## 1 Introduction

### 1.1 Asynchronous graph and fixable Boolean networks

A Boolean network (network for short) is a finite dynamical system usually defined by a function

Boolean networks have many applications. In particular, since the seminal papers of McCulloch and Pitts [21], Hopfield [16], Kauffman [17, 18] and Thomas [28, 29], they are omnipresent in the modeling of neural and gene networks (see [8, 20] for reviews). They are also essential tools in computer science, for the network coding problem in information theory [3, 14] and memoryless computation [9, 10, 15].

The “network” terminology comes from the fact that the interaction graph of is often considered as the main parameter of : it is the directed graph with vertex set and an arc from to if depends on , that is, if there exist that only differ in the component such that .

In many applications, for modelling gene networks in particular, the dynamics derived from is not the synchronous dynamics, which is simply described by the successive iterations of , but the asynchronous dynamics [1]. The latter is usually represented by the directed graph , called asynchronous graph of and defined as follows. The vertex set of is , the set of all the possible states, and there is an arc from to if and only if and differs in exactly one component, say , and . An example of a network with its interaction graph and its asynchronous graph is given in Figure 1.

Before going on, let us review a few basic properties of the asynchronous graph. First, completely determines and vice versa. In particular, is a fixed point of (i.e. ) if and only if it is a sink of (i.e. it has no outgoing arcs). Second, since arcs of link states that differs in exactly one coordinate, is a directed subgraph of the hypercube (where cycles of length two are allowed). The converse obviously holds: any directed subgraph of the hypercube is the asynchronous graph of a network.

can be naturally regarded as a deterministic finite automaton where the set of states is , the alphabet is , and where

is the result of the action of a letter on a state . This action is extended to words on the alphabet in the natural way: the result of the action of a word on a state is inductively defined by or, equivalently,

This interpretation is illustrated in Figure 2.

In this paper, we are interested in words that fix , that is, such that is a fixed point of for every . This corresponds to the situation where the start state is undetermined, and the accepting states are exactly the fixed points. As such, there is an obvious connection between fixing word and synchronizing words: if has a unique fixed point, then is a synchronizing word for if and only if fixes . If admits a fixing word we say that is fixable. For instance, the network in Figure 1 is fixed by , and is hence fixable. It is rather easy to see that is fixable if and only if there is a path in from any initial state to a fixed point of .

Trivially, if is fixable, then it has at least one fixed point. Interestingly, Bollobás, Gotsman and Shamir [7]

showed that, considering the uniform distribution on the set of

-component networks, the probability for

to be fixable when has at least one fixed point tends to when . Thus almost all networks with a fixed point are fixable. In turn, this shows that for large, a positive fraction of all -component networks are fixable.For any fixable network , the fixing length of is the length of a shortest word fixing and is denoted as . For instance, we have seen that fixes the network in Figure 1, thus this network has fixing length at most , and it is easy to see that no word of length three fixes this network, and thus it has fixing length exactly . It is easy to construct a fixable -component network such that is exponential in , as follows. Let be a Gray code ordering of , i.e. and only differ by one coordinate for all . Then let for and for . It is clear that the asynchronous graph of such a , illustrated below, only contains a directed hamiltonian path of , and hence .

We are then interested in networks which can be fixed in polynomial time, i.e. is bounded by a polynomial in . More strongly, we extend our concepts to entire families of -component networks. We say that is fixable if there is a word such that fixes for all , which is clearly equivalent to: all the members of are fixable. The fixing length is defined naturally as the length of a shortest word fixing . We are then interested in families which can be fixed in polynomial time. For each family, we aim to derive an upper bound on and a lower bound on the maximum amongst all . Up to our knowledge, the only result of this kind was given in [13], where it is shown that the word repeated times fixes any -component symmetric threshold network with weights in . This family of threshold networks has thus a cubic fixing length. We could also mention somewhat less connected work concerning the minimal, maximal and average convergence time toward fixed points in the asynchronous graph for some specific fixable networks [4, 22, 23, 12, 11].

### 1.2 Results

#### 1.2.1 Acyclic networks

We first consider the family of -component networks with an acyclic interaction graph. Let be a member of this family, and consider a topological sort of the interaction graph (no arc from to for ). A first observation is that the word fixes . As a consequence, the fixing length of is exactly . We will prove, more generally, that a word fixes if at least one topological sort of is a subsequence of . As a consequence, fixes the whole family if contains all the permutations of as subsequences, and this sufficient condition is actually also necessary. Interestingly, such a word appears naturally in many combinatorial contexts. It is called an -complete word. The minimal length of an -complete word is not exactly known, but it is quadratic: Adleman [2] proved that (see [24] and the references therein for improvements) and Kleitman and Kwiatkowski [19] proved that, for all , there exists a positive constant such that . Thus, to sum up, for the family , the picture is very clear:

###### Theorem 1.

For every , the fixing length of any member of is , and the fixing length of the family is , and as .

#### 1.2.2 Increasing networks

A network is increasing if for every , where is applied componentwise. We denote by the set of -component increasing networks. A simple exercise shows that the number of increasing networks is doubly exponential: . It is then remarkable that, even though some increasing networks require quadratic time to be fixed, they can all be fixed together in quadratic time still:

###### Theorem 2.

For any and sufficiently large, there exists of fixing length at least , and, for , the fixing length of is .

The dual of a network is defined as . It is easily checked that a word fixes if and only if it fixes . Since a network is increasing if and only if its dual is decreasing ( for all ), Theorem 2 also holds for decreasing networks.

#### 1.2.3 Monotone networks

The main topic of this paper is the family of monotone networks, that is, the -component networks such that for all . We denote by the set of -component monotone networks. The fact that monotone networks are fixable is not obvious and is proved in [25, 22]. Our first contribution concerning monotone network is that some monotone networks have a quadratic fixing length.

###### Theorem 3.

For any and sufficiently large, there exists an -component monotone network with fixing length at least .

The main step in the proof consists in proving that there exists a sub-exponential collection of permutations of such that any word containing all these permutations as subsequences is of quadratic length. The proof uses Baranyai’s theorem (if divides , then there exist partitions of in -sets such that each -set of appears in exactly one of these partitions, see [30]).

###### Theorem 4.

For any and sufficiently large, there exists a set of at most permutations of such that any word containing all these permutations as subsequences is of length at least .

We then show that there is a word of cubic length, based on the concatenation of -complete words for , that fixes all monotone networks.

###### Theorem 5.

For all , there is a word of length that fixes .

#### 1.2.4 Refinements and extensions

We also refine and extend our results for monotone networks in three different fashions.

Firstly, some accurate results can be obtained for subfamilies of monotone networks. We say that the network is conjunctive if for all there exists such that . We denote by the set of -component conjunctive networks.

###### Theorem 6.

For , the maximum fixing length of a member of is .

Secondly, we refine the cubic upper bound for the word fixing all monotone networks by considering the interaction graph. We denote the set of monotone networks whose interaction graph is contained in a directed graph as . Recall that a feedback vertex set of is a set of vertices intersecting every cycle; the minimum size of a feedback vertex set is usually called the transversal number of and denoted . We then prove that the family can be fixed by a word of length . Thus, when we bound the transversal number, the fixing length becomes linear.

###### Theorem 7.

Let be a directed graph on with transversal number . There is a word of length that fixes .

On one hand, if is the complete directed graph on , with arcs, then and we obtain a cubic bound, as in Theorem 5. On the other hand, if is acyclic, then , and we obtain an upper-bound of , which is tight according to Theorem 1.

Thirdly, we consider balanced networks. The signed interaction graph is a refined description of the interaction graph, which not only indicates that has an influence on , but also whether that influence is positive ( is a non-decreasing function of ), negative ( is a non-increasing function of ) or null (other cases). Monotone networks are exactly the networks such that all the arcs in the signed interaction graph are signed positively (all the interactions are positive). A network is called balanced if every cycle of its signed interaction graph is positive. Here, the sign of a cycle is defined as the product of the signs of its arcs. Thus monotone networks are a special case of balanced networks. We denote the set of all -component balanced networks as . We once again show that the family of balanced networks can be fixed in polynomial time.

###### Theorem 8.

For all , there is a word of length that fixes .

Our results are summarised in Table 1. A dash means that we did not find any nontrivial result for the given entry: the tightest upper-bound on is that of , the tightest lower bound on is that of .

Networks | |||
---|---|---|---|

Acyclic | |||

Path | |||

Increasing | |||

Monotone | |||

Conjunctive | – | ||

-monotone | – | ||

Balanced | – |

## 2 Preliminaries

### 2.1 Notation

Let be a word. The length of is denoted . If with , then we shall sometimes use the notation ; if , then , where is the empty word. Any such is a subsequence of . Moreover, for any integers we set and hence if and if . Any such is a factor of . For any word and any , the word is obtained by repeating exactly times; is the empty word.

Graphs are always directed and may contain loops (arcs from a vertex to itself). Paths and cycles are always directed and without repeated vertices. We denote by the subgraph of induced by a set of vertices . If is a set, a graph on is a graph with vertex set . We refer the reader to the authoritative book on graphs by Bang-Jensen and Gutin [5] for some basic concepts, notation and terminology.

We denote by the family of -component monotone networks and by the fixing length of . More generally, if is any family of -component fixable networks, then is the fixing length of . If is a graph on , then denotes the set of -component networks such that the interaction graph of is a subgraph of . Then, and is the fixing length of .

Let be an -component network. We set and, for any integer and , we define as in the introduction if , and if . This extends the action of letters in to letters in , and by extension, this also defines the action of a word over the alphabet . Let be a graph on . The conjunctive network on is the unique conjunctive network whose interaction graph is . Namely, it is the -component network defined as follows: for all ,

where is the set of in-neighbors of in , and if is empty.

For all , we denote as the

-th unit vector, i.e.

with the in position . Given two states and , is applied componentwise and computed modulo two. For instance and only differ in the th position. The state containing only s is denoted , and the state containing only s is denoted . Hence, if is a strongly connected graph (strong for short) on , then and are the only two fixed points of the conjunctive network on . The Hamming weight of a state , denote , is the number of s in .### 2.2 Acyclic networks

Recall that is acyclic if its interaction graph is acyclic, and that denotes the set of -component acyclic networks. An important property of acyclic networks is that they have a unique fixed point [26] and that they have an acyclic asynchronous graph [27]. This obviously implies that is fixable. We show here that the fixing length of acyclic networks are rather easy to understand. The techniques used will be useful later, for analyzing the fixing length of monotone networks.

###### Lemma 1.

Let be an acyclic graph on and . If a word contains, as subsequence, a topological sort of , then fixes . Furthermore, .

###### Proof.

Let be the unique fixed point of . Let be topological sort of , and let be any word containing as subsequence. Hence, there is a increasing sequence of indices such that . Let be any initial state, and for all , let be obtained from by updating , that is, . Equivalently, . Let us prove, by induction on , that for all . Since is a source of the interaction graph, is a constant. Thus for all , and since , we deduce that for all . Let . Since only depends on components with , and since, by induction, for all and , we have for all . Since we deduce that for all , completing the induction step. Hence, for any initial state , thus fixes .

We deduce that, in particular, any topological sort of fixes , thus . Conversely, if a word fixes , then , and hence at least asynchronous updates are required, that is, the length of is at least . Thus . ∎

The converse of the previous proposition is false in general (for instance if is the -component network defined by , and , then fixes while is the unique topological sort of the interaction graph of ) but it holds for conjunctive networks.

###### Lemma 2.

Let be an acyclic graph on and let be the conjunctive network on . A word fixes if and only if it contains, as subsequence, a topological sort of .

###### Proof.

According to Lemma 1, it is sufficient to prove that if fixes then contains, as subsequence, a topological sort of . Let and for all . Since fixes and since is the unique fixed point of , we have . Thus for each , there exists such that and for all . We have, obviously, . Let be the enumeration of the vertices of such that is increasing. In this way is a subsequence of , and it follows the topological order. Indeed, suppose that has an arc from to . Since , we have , and thus , that is, is before in the enumeration. ∎

As an immediate application we get the following characterization.

###### Proposition 1.

Let be an acyclic graph on . A word fixes if and only if it contains, as subsequence, a topological sort of .

A word is complete for a set (or -complete) if it contains, as subsequence, all the permutations of . An -complete word is a -complete word. Let be the length of a shortest -complete word. Interestingly, is unknown. Let be permutations of (not necessarily distinct). Then the concatenation clearly contains all the permutations of . Thus . Conversely, if contains all the permutations of , then is at least and we deduce that (this simple counting argument will be reused later). This shows that the magnitude of is quadratic. We have however the following tighter bounds:

where and where is a positive constant that only depends on . Thus .

Let be the set of path networks on , i.e. the conjunctive networks on the paths of length with vertex set . Thus, for each permutation of , there exists exactly one such that and for all and . We show below that the family has a quadratic fixing length.

###### Lemma 3.

A word fixes if and only if it is -complete. Hence .

###### Proof.

As an immediate consequence, we get the following proposition, which implies, with Lemma 1, Theorem 1 stated in the introduction.

###### Proposition 2.

A word fixes if and only if it is -complete. Hence .

Therefore, it is as hard to fix as to fixe : these two families have the same quadratic fixing length, while is much smaller than (the former has members while the latter has members). We shall use this to our advantage when designing a monotone network with quadratic fixing length in Section 3.1.

### 2.3 Increasing networks

Recall that a network is increasing if for all . Those networks are also relatively easy to fix collectively, as seen below. We shall use this fact when constructing a cubic word fixing all monotone networks in Section 3.2.

###### Lemma 4.

Let be an -component network and . If for any words and , then is a fixed point of for any word containing all the permutations of . Similarly, if for any words and , then is a fixed point of for any word containing all the permutations of .

###### Proof.

Suppose that for any words and , and that be -complete, with . Let and for all . By hypothesis, . Suppose for the sake of contradiction that is not a fixed point, i.e. there is such that . Let be the set of positions such that , and let for all . Clearly, for every . Setting , we have , thus for all . We deduce that does not appear in or, equivalently, is the first position of in . Similarly, we have , thus for all and we deduce that does not appear in . Since is -complete, the sequence appears in , say at positions . Since is the first position of in , we have for all . In particular, , thus appears in which is the desired contradiction. If for any words and the proof is similar. ∎

###### Proposition 3.

A word fixes if and only if it is -complete. Hence .

###### Proof.

If is -complete, then fixes by Lemma 4. Conversely, suppose that fixes all -component increasing networks and let be any permutation of . Let and for all . Then is chain from to in the hypercube . Let be the -component increasing network defined by

Then is the unique fixed point of reachable from in the asynchronous graph, and it is easy to check that if and only if is a subsequence of . Thus is -complete. ∎

On the other hand, there exists an increasing network requiring quadratic time to be fixed (this and the Proposition 3 above give the Theorem 2 stated in the introduction).

###### Theorem 9.

For any and sufficiently large, there exists such that .

## 3 Monotone networks

### 3.1 A monotone network with quadratic fixing length

The aim of this section is to exhibit a monotone network with quadratic fixing length. As we saw in Section 2.2, the family of path networks has quadratic fixing length. Therefore, our strategy is to “pack” many of these path networks in the same network . As an illustration of this strategy, we first describe a monotone network with fixing length of order .

Let where , and let us write . There is then a surjection where is the set of states in with Hamming weight . The -component network then views the last components as controls, that decide, through , which network in to choose on the first components. More precisely, by identifying with , we define as follows:

The first and third cases are there to guarantee that is indeed monotone. Since any network in can appear, a word fixing must fix . Thus a word fixing is -complete, and hence has length . Choosing then yields .

The network above reached a fixing length of because it packed all possible networks in . However, it did not reach quadratic fixing length because had to be in order to embed all networks of in . Thus, we show below that only a subexponential subset of is required to guarantee . This is equivalent to prove that there exists a subexponential set of permutations of such that any word containing these permutations as subsequences is of length . In that case, we can use , and hence reach a fixing length of .

The main tool is Baranyai’s theorem, see [30].

###### Theorem 10 (Baranyai).

If divides , then there exists a collection of partitions of into sets of size such that each -subset of appears in exactly one partition.

###### Lemma 5.

Let and be positive integers, and . There exists a set of permutations of such any word containing all these permutations as subsequences is of length at least

###### Proof.

According to Baranyai’s theorem, there exists a collection of partitions of into sets of size , such that each -subset of appears in exactly one partition. Let be these partitions. For each , we set

Then, for all and we set and

where addition is modulo . So, the form a set of ordered partitions of in sets of size . The interesting point is that, for all fixed and fixed , the sequence is a permutation of (namely ). Since each -subset of appears in exactly one , we deduce that, for any fixed , the set of is exactly the set of -subsets of .

Given an -subset of and a permutation of , we set , where is an enumeration of the elements of in the increasing order. Let be an enumeration of the permutations of . For all , , , we set

The form a collection of permutations of . The interesting property is that, for fixed, the set of is exactly the set of words in without repetition, simply because, for fixed, the set of is exactly the set of -subsets of , as mentioned above. In particular, for fixed, the are pairwise distinct.

Let be a shortest word containing all the permutations as subsequences. We know that . Let

be the profile of , defined recursively as follows: and, for all , is the smallest integer such that is a subsequence of the factor

Since and for all , there are at most possible profiles. Thus there exist at least

permutations with the same profile. Let be these permutations, and let be their profile. For all , let

By construction, contains, as subsequences, each of . Since these elements of are pairwise distinct (because, for fixed , all the are pairwise distinct), this means that contains at least distinct subsequences of length , and thus

We deduce

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