a, b, c } and I = { α, β, γ } , let M be a modulewith local functions f a ( x, i ) = ¬ b ∨ α , f b ( x, i ) = a ∨ ¬ c ∨ β ∨ ¬ α , and f c ( x, i ) = ¬ c ∧ ¬ γ . Let also be S ′ = { d, e } , I ′ = { δ } and M ′ another module with localfunctions f ′ d ( x, i ) = ¬ d ∨ e ∨ δ and f ′ e ( x, i ) = ¬ e ∨ d . Let ω : I ′ → S be thefunction such that ω ( δ ) = b . The result of the non-recursive wiring M ω M ′ is the module with sets S ′′ = { a, b, c, d, e } and I ′′ = { α, β, γ } with local functions f ′′ a ( x, i ) = ¬ b ∨ α , f ′′ b ( x, i ) = a ∨¬ c ∨ β ∨¬ α , f ′′ c ( x, i ) = ¬ c ∧¬ γ , f ′′ d ( x, i ) = ¬ d ∨ e ∨ b and f ′′ e ( x, i ) = ¬ e ∨ d . (See an illustration in Figure 2.) Acyclicity is a property of the interaction digraph of the considered AN; itmeans that no node of the network influences itself, neither by a direct loop northrough the action of any cycle that would include this node. Acyclic ANs havebeen one of the first families of ANs to be studied and characterised [22]. Theirdynamical behavior is trivial: there is only one fixed point, which attracts everyother configuration. This is true under the parallel update schedule as wellas any other schedule which would eventually update every node a minimumamount of time for the stabilisation to propagate. This early result led to thesimple conclusion that cycles are essential to the complexity of their dynamics.
Acyclicity.
A module M is acyclic if its interaction digraph is acyclic. Example 3.1.
For
Λ = B , S = { a, b, c } and I = { α, β, γ } let M be a modulewith local functions f a ( x, i ) = α , f b ( x, i ) = a ∨ β ∨¬ α , and f c ( x, i ) = ¬ b ∧ a ∧¬ γ . M is acyclic. (See an illustration in Figure 3.) cb ααβ γ a cb ααβ γ Figure 3: Interaction digraph of (left) the module of Example 2.4, (right) theacyclic module of Example 3.1.The dynamics of this family of objects is simple enough to be studied, andcomplex enough to provide insights into the general dynamics of ANs. It isindeed clear that every AN can be decomposed into a recursively wired acyclicmodule. This can be done by taking a feedback arc set of the interaction digraphof the network, and producing a module that replaces every arc in the set byan input.As an acyclic module has no loop or cycle in its influences, it can support nolong lasting memory used for computation. As such the behavior of any nodein the network can be understood as a function of only the evaluation of theinputs in its last iterations. This function is called an output function and howmuch it must look in the past to make its prediction is called the delay of thefunction.For M a module with k inputs, an output function O with delay m is afunction defined over a sequence of inputs ( i , i , . . . , i m ). Each node of a net-work defines its own output function, similarly to how it defines a local function.The output function of a node always has minimal delay and will depend onthe output functions defined by the nodes which influence it. In other terms,if node a influences node b , then whatever output function which predicts thevalue of a based only on inputs will be useful to predict the evaluation of b oneiteration later. As such b does not directly depend on the output function of a ,but on the output function of a with incremented delay.Output functions are sufficient to describe the behavior of the entire modulefrom the inputs after a given amount of time. This fact is illustrated by theProperty 3.1 below. Node output.
Let M be an acyclic module. For every s ∈ S , we define theoutput function of s , denoted O s , as the output function with minimal delay m such that for any sequence of inputs J = ( i , i , . . . , i m ) and any configuration x , M ( x, J ) s = O s ( J ) . Example 3.2.
For
Λ = B , S = { a, b, c } and I = { α, β, γ } let M be a modulewith local functions f a ( x, i ) = α , f b ( x, i ) = a ∨ β ∨¬ α , and f c ( x, i ) = ¬ b ∧ a ∧¬ γ .The module M verifies the following output functions : O a = α , which has delay , O b = α ∨ β ∨ ¬ α , which has delay , and O c = ¬ α ∧ ¬ β ∧ α ∧ α ∧ ¬ γ ,which has delay . Property 3.1.
Let M be an acyclic module. For every s ∈ S , s has one andonly one output function O s . roof. We first claim that there always exist an output function for any node:
Claim 3.1.
Let M be an acyclic module with k inputs. For every s ∈ S , thereexists an output function O s with delay m which for any sequence of inputs J = { i , i , . . . , i m } and any initial configuration x verifies M ( x, J ) s = O s ( J ) . Let us define the incrementation of an output function.
Output function incrementation.
Let O be an output function of delay m .The incrementation of O is the output function of delay m + 1 denoted O +1 such that O +1 ( i , i , . . . , i m +1 ) = O ( i , i , . . . , i m +1 ) for any sequence of inputconfigurations ( i , i , . . . , i m +1 ) . To prove 3.1, see that M is an acyclic module, therefore there exists a node s ∈ S such that s is not influenced by any node in S (but possibly by someinputs). As a consequence, there exists an output function O s which simplyequals f s , and has a delay of j s = 1. Now for the induction, consider themodule M without some set of nodes S ′ ⊂ S such that for each node s ′ ∈ S ′ we have already defined an output function O s ′ with delay j s ′ . Since it is stillacyclic there exists a node s ∈ S \ S ′ such that s is not influenced by any nodein S \ S ′ (but possibly by some inputs and some nodes in S ′ ). As a consequence,there exists an input function O s which computes the local function f s , replacingthe evaluation of any node s ′ ∈ S ′ by the incrementation of the output function O s ′ , and has a delay of j s = 1 + max { j s ′ | s ′ ∈ S ′ } .We now make the following claim: Claim 3.2.
Let M be an acyclic module. Let s ∈ S , and O s and O ′ s be twooutput functions with respective delays m and m ′ such that for any two sequencesof inputs J, J ′ of size m and m ′ respectively and any initial configuration x , M ( x, J ) s = O s ( J ) and M ( x, J ′ ) s = O ′ s ( J ) . If m = m ′ , then O s = O ′ s . To see this is true, suppose m = m ′ . This implies that J and J ′ are ofthe same size. For any J such that J = J ′ , we verify O s ( J ) = M ( x, J ) s = M ( x, J ′ ) s = O ′ s ( J ′ ) = O ′ s ( J ). Therefore O s = O ′ s .We conclude by stating that any two different minimal output function for s would provide a contradiction with claim 3.2.Property 3.1 can be further refined to propose the following result, whichstates that two networks have the same attractors if and only if the modules theycan be decomposed into have the same number of inputs and the same outputfunctions on the nodes on which those inputs are wired. As such, modules canbe considered as black boxes which are to be considered equivalent in their limitbehavior, as long as they share the same output functions, according to somebijection between their inputs. Theorem 1.
Let M and M ′ be two acyclic modules, with T and T ′ subsets oftheir nodes such that | T | = | T ′ | . If there exists g a bijection from I to I ′ and a bijection from T to T ′ such that for every s ∈ T , O s and O ′ h ( s ) have samedelay, and for every input sequence J with length the delay of O s , O s ( J ) = O ′ h ( s ) ( J ◦ g − ) then for any function ω : I → T , the networks (cid:8) ω M and (cid:8) h ◦ ω ◦ g − M ′ haveisomorphic attractors (up to the renaming of automata given by h ).Proof. First remark that ω has domain I hence it wires all inputs of M andtherefore (cid:8) ω M is an automata network with a dynamics and attractors. Fur-thermore g is a bijection from I to I ′ hence the same applies to (cid:8) h ◦ ω ◦ g − M ′ .Let us denote F = (cid:8) ω M and F ′ = (cid:8) h ◦ ω ◦ g − M ′ for simplicity, with G and G ′ the dynamics restricted to their respective attractors. We want to show that G and G ′ are isomorphic.For x ∈ Λ S , we define the input sequence of length k generated by x , denotedˆ J ( x ) k , as the sequence which verifiesˆ J ( x ) kℓ = F ℓ − ( x ) | T ◦ ω , for 1 ≤ ℓ ≤ k. Intuitively, the sequence ˆ J ( x ) k records the evaluation of the network’s outputson T , over k consecutive updates, starting with configuration x . Claim 3.3.
Let k be such that ∀ s ∈ S with d s ≤ k , for d s the delay of theoutput function O s in M . For J an input sequence of length k , the evaluationof M ( x, J ) is always the same, regardless of the starting configuration x ∈ Λ S . To see that this is true, apply Property 3.1 and consider that M ( x, J ) s = O s ( J ), for every s ∈ S . This computation is properly defined as per the defini-tion of the length of J . It follows that the computation of M ( x, J ) only dependson J . Based on this fact, we will denote M ( x, J ) = M ( J ) in the rest of thisdemonstration when applicable, that is, when no output function of M has adelay greater than k . Claim 3.4.
Let J be an input sequence of length k such that the configuration M ( J ) is defined. ˆ J ( M ( J )) k = J ⇒ M ( J ) ∈ V ( G ) . This Claim states that if the configuration M ( J ), which is obtained by up-dating any configuration x in M with the input sequence J , generates the inputsequence J , then M ( J ) is a configuration which belongs to an attractor of F .Let us denote x = M ( J ). By hypothesis, ˆ J ( x ) k = J . It follows that: F k ( x ) = F ( F k − ( x )) = M ( F k − ( x ) , F k − ( x ) | T ◦ ω )= M ( M ( . . . M ( x, F ( x ) | T ◦ ω ) . . . , F k − ( x ) | T ◦ ω ) , F k − ( x ) | T ◦ ω )= M ( M ( . . . M ( x, ˆ J ( x ) k ) . . . , ˆ J ( x ) kk − ) , ˆ J ( x ) kk )= M ( x, ˆ J ( x ) k ) = M ( x, J ) = M ( J ) = x which implies that F k ( x ) = x and x is part of an attractor which length divides k , hence the Claim holds. 9 laim 3.5. Let x ∈ V ( G ) . There exists x ′ ∈ V ( G ′ ) such that ˆ J ( x ) k ◦ g − =ˆ J ( x ′ ) k , for every k ∈ N . This Claim implies that, for any configuration x in an attractor of F , thereexists a configuration x ′ in an attractor of F ′ which generates an input sequenceˆ J ( x ′ ) k equivalent to the input sequence ˆ J ( x ) k up to the bijection g , and thatholds for any length k .To prove it, consider x ∈ V ( G ) and let us take k greater than the the delayof any output function in M and M ′ ; and such that F k ( x ) = x . We consider theinput sequences ˆ J ( x ) k and ˆ J ( x ) k ◦ g − . Claim 3.3 implies that M ′ ( ˆ J ( x ) k ◦ g − )is a well defined configuration over M ′ , which we shall denote x ′ . Let us provethat ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k . By definition we know thatˆ J ( x ) k ◦ g − = F ( x ) | T ◦ ω ◦ g − = x | T ◦ ω ◦ g − while ˆ J ( x ′ ) k = F ′ ( x ′ ) | T ′ ◦ h ◦ ω ◦ g − = x ′ | T ′ ◦ h ◦ ω ◦ g − = M ′ ( ˆ J ( x ) k ◦ g − ) | T ′ ◦ h ◦ ω ◦ g − . Let us note that for any s ′ ∈ T ′ , M ′ ( ˆ J ( x ) k ◦ g − ) s ′ = O ′ s ′ ( ˆ J ( x ) k ◦ g − ) whichequals O h − ( s ′ ) ( ˆ J ( x ) k ) by the hypothesis of the Theorem. It follows that M ′ ( ˆ J ( x ) k ◦ g − ) | T ′ ◦ h = M ( ˆ J ( x ) k ) | T ◦ h − ◦ h = x | T and this implies that ˆ J ( x ′ ) k = x | T ◦ ω ◦ g − = ˆ J ( x ) k ◦ g − therefore ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k .This marks the first step of the induction to prove ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k . Letus state the induction hypothesis thatˆ J ( x ) k [1 ,ℓ ] ◦ g − = ˆ J ( x ′ ) k [1 ,ℓ ] , for ℓ < k. We now prove that it implies ˆ J ( x ) k [1 ,ℓ +1] ◦ g − = ˆ J ( x ′ ) k [1 ,ℓ +1] . To prove it, weonly need to prove ˆ J ( x ) kℓ +1 ◦ g − = ˆ J ( x ′ ) kℓ +1 . Let · denote the concatenation oftwo sequences. We know thatˆ J ( x ) kℓ +1 ◦ g − = F ℓ ( x ) | T ◦ ω ◦ g − = M ( x, ˆ J ( x ) k [1 ,ℓ ] ) | T ◦ ω ◦ g − = M ( M ( ˆ J ( x ) k ) , ˆ J ( x ) k [1 ,ℓ ] ) | T ◦ ω ◦ g − = M ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) | T ◦ ω ◦ g − J ( x ′ ) kℓ +1 = F ′ ℓ ( x ′ ) | T ′ ◦ h ◦ ω ◦ g − = M ′ ( x ′ , ˆ J ( x ′ ) k [1 ,ℓ ] ) | T ′ ◦ h ◦ ω ◦ g − = M ′ ( x ′ , ˆ J ( x ) k [1 ,ℓ ] ◦ g − ) | T ′ ◦ h ◦ ω ◦ g − = M ′ ( M ′ ( ˆ J ( x ) k ◦ g − ) , ˆ J ( x ) k [1 ,ℓ ] ◦ g − ) | T ′ ◦ h ◦ ω ◦ g − = M ′ (( ˆ J ( x ) k ◦ g − ) · ( ˆ J ( x ) k [1 ,ℓ ] ◦ g − )) | T ′ ◦ h ◦ ω ◦ g − = M ′ (( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) ◦ g − ) | T ′ ◦ h ◦ ω ◦ g − . As the sequence ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) ◦ g − is at least of length k , we can use it tocompute the result of output functions. From the hypothesis of the Theorem itfollows that for every s ′ ∈ T ′ , M ′ (( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) ◦ g − ) s ′ = O ′ s ′ (( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) ◦ g − )= O h − ( s ′ ) ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] )= M ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) h − ( s ′ ) which, using again the hypothesis of the Theorem to relate M and M ′ , impliesˆ J ( x ′ ) kℓ +1 = M ′ (( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) ◦ g − ) | T ′ ◦ h ◦ ω ◦ g − = M ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) T ◦ h − ◦ h ◦ ω ◦ g − = M ( ˆ J ( x ) k · ˆ J ( x ) k [1 ,ℓ ] ) T ◦ ω ◦ g − = ˆ J ( x ) kℓ +1 ◦ g − and concludes the induction, therefore ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k . It follows thatˆ J ( M ′ ( ˆ J ( x ) k ◦ g − )) k = ˆ J ( x ′ ) k = ˆ J ( x ) k ◦ g − , which implies by Claim 3.4 that x ′ ∈ V ( G ′ ), and that x ′ is in an attractor which size divides k , just like x .This concludes our proof of Claim 3.5 for k big enough, but remark that as aconsequence it holds for any k ∈ N .Observe a symmetric sequence of arguments to prove that for every x ′ ∈ V ( G ′ ), there exists x ∈ V ( G ) such that ˆ J ( x ′ ) k ◦ g = ˆ J ( x ) k . It follows that forany x ∈ V ( G ), there exists a unique x ′ ∈ V ( G ′ ) such that the above relationholds. This is true since if there existed x ′ , x ′′ ∈ V ( G ′ ) such that ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k , ˆ J ( x ) k ◦ g − = ˆ J ( x ′′ ) k , and x ′ = x ′′ , then it would follow thatˆ J ( x ′ ) k = ˆ J ( x ) k ◦ g − = ˆ J ( x ′′ ) k ◦ g ◦ g − = ˆ J ( x ′′ ) k . Since x ′ , x ′′ ∈ V ( G ′ ), for alarge enough k multiple of the sizes of the attractors containing x ′ and x ′′ , wewould have x ′ = M ′ ( ˆ J ( x ′ ) k ) = M ′ ( ˆ J ( x ′′ ) k ) = x ′′ , a contradiction.Let us therefore denote ˆ h : V ( G ) → V ( G ′ ) the bijection which to any x ∈ V ( G ) associates x ′ ∈ V ( G ′ ) such that ˆ J ( x ) k ◦ g − = ˆ J ( x ′ ) k . This implies thatˆ h ( x ) = M ′ ( ˆ J ( x ) k ◦ g − ), for k larger than the delay of any output function in M and M ′ , and multiple of the size of the attractors which contain x and ˆ h ( x ).11et us prove that ˆ h is an isomorphism from G to G ′ . We need to prove that,for any x ∈ V ( G ), ˆ h ( F ( x )) = F ′ (ˆ h ( x )).Let x ∈ V ( G ) and k a multiple of the length of the attractor x is part of,such that k is greater than any delay of any output function in both M and M ′ .It follows thatˆ h ( F ( x )) = M ′ ( ˆ J ( F ( x )) k ◦ g − )= M ′ (( F ( F ( x )) | T ◦ ω, F ( F ( x )) | T ◦ ω, . . . , F k − ( F ( x )) | T ◦ ω ) ◦ g − )= M ′ (( F ( x ) | T ◦ ω, F ( x ) | T ◦ ω, . . . , F k ( x ) | T ◦ ω ) ◦ g − )= M ′ (( F ( x ) | T ◦ ω ◦ g − , F ( x ) | T ◦ ω ◦ g − , . . . , F k ( x ) | T ◦ ω ◦ g − )) . Let us consider an individual element of the above sequence, F ℓ ( x ) | T ◦ ω ◦ g − .It follows that for every s ∈ S , F ℓ ( x ) s = M ( x, ˆ J ( x ) ℓ ) s = M ( M ( ˆ J ( x ) k ) , ˆ J ( x ) ℓ ) s = M ( ˆ J ( x ) k · ˆ J ( x ) ℓ ) s = O s ( ˆ J ( x ) k · ˆ J ( x ) ℓ )= O ′ h ( s ) (( ˆ J ( x ) k · ˆ J ( x ) ℓ ) ◦ g − )= M ′ (( ˆ J ( x ) k · ˆ J ( x ) ℓ ) ◦ g − ) h ( s ) = M ′ ( M ′ ( ˆ J ( x ) k ◦ g − ) , ˆ J ( x ) ℓ ◦ g − ) h ( s ) = M ′ (ˆ h ( x ) , ˆ J ( x ) ℓ ◦ g − ) h ( s ) = F ′ ℓ (ˆ h ( x )) h ( s ) which implies that F ℓ ( x ) | T ◦ ω ◦ g − = F ′ ℓ (ˆ h ( x )) | T ′ ◦ h ◦ ω ◦ g − . This, appliedto the previous development, givesˆ h ( F ( x )) = M ′ (( F ′ (ˆ h ( x )) | T ′ ◦ h ◦ ω ◦ g − , F ′ (ˆ h ( x )) | T ′ ◦ h ◦ ω ◦ g − , . . .. . . , F ′ k (ˆ h ( x )) | T ′ ◦ h ◦ ω ◦ g − ))= M ′ ( ˆ J ( F ′ (ˆ h ( x ))))= F ′ (ˆ h ( x ))and concludes the proof of the Theorem.Output functions are a characterisation of the behavior of acyclic moduleswhich is enough to understand their limit dynamics under parallel schedule.This characterisation behaves in expected ways under non-recursive wirings.Taking two acyclic modules and wiring them non-recursively makes a modulewhose output functions are deducible from the output functions of the initialacyclic module. We now state a result which provides an upper bound on thenumber of attractors of each size of an AN, which is wired from a module with k inputs. 12 heorem 2. Taking an acyclic module with k inputs and wiring all inputsrecursively gives an AN. Let us denote a ( k, c ) the number of attractors of size c of its dynamics. We state a ( k, c ) ≤ A ( k, c ) , with: A ( k,
1) = | Λ | k and A ( k, c ) = | Λ | kc − X c ′ Let us consider an acyclic module M with k inputs. Consider a wiring ω over M such that dom( ω ) = I , for I the set of k inputs of M . Finally considerthe dynamics of the Automata Network F = (cid:8) ω M . Let us denote ω ( I ) and calloutput set the set defined ω ( I ) = { ω ( α ) | α ∈ I } . We remark the following fact: | ω ( I ) | ≤ | I | = k (1)Let us consider an attractor X = { x , x , . . . , x c } over F . By definition F ( x i ) = x i +1 for i < c and F ( x c ) = x . For R ⊆ S , and x a vector over S withvalues in Λ, we define x | R the projection of this vector over R . By extension, X | R denotes the projection of the attractor X . Provided another such attractor X ′ of same size, we make the following claim. Claim 3.6. X | ω ( I ) = X ′ | ω ( I ) ⇒ X = X ′ . To see this is true, let us assume that X | ω ( I ) = X ′ | ω ( I ) . Since M is acyclicby definition, we know that there exists a non empty set of nodes S ⊆ S suchthat every s ∈ S is only influenced by inputs and not by any other node. Thismeans that assuming X | ω ( I ) = X ′ | ω ( I ) implies X | ω ( I ) ∪ S = X ′ | ω ( I ) ∪ S . Nowconsider that, after the same acyclicity hypothesis, there exists a non-empty set S ⊆ S of nodes which are only influenced by inputs, and nodes in S , whichimplies X | ω ( I ) ∪ S ∪ S = X ′ | ω ( I ) ∪ S ∪ S . The claim follows by induction.This claim allows us to prove that there can only be as many attractors ofsize c in F as there is distinct X | ω ( I ) . This provides us with a weaker form ofthe result: a ( k, c ) ≤ | Λ | kc (2)Let X be one of the | Λ | kc possible sequence of c configurations. Let us assumethat F ( x i ) = x i +1 for i < c and F ( x c ) = x . By definition, if there exists i, j such that i = j and x i = x j , the sequence X will be periodic. This impliesthe existence of a smaller sequence X ′ such that X = X ′ q for q ∈ N . Inother words, for every possible proper attractor X ′ such that the size of X ′ divides c , there exists a sequence X = X ′ c | X ′| which is not an attractor of F bydefinition. Using this fact and 2, we conclude that a ( c, k ) is not greater than | Λ | kc − P c ′ Let M be a one-to-one module. The one-to-one module M min with a minimum number of nodes and which defines the same output functionas M is of size d , for d the delay of the output function of M .Proof. First we can prove that we cannot construct a module with a size smallerthan the delay of its output function. This is easily shown as there need to bea line of at least d in size in the network’s interaction digraph.To prove that such a minimal network always exists, simply construct it byusing d − d .The last node computes the desired output function and takes the values fromthe input directly for a delay of 1, or from the rest of the network for a delayfrom 2 to d .An example of the application of Theorem 3 is illustrated in Figure 4. Thisconstruction is polynomial in time, and bears strong resemblances with theobjects known as Feedback Shift Registers [12].14 Complexity Results This section presents complexity results that have been obtained around out-put functions, and the difficulty of the analysis of the dynamics of acyclicmodules after being recursively wired into a complete network. Remark thatthese questions have been widely addressed in the context of threshold BooleanANs [2, 14, 19]. Such a wiring will sometimes be denoted as a complete recur-sive wiring of the module. A module is encoded into the input of a decisionproblem as the list of its local functions written in propositional logic. As suchthe computation of the output functions of an acyclic module is comparable tothe computation of a circuit.Let us provide a few decision problems on the dynamics of a network obtainedfrom a recursively wired acyclic module. ◮ Acyclic Module Attractor Problem Input: An acyclic module M with k inputs and n nodes, a function ω which defines a complete recursive wiring over M , and anumber c encoded in unary. Question: Does there exist an attractor of size c in the dynamics of (cid:8) ω M ? ◮ One-to-one Module Attractor Problem Input: A one-to-one module M with n nodes, a function ω whichdefines a complete recursive wiring over M , and a number c encoded in unary. Question: Does there exist an attractor of size c in the dynamics of (cid:8) ω M ? ◮ Acyclic Module Fixed Point Problem Input: An acyclic module M with k inputs and n nodes, and a func-tion ω which defines a complete recursive wiring over M . Question: Does there exist a configuration x such that (cid:8) ω M ( x ) = x ? ◮ One-to-one Module Fixed Point Problem Input: A one-to-one module M with n nodes, and a function ω whichdefines a complete recursive wiring over M . Question: Does there exist a configuration x such that (cid:8) ω M ( x ) = x ?Those four problems are variations of the same question under differentsets of constraints. The first problem, the Acyclic Module Attractor Problem,generalises the other three decision problems, while the One-to-one ModuleFixed Point Problem is a specific case of the other three decision problems. Weprovide our complexity analysis of those problems in a way that mirrors thisdiamond-like structure. Theorem 4. The Acyclic Module Attractor Problem can be solved in time O ( f ( k × c ) q ( n )) for some function f and q a polynomial, i.e. it is fixed pa-rameter tractable. roof. We construct an algorithm which iterates all of the possible input se-quences of size c . We then execute the network on each sequence and checkif the outputs correspond to the given input. This process scales polynomiallywith the size of the network, but exponentially with the size of the attractorand the number of inputs.Formally, this algorithm checks all of the | Λ | k × c possible sequences of inputconfigurations for k inputs and of length c . To check if an input configuration J describes an attractor of size c in (cid:8) ω M , simply update module M with aninput sequence composed as the repetition of the sequence J until the obtainedsequence is at least as long as the largest delay in an output function of M . Anattractor in (cid:8) ω M will be obtained if for every input α , the sequence of valuesof the node ω ( α ) over time is identical to the sequence of values of the input α . This procedure only requires in the worst case the evaluation of the entirenetwork c times and k checks at each step, which is polynomial in n × k × c .Similarly, every possible attractor of size c in (cid:8) ω M has a correspondinginput sequence in M . To see that this is true, simply construct an input sequence J which for every input α defines the i -th evaluation of input α as the evaluationof node ω ( α ) in the i -th configuration of the attractor.By checking every possible input sequence for k inputs and of length c , weconclude on the existence of an attractor of size c in (cid:8) ω M . This algorithmis of complexity O ( | Λ | k × c r ( n × k × c )), for r a polynomial, which implies thatthere exists f a function and q a polynomial such that the complexity of thisalgorithm is O ( f ( k × c ) q ( n )). Theorem 5. The One-to-one Module Attractor Problem is NP-complete.Proof. In this proof we provide a reduction from the SAT problem which forany formula with m variables, constructs a module of size 3 m + 1. The first 3 m nodes encode the input and the last node checks the evaluation. If at any pointthe formula is evaluated at false or if the encoding is wrong, the whole networkstabilises to a fixed point. If the encoding is correct and the evaluation positive,the configuration will shift in the network, providing an attractor of size 3 m + 1.The existence of this attractor is proven equivalent to the satisfiability of theformula.First see that this problem is in NP as, providing any configuration, we canverify that it is part of a cycle of size c by updating the network c times (eachupdate requires to evaluate n local functions) and making at most c comparisonsper step, for an overall polynomial time in the input size.To prove that the problem is NP-hard, we present a reduction from the SATproblem. Given a formula f on m variables v , . . . , v m , we will construct a one-to-one module on m + e + 1 nodes (for some e upper bounded by a constant)such that, when the output is wired to the input, there exists a cycle of size c = m + e + 1, if and only if there exists a valuation satisfying f .The one-to-one module, denoted M , is composed of two parts. The firstpart is a shifting tape , which is composed of m + e nodes t , . . . , t m + e with e the smallest number such that m + e + 1 is a prime number (the value of e is16 t t . . . t m t m +1 . . . t m + e q Figure 5: Module M in the proof of Theorem 5. If f has a satisfying valuationthen node q can let the shifting tape of size m + e become a rotating tape ofsize m + e + 1, otherwise f q evaluates to 0 and any configuration converges tothe fixed point 0 m + e +1 .at most 2 m according to the Bertrand–Chebyshev theorem [24], and one canfind it in polynomial time thanks to the well-known algorithm from [1]). For1 < k ≤ m + e we define the local functions f t k ( x ) = t k − , and f t ( x ) = α with α the only input of the network. For i ∈ { , . . . , m } the state of node t i encodesthe evaluation of variable v i .The second part of the network is composed of a unique node denoted q ,the output node to be wired to input α , which has the role of either letting theshifting tape of size m + e become a shifting tape of size m + e + 1, or stoppingthe process and make the configuration converge to 0 m +1 . Its local function is: f q ( x ) = x t m + e if nodes t , . . . , t m of the shifting tape encodea valuation satisfying f or a shift may encode a valuation satisfying f ,0 otherwise.Since module M is acyclic node q cannot know its own state, but it knows thestate of all other nodes. Therefore the second condition of the disjunction ischecked as follows: node q tries, for x q = 0 and for x q = 1, and for any k from 1to m + e , whether cyclically shifting the configuration (considering that q follows t m + e and preceeds t ) by k units can give a shifting tape encoding a valuationsatisfying f on the states of nodes t , . . . , t m ; if any combination of state for x q and shift k gives a shifting tape encoding a valuation satisfying f then thecondition “a shift may encode a valuation satisfying f ” is true.This construction is illustrated in Figure 5. It has polynomial size, as thelocal functions of the c = m + e + 1 nodes can be expressed with propositionalformulas of size polynomial in f and m + e (naively for f q with a disjunction of m + e + 1 terms, each containing a copy of f ).If f has a satisfying valuation, then some configuration x encoding thisvaluation on nodes t , . . . , t m of the shifting tape belongs to a cycle of size c .Indeed, in this case x is cyclically shifted by one unit at each step along the c = m + e + 1 nodes of (cid:8) M , and by taking x q = x configuration x cannot bea fixed point therefore m + e + 1 prime implies that F c ′ ( x ) = x for all c ′ < c .17f f has no statisyfing valuation, then f q ( x ) = 0 for any x and (cid:8) M has onlyone attractor which is a fixed point, 0 m + e +1 . Theorem 6. The Acyclic Module Fixed Point Problem is NP-complete.Proof. This proof provides a reduction from the SAT problem. In this reduction,the obtained module will stabilise only if a given node, which computes a SATformula, has constant value 1.First see that this problem is in NP since, given any configuration, verifyingthat it is a fixed point can be done by updating the whole network once, whichis done in polynomial time in the size of its encoding.To see this problem as NP-hard we present a reduction from the SAT prob-lem. given a formula f , we construct a module with one node for each variablein f . Each of these nodes are wired to themselves by the wiring ω , formingidentity local functions of the form f a ( x ) = a . Then we add two other nodesto the module. One, named solver , computes f from the states of nodes cor-responding to variables. The second, named oscillator , has local function f oscillator ( x ) = ¬ solver ∧ ¬ oscillator . This is constructed via an inputwhich is wired onto oscillator by ω .Every node except solver and oscillator have a fixed state, thereforethe existence of a fixed point only depends on the evaluation of solver and oscillator . The solver node has a fixed state after one iteration, correspond-ing to the evaluation of formula f according to the states of variables nodes.Consequently the existence of a fixed point only depends on the behavior of the oscillator node, which by definition will oscillate as long as the evaluation ofthe solver node is 0. We conclude that the existence of a fixed point in the ANobtained by wiring this module according to ω is equivalent to the existence ofa positive evaluation of the formula f . This construction being polynomial inthe size of the formula, we conclude that the problem is NP-hard. Corollary 4.1. The One-to-one Module Fixed Point Problem is in P.Proof. This is an application of Theorem 4.The above stated results imply that the size of the network is not a mean-ingful parameter in the difficulty of the task of finding attractors. Thereom 4shows that the two parameters which apply this effect are the size of the desiredattractor and the number of inputs the network bears when seen as an acyclicmodule. In other terms this second parameter is the level of interconnectivityof the network. Theorems 5 and 6 prove that this caracterisation is tight. To-gether, these four theorems provide a new perspective on a known fact; thatcycles in ANs are crucial for complexity to arise.18 Acyclic Module Output Construction Problem Input: A set { M , M , . . . , M ℓ } of acyclic modules, and O an outputfunction encoded in a lookup table. Question: Does there exist a set of non-recursive wirings ω which canconstruct an acyclic module from M , M , . . . , M ℓ such that O is an output function of the obtained module? Theorem 7. The Acyclic Module Output Construction Problem is NP-complete.Proof. We provide a reduction from the SAT problem. We ask for the construc-tion of an output function via the wiring of two modules with a unique constantfunction ‘0’ and ‘1’ respectively, and a bigger module which executes a computa-tion from its inputs based on the formula, such that the target output functionis obtained by non-recursive wirings if and only if the formula is satisfiable.This decision problem is in NP since, given the non-recursive wiring and thenode which carries the target output function, the verification can be done inpolynomial time. Note that the target output function is provided as a lookuptable, and that checking the egality of two functions given as lookup tables canbe done in a single pass, which is polynomial in time.To prove that this problem is NP-hard, take a SAT formula f , and constructthe following instance of the present decision problem: the set of modules is { M , M , M f } . Modules M and M have no input and only one node whosefunction is the constant 0 and 1 respectively. The module M f has as manyinputs as the formula f as variables, plus one denoted α , and only one nodewhich computes f ∧ α using inputs corresponding to variables to compute f .The target output function O is the identity (on one input) with delay 1.For this instance to be positive, there has to be some wirings reducing thefunction f ∧ α to the identity (modules M and M have no input hence cannotproduce O ), meaning that the formula is satisfiable: either α is not wired and f reduces to 1; or α is wired (to 1) and f reduces to the identity on one variable,hence evaluating this last variable to 1 satisfies f .Conversely, if f is satisfiable then wiring inputs corresponding to variablesaccording to a satisfiable assignment reduces the local function of module M f to α , i.e. this node has the target output function O . Automata Networks are complex systems, the exhaustive study of which requiresan amount of resources exponential in the size of the network. By defining andstudying acyclic modules we propose an innovative way of approaching thisquestion. Theorem 1 proposes the reduction of the limit dynamic of a networkto the output functions of an acyclic module which composes it. We thinkthat this result, alongside with Theorem 3 which is a direct application of it,provides an interesting way of categorising networks depending on their outputfunctions. Also presented are Theorem 2 which proposes a bound on the total19umber of attractors depending on the number of inputs in an acyclic module,and the results listed in Section 4, which state a range of complexity resultson acyclic modules. The set of results proposed in this paper describe, in ouropinion, a good picture of the limits and possibilities that come from studyingacyclic modules.In future works, we plan to expand this formalism to more general updateschedules, and to propose a version of Theorem 3 which would generalise tomodules with more than one input and one output. We also plan to apply thosetools to optimise large automata networks, such as those designed and studiedin biology applications. Acknowledgments. The works of K´evin Perrot and Sylvain Sen´e were fundedmainly by their salaries as French State agents, affiliated to Aix-Marseille Univ.,Univ. de Toulon, CNRS, LIS, UMR 7020, Marseille, France (both) and toUniv. Cˆote d’Azur, CNRS, I3S, UMR 7271, Sophia Antipolis, France (KP), andsecondarily by ANR-18-CE40-0002 FANs project, ECOS-Sud C16E01 project,STIC AmSud CoDANet 19-STIC-03 (Campus France 43478PD) project. References [1] M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Ann. Math. 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