Bisimilar Conversion of Multi-valued Networks to Boolean Networks
aa r X i v : . [ c s . D M ] J a n Bisimilar Conversion ofMulti-valued Networks to Boolean Networks
Franck Delaplace ˚ , Sergiu Ivanov IBISC - Paris-Saclay University, Univ. Evry
Abstract
Discrete modelling frameworks of Biological networks can be divided intwo distinct categories: Boolean and Multi-valued. Although Multi-valuednetworks are more expressive for qualifying the regulatory behaviours mod-elled by more than two values, the ability to automatically convert themto Boolean network with an equivalent behaviour breaks down the funda-mental borders between the two approaches. Theoretically investigatingthe conversion process provides relevant insights into bridging the gap be-tween them. Basically, the conversion aims at finding a Boolean network bisimulating a Multi-valued one. In this article, we investigate the bisimilarconversion where the Boolean integer coding is a parameter that can befreely modified. Based on this analysis, we define a computational methodautomatically inferring a bisimilar Boolean network from a given Multi-valued one.
Keywords:
Boolean Network, Multi-valued network, Bisimulation,Biological network modelling, Automatic conversion inference
1. Introduction
Discrete network based modelling frameworks, seminally introduced byS. Kauffman [5, 4] and R. Thomas [8, 7] for regulation network modellingcan be divided in two distinct categories: Boolean networks and Multi-valued networks. In the former, the states of genes are modelled by Boolean ˚ Corresponding author
Email addresses: [email protected] (Franck Delaplace), [email protected] (Sergiu Ivanov) alues, with propositional logic as the modelling framework, whereas in thelatter the state is extended to the integer domain, also called Multi-valued,using Presburger arithmetic as a modelling framework. It is often admittedthat Multi-valued networks provide more expressiveness for modelling geneexpression behaviour by distinguishing between more than two states ( i.e. , off or on ) for specifying the regulatory activity. However, the abilityto automatically convert a Multi-valued network to a Boolean one withthe same dynamical behaviour weakens this distinction from an analyticalstandpoint since the analysis of the dynamics can be performed on theBoolean network directly.More generally, the Boolean conversion of a Multi-valued network offersthe opportunity to bridge the gap between the two modelling formalismsthat enables to inherit, adapt and extend the theoretical results defined in aframework to the other [9]. Moreover, this allows the use of software basedon propositional logic that could prove computationally more efficient thanthe algorithms developed for Presburger arithmetic for the same problem.In particular, a wide spectrum of problems in modelling regulatory networksby symbolic characterization of stable states can be formalized as problemsof logical valuation of variables satisfying a formula in the Boolean case (theSAT problem) or finding solutions complying to a set of linear constraints forthe integer case (integer linear programming, ILP). The work [1] providesan experimental comparison of ILP and SAT solvers applied to the SATproblem.By considering these opportunities, the issue is thus to investigate meth-ods for converting Multi-valued networks to Boolean, while preserving thedynamical behaviour. This conversion is primarily based on an encodingof integers by Boolean profiles, establishing the equivalence between thetwo kinds of values. The challenge is to extend this equivalence to statetransitions in order to certify the behavioural integrity.In [2], G. Didier, E. Remy and C. Chaouya extensively study the con-ditions for the conversion of Multi-valued networks to Boolean ones usingVan Ham code [10] (Section 4). To overcome the potential limitation ofVan Ham code restraining the dynamics to a sub-region of the Booleanstate space, A. Faur´e and S. Kaji study the conversion based on Summingcode (Section 4), which provides several alternative Boolean profiles for en-coding an integer, such that the resulting Boolean dynamics is deployed onthe whole Boolean state space [3]. Following similar motivations, E. Tonelloalso studies the conversion based on this code [9].2hese research works elegantly pave the formal foundation of the Multi-valued to Boolean network conversion. However, the results are intrinsicallydependent on a specific coding, the Summing code, and are mainly designedfor the asynchronous mode. Therefore, it appears interesting to generalisethis approach by distinguishing the properties that purely relate to theconversion process from those depending on the code for highlighting thefoundations of this process.The behavioural equivalence is formally defined by the reachability preser-vation property, namely: whenever an integer state is reachable from an-other one, the equivalent Boolean state of the former is also reachable by theequivalent Boolean state of the latter, and conversely. Reachability preser-vation relies on the existence of a bisimulation [6] between both networks,parametrised by the Boolean-to-integer coding. While preserving the reach-ability is essential, it also appears desirable to extend the preservation tostructural properties of the interactions and other properties related to equi-librium. In this article, we study the network conversion by regarding it asa bisimulation process applied to any Boolean coding of the integers. Basedon this study, we propose an algorithm inferring the formulas of a Booleannetwork behaviourally equivalent to the input Multi-valued network.After recalling the main notions of Multi-valued networks (Section 2), weexamine the bisimulation properties between the dynamics of the networksand the admissibility conditions for stating a bisimulation between a Multi-valued and Boolean networks (Section 3) with regard to different codings(Section 4). Then, we study the extension of the properties preserved byconversion (Section 5). Finally we define a method inferring a Booleannetwork bisimilar to the Multi-valued one whatever the coding procedure(Section 6).
Notations.
We use the following notations:
Set:
The complement of a subset E z E , E Ď E is denoted by ´ E . A singleton t e u is denoted by its element e . The set of parts of E isnoted E “ t E | E Ď E u . State: A state s is an application from variables Y to a domain of values D , i.e. , s “ t y ÞÑ d , . . . , y n ÞÑ d n u and D Y “ p Y Ñ D q denotes the statespace defined on variables Y . The restriction/projection of a state s P D Y on W Ď Y is denoted s W P D W . This notation also holds forfunction on states, i.e. , dom g W “ D W . A substitution within a state s is the replacement of the value of a variable of s by another value,3ormally defined as: s r y ÞÑ v s “ s zt y ÞÑ s y u Y t y ÞÑ v u . The distance onstates is defined as: d p s, s q “ ř ni “ | s y i ´ s y i | .
2. Multi-valued networks A Multi-valued network x g, Y y defined on a set of variables Y is a dy-namical system on integer states where the evolution function g : N Y Ñ N Y is composed of a collection of local evolution functions g “ p g , ¨ ¨ ¨ , g n q , n “| Y | . The evolution function is defined for a variable y i P Y as follows: g i p s q “ $’’’’’’’’&’’’’’’’’% if C p s q¨ ¨ ¨ l if C l p s q¨ ¨ ¨ L if C L p s q otherwise (1)where C l is the guard of level l . The application g i p s q equals level l if andonly if the guard C l p s q is the first satisfied condition with respect to thereading order. Model of dynamics.
The model of dynamics of a Multi-valued network x g, Y y is formalized by a labelled transition system x N Y , M, ÝÑ g y wherethe labels are sets of variables that determine which variables are updatedjointly during a transition. The mode M Ď Y describes the organization ofthe joint updates per transition. For example, in the asynchronous mode, Y “ tt y i uu y i P Y , the state of one variable only is updated per transition andin the parallel or synchronous mode t Y u , all the variables are updated to-gether. The mode is also introduced in the network specification if needed, i.e. , x g, Y, M y .Thus, only the state of the variables in m P M can be updated by atransition s m ÝÑ g s whereas the state of the other variables remains un-changed i.e. , s “ p g m p s q Y s ´ m qq . A transition that does not change thestate, s m ÝÑ g s , is called a self-loop . The global transition relation corre-sponds to the union of all transition relations labelled by the componentsof the mode: ÝÑ g “ Ť m P M m ÝÑ g . Hereafter, f : B X Ñ B X , B “ t , u , always stands for a Boolean func-tion, g : N Y Ñ N Y designates a Multi-valued/integer function, whereas4 “ $’’’’’’’’’’’&’’’’’’’’’’’% x “ y ě otherwise y “ $’’’&’’’% x “ ^ y ě p x “ ^ y “ q _ p x “ ^ y “ q p x “ ^ y “ q _ p x “ ^ y “ q otherwise x y ` ` `
00 13010203 101112 y x y x y xx yyy Figure 1: A Multi-valued network with the interaction graph (below) and the asyn-chronous dynamics (right), with the self-loops removed. Y always corresponds to a set of integer variables. w P B X represents aBoolean state whereas s P N Y a Multi-valued one. The Multi-valued dy-namics where in which transitions modify the current level by 1 only ( i.e. , @ s ÝÑ s , @ y i P Y : d p s y i , s y i q ď
1) is said unitary stepwise . Equilibrium.
A state s is an equilibrium , if it can be reached infinitely oncemet: @ s P N Y : s ÝÑ ˚ s ùñ s ÝÑ ˚ s. (2)An attractor is a set of equilibria that are mutually reachable and a stablestate is an attractor of cardinality 1.Figure 1 shows an example of a Multi-valued network and the result-ing dynamics for the asynchronous mode with two stable states that arerespectively 13 and 00. Interaction graph. An interaction graph x Y, y portrays the interdepen-dencies of the variables in the network x g, Y y . An interaction y i y j existswhenever changing the value of y i may lead to a change in the value of y j : y i y j def “ D s, s P N Y : s y i ‰ s y i ^ s ´ y i “ s y i ^ g j p s q ‰ g j p s q . (3) ÝÑ ˚ denotes the reflexive and transitive closure of ÝÑ . signed interaction graph x Y, , σ y refines the nature of the in-teractions by signing the arcs with σ : p q Ñ t´ , , u to represent amonotone relation between the source and target variables of the interac-tion (4); either increasing (label 1, denoted ` ), or decreasing (label ´ ´ ), or neither (label 0, denoted ˘ ), and formally defined as: y i ` y j def “ y i y j ^@ s, s P N Y : s y i ď s y i ^ s ´ y i “ s y i ùñ g j p s q ď g j p s q y i ´ y j def “ y i y j ^@ s, s P N Y : s y i ď s y i ^ s ´ y i “ s y i ùñ g j p s q ě g j p s q (4)
3. Network bisimulation
By definition [6], bisimulation between the dynamics of networks pre-serves the reachability, thereby maintaining the trajectories and the attrac-tors in both ways. Definition 1 illustrated in Figure 2 formally definesfunctional bisimulation, which depends on a partial function ψ : B X Ñ N Y decoding a Boolean state to an integer state. Definition 1.
Given a Boolean network B “ x f, X, M X y and a Multi-valuednetwork N “ x g, Y, M Y y , a pair of functions p ψ, µ q , with ψ : B X Ñ N Y apartial function and µ : M X Ñ M Y a total function, form a bisimulation ifand only if the following properties hold: (forward simulation) for any two Boolean states w, w P dom ψ and m P M X , w m ÝÑ f w implies ψ p w q µ p m q ÝÑ g ψ p w q : @ w, w P dom ψ, @ m P M X : w m ÝÑ f w ùñ ψ p w q µ p m q ÝÑ g ψ p w q ;2. (backward simulation) for any two Multi-valued states s, s P N Y , forany w P B X such that ψ p w q “ s , and for any n P M Y , s n ÝÑ g s impliesthat there exists a w P B X and an m P M X such that ψ p w q “ s , µ p m q “ n and w m ÝÑ f w : @ s, s P N Y , @ w P B X , @ n P M Y : ψ p w q “ s ^ s n ÝÑ g s ùñ ` D w P B X , D m P M X : µ p m q “ n ^ ψ p w q “ s ^ w m ÝÑ f w ˘ . Two networks B and N complying to Definition 1 with respect to ψ aresaid bisimilar , noted B „ ψ N . Although, (1.2) and (1.1) are similar in their6 w m ψ p w q ψ p w q µ p m q (a) Forward simulation ss n @ w P ψ ´ p s qD w P ψ ´ p s q D m P µ ´ p n q (b) Backward simulation Figure 2: Illustration of a bisimulation p ψ, µ q between a Boolean and a Multi-valuednetwork. ψ ´ p s q and µ ´ p n q denote the preimages of s and n under ψ and µ respectively. definition, it is worth noticing that they however differ in the following point:all the transitions on the integer state space should fulfill (1.2) whereas onlythe transitions defined on the domain of ψ , dom ψ , should comply to (1.1). dom ψ circumscribes the admissible region [3, 9], where each Boolean stateencodes an integer state and each transition is bisimilar to a Multi-valuedone. Hence, no transitions from a state located in the admissible regioncan escape from this region, thus avoiding aberrant cases exemplified in [9].From (1.2), we deduce that ψ is a surjective partial function defined on B X but it is not necessary injective and thus not bijective. Hence, the preimageof an integer state is a set: ψ ´ p s q “ t w P B X | ψ p w q “ s u . The issue is to determine the conditions on a Boolean network enablinga bisimulation with a Multi-valued network. These conditions depend ona general relation between the integer and Boolean function including themode.
Integer states are coded by the Boolean states in which the Boolean vari-ables storing the code constitute the support of the integer variables. The support function associates each subset of integer variables to its support: p ¨ : Y Ñ X . This function has the following properties: 1) the Booleanvariables are exactly the supports of the integer variables, 2) the supportsare pairwise disjoint, and 3) they are modular in the sense that the unionof the supports is the support of the union of the integer variables:1 q X “ p Y and,2 q @ y i , y j P Y : y i ‰ y j ðñ p y i X p y j “ H and,3 q @ Y , Y ” Ď Y : { Y Y Y ” “ x Y Y x Y ” . (5)For example, the state s p y ,y q “ p , q is encoded by w “ p , q by usingthe classical binary code or the Gray code. The variables of the Boolean7etwork will be therefore the variables supporting the Boolean code of theinteger variables of Y ( X “ p Y ). The states of said Boolean variables arerespectively: w x y “ , w x y “ , w x y “ , w x y “
1. Note that there aretwo kinds of indices: one for the multivalued variables, and the other forthe Boolean variables of the corresponding supports.We consider henceforth that ψ fits all supports, i.e. , ψ P Ť W Ď Y ` B x W Ñ N W ˘ .The function ψ transforms the Boolean state of the support into an integerstate in a modular manner, by decoding distinct sub-parts of a Booleanstate separately, so that the decoding of the whole integer state is the unionof the local decoding results: @ W Ď Y, @ w P B p Y : ψ p w x W q “ ψ p w q W . (6)From (5,6) we deduce the following relation on two disjoint sets of variablesrepresenting the modularity of the decoding: ψ p w { W Y W q “ ψ p w x W qY ψ p w y W q “ ψ p w q W Y ψ p w q W , W X W “ H , W, W Ď Y. Moreover, the mode of the converted Boolean network must be compat-ible with the modularity of the coding. A mode is local-to-support whenthe parallel updates of the Boolean variables operate inside supports only,namely M is local-to-support if and only if: @ m P M, D y i P Y : m Ď p y i . The asynchronous mode is always local-to-support and the parallel local-to-support mode is the gathering of the supports: t p y i u y i P Y . Within the framework, inferring a Boolean network bisimilar to a multi-valued one is reduced to the discovery of a Boolean network in bisimulationwith a local multivalued network x g i , Y y where only the state of a single vari-able evolves. A Boolean network in bisimulation with the entire multivaluednetwork results from the union of Boolean networks in bisimulation with lo-cal multivalued networks (Proposition 1). Hence, for each g i , we focus onthe discovery of the appropriate evolution function of the support f p y i andthe determination of the admissible modes for enabling the bisimulation. Proposition 1.
Consider the multivalued network N “ x g, Y, Y y and thefamily p B i q y i P Y of Boolean networks over the supports of the variables in Y : B i “ x f p y i , X, M y i y , M y i Ď p y i . Then the following holds: ` @ y i P Y : x f p y i , X, M y i y „ x g i , Y, y i y ˘ ùñ x f, X, ď y i P Y M y i y „ x g, Y, Y y . where f “ p f p y i q y i P Y and g “ p g i q y i P Y are the global evolution functionscollecting their respective local evolution functions. f and a Boolean transition function g , it suffices that the local in-teger evolution function applied to the decoding g i ˝ ψ p w q coincide with theBoolean evolution function ψ ˝ f p y i p w q or, more generally, with the evolutiontaken under a local-to-support mode: ψ ˝ f “ g ˝ ψ Global function ψ p f m p w q Y w p y i z m q “ g i ˝ ψ p w q , m Ď p y i Local-to-support mode (7)If ψ is a bijective function then (7) is expressed as f “ ψ ´ ˝ g ˝ ψ , whichcorresponds to the conjugated evolution function defined in [2].Theorem 1 shows that Property (7) is necessary and sufficient to as-certain that a multivalued network bisimulates a Boolean network with alocal-to-support mode. Theorem 1.
Let N “ x g i , Y, y i y be a multivalued network, B “ x f p y i , p Y , M y a Boolean network with M a local-to-support mode, and ψ : B p y i Ñ N y i , asurjective function, Property 7 is met between the evolution functions of B and N if and only if B „ ψ N .3.2. Bisimulation admisibility Theorem 1 states the equivalence between bisimilarity and Property 7for local-to-support modes, including the parallel mode updating all thevariables of the support. In this section, we extend this result to a largerfamily of modes that are admissible with respect to the parallel mode. In-formally, a modality m is m -admissible if, for any Boolean state w , running f on w under m or under m yields (possibly different) states belonging tothe same preimage under ψ . Definitions 2 and 3 detail this compatibilityformally. Both definitions assume the Boolean network B “ x f, p Y , M y operating in mode M and the Multi-valued network N “ x g i , Y, y i y . Definition 2.
A mode component m P M is m -admissible with respectto the functional bisimulation B „ ψ N denoted by adm ψ p m, m q , if thefollowing holds: @ w P B p Y : ψ p f m p w q Y w ´ m q “ ψ p f m p w q Y w ´ m q . Notice that it follows directly from the definition that admissibility is anequivalence relation on modalities. Admissibility is defined on the modal-ities composing a mode and can be lifted from modalities to modes in anatural way: 9 efinition 3.
A mode M is M -admissible with respect to the functionalbisimulation B „ ψ N , denoted by adm ψ p M, M q , iff the following conditionshold: @ m P M , D m P M : adm ψ p m, m q ; @ m P M, D m P M : adm ψ p m, m q . w w w f m f m ψ p w q ψ p w q “ ψ p w q ψ ψ Figure 3: Illustration of m -admissibility adm ψ p m, m q of a modality m with respect tothe bisimulation B „ ψ N . According to the definition, a mode M is M -admissible if, for everymodality m P M , there exists a modality m P M such that m is m -admissible. Note that this requirement does not imply the existence of abijection between M and M : two functions M Ñ M and M Ñ M areindeed required by, respectively, clauses (1) and (2) of Definition 3, but theymay not be the inverse of each other. Lemma 1.
The relation of admissibility with respect to the functional bisim-ulation B „ ψ N , defined on all possible modes of B , is an equivalence rela-tion. It turns out that switching update modes within an admissibility classof modes preserves bisimulation.
Theorem 2.
Given the functional bisimulation B „ ψ N between the Booleannetwork B “ x f, p Y , M y and the Multi-valued network N “ x g i , Y, y i y , anyBoolean network B “ x f, p Y , M y with adm ψ p M, M q functionally bisimu-lates N as well: @ M Ď p Y : B „ ψ N ^ adm ψ p M, M q ùñ B „ ψ N. This result allows us to prove the bisimilarity of a network with anothermode providing that Property (7) holds and the mode is admissible.10 orollary 1.
Let B “ x f, p Y , M y , B “ x f, p Y , M y be two Boolean networks, ψ : B p Y Ñ N Y a surjective function, and N “ x g i , Y, y i y a Multi-valuednetwork. If Property 7 holds for B , N , and ψ , and if adm ψ p M, M q , then B „ ψ N. Proof.
If Property 7 holds then we deduce that B „ ψ N from Theorem 1.As B „ ψ N and M is an admissible mode we conclude from Theorem 2that B „ ψ N .
4. Boolean coding
The coding procedure characterizes a function ψ mapping a Booleanprofile to an integer. We study two fundamental codes that are suitablefor asynchronous Boolean dynamics: the Summing code and the
Gray code .Table 1 shows both codings for encoding levels ranging from 0 to 3.
Summing code.
For the Summing code, the integer corresponding to aBoolean state w is the sum of the states of the Boolean support variables: ψ p w p y i q “ ÿ p y ik P p y i w p y ik . The size of the support is linear in the maximal level, | p y i | “ L , and differentencodings are possible for the same integer. The number of different codesfor an integer 0 ď l ď L is ` Ll ˘ . Van Ham code [10] is a sub-case of theSumming code in which the unitary stepwise evolution restricts the fillingof 1 from left to right. This code is emphasized in bold in Table 1. Gray code.
The Gray code associates Boolean states differing in only oneposition to consecutive integers. The coding function is bijective and con-structs the integer value from a Boolean state by first transforming a Graycode profile into its equivalent binary code and then by computing the in-teger from this coding : ψ p w p y i q “ | p y i | ÿ k “ | p y i |´ k . k à j “ w p y ij . The support size is logarithmic in the maximal level: | p y i | “ r log p L ` q s . ‘ is the exclusive or , xor .
11 1 2 3 p , , q p , , qp , , q p , , q p , , q p , , qp , , q p , , q - Summing code - p , q p , q p , q p , q - Gray code - Table 1: Example of codes for levels ranging from 0 to 3. The states correspond to thevariable profiles p p y i , p y i , p y i q . The links connect codes differing by 1, and the codes inbold correspond to Van Ham sequence. The Summing code is defined on the whole Boolean state space ( i.e. , dom ψ “ B p Y ). The Gray code can be also defined on the whole Booleanspace when the maximal number of levels is L “ k ´
1. Van Ham code,on the other hand, never covers the entire Boolean space, except when themaximal level is 1. All these codings associate the integer 0 to the 0 Booleanprofile. Furthermore, they all fulfil the neighbourhood preserving property(8) defined in [2] and stressing that the distance of 1 between two integerstates should map to a distance of 1 between the corresponding Booleanstates, and conversely: @ s, s P N Y : d p s, s q “ ùñ D w P ψ ´ p s q , D w P ψ ´ p s q : d p w, w q “ ^@ w, w P dom ψ : d p w, w q “ ùñ d p ψ p w q , ψ p w qq “ . (8)These codes are individual representatives of families of linear and log-size codes which can be obtained by a permutation π on the integer states, i.e. , ψ “ π ˝ ψ . This permutation may notably relax the neighbourhood pre-serving property. In the literature, the study of the Multi-valued-to-Booleannetwork conversion has been carried out extensively for the Summing andVan Ham codes [2, 3, 9]. Although the Gray code is bijective and pro-vides the most compact binary representation of integers, it has never beenstudied for the conversion purposes according to our knowledge.12 . Extensions of property preservation Although bisimulation preserves the essential property of reachability, itappears desirable to preserve additional properties for performing an accu-rate analysis of dynamics on the Boolean network directly. These additionalproperties pertain to the nature of equilibria and the interaction graph.
By definition of the bisimulation, the equilibria of a Multi-valued net-work match with the equilibria of a bisimilar Boolean network, and con-versely. However, when some equilibria are stable states, their nature maydiffer: a stable state of the Multi-valued network can be represented by acyclic attractor over Boolean profiles, all coding for the same integer (Fig-ure 4). Nevertheless, any cyclic attractor will still be bisimulated by a cyclic y “ $’’’&’’’% y “
32 1 ď y ď y “ otherwise $’&’% p y “ p y “ p y _ p y _ p y p y “ p y ^ p y ^ p y $’&’% p y “ p y _ p y p y “ p y _ p p y ^ p y q p y “ p p y ^ p y q _ p y Multi-valued network Bisimilar Boolean networkpreserving stability Bisimilar Boolean networklosing stability
Figure 4: Stability loss during bisimulation – synchronous mode. attractor since, by definition of coding, a transition between two differentinteger states is always simulated by a transition with two different Booleanprofiles. Figure 4 shows an example where the self-loop of stable state 2 issimulated by a cyclic attractor over the three Boolean profiles coding for13t (right-hand side network). Indeed, for any integer level, the synchronousdynamics allows reaching any of its codings from any other one. This caseis however not encountered for stable state 3 coded by a single Boolean pro-file. The occurrence of such situations also depends on the concrete Booleanfunction, as shown by the middle network that preserves the stability. Eventhough in the former case the stable state is represented by a cyclic attrac-tor, it is worth noticing that the original level 2 can be recovered from thestates encompassed by the attractor since t ψ p q , ψ p q , ψ p qu “ t u .Maintaining the stability matters for the analysis performed on theBoolean dynamics. In particular, the symbolic computation of stable stateswill fail to find state 2 as equilibrium from the Boolean network since thisstate is represented by a cyclic attractor. Therefore, such cases should beruled out to ensure a matching analysis of the dynamics on the two net-works. To preserve the stability of equilibria, we basically have to preventreaching a code of an integer level l from another code also encoding for l .This depends on the mode and on the Boolean function ( cf. Figure 4). Theexpected outcome can be expressed as follows for a mode M : @ w P dom ψ, @ m P M : w m ‰ f m p w q ùñ ψ p w q ‰ ψ p f m p w q Y w ´ m q . (9)We examine two effective conditions for satisfying (9) that are inde-pendent of the specification of the Boolean network. A simple one workingwhatever the mode and the Boolean function is to remove the self-loops, andthus establish bisimulation between reflexive reductions of the state graphsof both networks, instead of operating on the original state graphs. Theequilibrium stability then remains preserved since no circuits can simulatea self-loop and the important features of the reachability are not altered.Another more explicit condition, based on the code and the sizes of modali-ties, forbids the access by a transition to another Boolean profile coding forthe same integer. Proposition 2.
Let B “ x f, p Y , M y be a Boolean network bisimilar to aMulti-valued network N “ x g, Y, Y y , with M a local-to-support mode. Ifthe following holds: @ y i P Y, @ s y i P N y i , @ w, w P ψ ´ p s y i q : w ‰ w ùñ d p w, w q ą max t| m | | m P M u , then the equilibrium stability (9) is preserved. ψ is a bijection, the stability of equilibria is preservedsince every integer level is coded by a single Boolean profile. On the otherhand, the asynchronous mode preserves the stability under the Summingcode, since distances between two Boolean profiles coding for the sameinteger are at least 2. A Boolean network bisimulating a Multi-valued network is regulatory-preserving if it is possible to unambiguously recover the signed interactiongraph of the Multi-valued network ( migs ), x Y, , σ y , from the signedBoolean interaction graph of the bisimilar Boolean network ( bigs ), x p Y , , σ B y .Retrieving migs from bigs is divided in two steps: retrieving the interactiongraph and finding the signs. Interaction graph retrieval.
The structure of migs is retrieved from thequotient graph of bigs defined on the support of the integer variables, calledthe support interaction graph ( sig ) xt p y i u y i P Y , y , where an interactionbetween two Boolean support variables induces an interaction between thesupports they belong: p y i p y i def “ D p y ik P p y i , D p y jr P p y j : p y ik p y jr . (10)As a consequence, the topological structure of migs is the same as that of sig by merely replacing the supports by the integer variables they support(Proposition 3). In fact, sig essentially provides an intermediary represen-tation used for recovering the interactions of migs and their signs. Proposition 3.
Let N be a Multi-valued network and B a Boolean network.If N is bisimilar to B then migs p N q is isomorphic to sig p B q .Sign retrieval. The sign of an interaction is determined by bigs once theconversion is achieved (see Figures 5, 6). Therefore the issue is to deducefrom the signs of the interactions between the Boolean variables the signsof the corresponding interactions in migs . The recovery procedure is basedon a set of reference Boolean variables, considered as markers of sign , cov-ering all the supports such that the signs of the interactions between thesevariables are the same as the signs of the interactions between the integervariables they support. Hence the set of markers M p Y is a subset of Booleanvariables of p Y defined by: 15 efinition 4 (Markers of sign) . Let x g, Y y a Multi-valued network bisimu-lating a Boolean network x f, p Y y with x Y, , σ y and x p Y , , σ B y as theirrespective signed interaction graphs. M p Y Ď p Y is a set of markers of sign ifand only if: The sign σ of an interaction between any two Boolean variables in M p Y Ď p Y is the same as the sign of the interaction between the integervariables that they support: @ p y ik , p y jr P M p Y : p y ik σ p y jr ðñ y i σ y j . All integer variables have markers: @ y i P Y : M p Y X p y i ‰ H . To operationally identify the markers from a code, we define a code-based marker condition (11) directly linking the markers to the code for theasynchronous mode. This condition asserts the monotony of the coding formarkers with respect to the integer and Boolean orders by stipulating thatan integer coded by a Boolean profile is less than another coded by thisBoolean profile where a marker value is substituted by 1 (Lemma 2).
Lemma 2.
Let N “ x g, Y y be a Multi-valued network bisimulating an asyn-chronous Boolean network B “ x f, p Y , p Y y , and M p Y Ď p Y be a set of Booleanvariables complying to (4.2). If: @ p y ik P M p Y , @ w P dom ψ : ψ p w q ď ψ p w r p y ik ÞÑ s q (11) then M p Y fulfils Definition (4.1) and M p Y is a set of markers. Therefore, the goal is to determine for each integer variable the set ofmarkers by checking (11) for a given coding. For the Summing code all theBoolean variables are markers, and for the Gray code the variables storingthe most significant bit indexed by 1 ( p y i ) are the markers.16 heorem 3. Let N “ x g, Y y be a Multi-valued network in bisimulation withan asynchronous Boolean network B “ x f, p Y , p Y y . The sets of markers M p Y are respectively for the codes: • Summing code: M p Y “ p Y ; • Van Ham code: M p Y “ p Y ; • Gray code: M p Y “ t p y i | y i P Y u .
6. Inference of Boolean formulas
An analytical definition of the Boolean network function is given by(7). Although the function ψ ´ ˝ g ˝ ψ is closed on Boolean states when ψ is bijective characterizing a Boolean network, the Boolean formulas are notexplicitly defined. The lack of Boolean formulas makes the analysis harder inpractice, notably by preventing the characterization of the interaction graphdirectly from formula specifications. Moreover, the analytical definitiondoes not hold when ψ is not bijective, since ψ ´ returns a set of Booleanprofiles. To circumvent this limitation, the strategy is to infer the Booleannetwork bisimilar to a Multi-valued network directly from the specificationof the latter (1). As it is sufficient to find a bisimilar Boolean network foreach local Multi-valued evolution function g i (Proposition 1), the algorithmwill act on each function of integer variables independently. In this sectionwe define a method inferring the formulas f i,k for each support variable p y ik of y i such that the reflexive reduction of the resulting Boolean network isbisimilar to the reflexive reduction of the initial Multi-valued network, thecode being a parameter of this method. Due to the reflexive reductions,this method preserves the nature of the stable states (Section 5.1). Forsimplicity, the inference is presented for the asynchronous mode, but it canbe applied to any local-to-support mode. This point is discussed at the endof the section.The definition of a formula f i,k for a support variable is divided in twostages: The conversion of the guard into a Boolean form , and the derivationof the admissibility condition for guard validation . The examples use theSumming code which is the most complex coding for the inference.17 oolean conversion of the guard. Basically, the guard of level l must alsobe satisfied in the Boolean network to simulate a transition shifting thecurrent level l to l . The conversion of a Multi-valued guard to a Booleanguard gathers the codes of the state profiles fulfilling the conditions of level l , i.e. , C l ‹ y i “ t s p‹ y i q | C l p s qu where p‹ y i q is the set of regulatorsof y i . As all these integer states satisfy the guard C l , their Boolean codesshould also satisfy the Boolean guard C l B defined as : C l B “ ł s P C l yi ¨˝ ľ y j Pp‹ y i q ł w x yj P ψ ´ p s yj q minterm p w x y j q ˛‚ . (12)For example, in the case of the Multi-valued network from Figure 1, thestates fulfilling the conditions to reach level 2 for y are p x “ , y “ q for thetransition from level 1 to 2 for y , or p x “ , y “ q for the transition fromlevel 3 to 2. The code for x is ψ ´ x p q “ tp qu , ψ ´ x p q “ tp qu and the codesfor y are respectively for 1 and 3: ψ ´ y p q “ tp , , q , p , , q , p , , qu and ψ ´ y p q “ tp , , qu . Hence, the Boolean guard of level 2 for y is: C B “ p x “ ,y “ q ¨˚˝ p x loomoon minterm x p q ^ ¨˚˝ p^ p y ^ p y ^ p y q looooooooooomooooooooooon minterm y p , , q _ p p y ^ p y ^ p y q looooooooomooooooooon minterm y p , , q _ p p y ^ p y ^ p y q looooooooomooooooooon minterm y p , , q ˛‹‚˛‹‚ _ p x “ ,y “ q ¨˚˝ p x loomoon minterm x p q ^ p p y ^ p y ^ p y q looooooomooooooon minterm y p , , q ˛‹‚ Guard admissibility condition.
The generation of the Boolean guard is how-ever insufficient for obtaining the final formula because some support vari-ables shift to 0 during the transition even though the guard is satisfied,meaning that a direct evaluation of the Boolean guard would shift them The minterm of a state is a conjunction of the variables such that the unique inter-pretation satisfying it is the state itself, e.g. , minterm p x “ , x “ q “ x ^ x .
18o 1. For example, in Figure 1, shifting from 3 to 2 for y is bisimilar to p , , q p y ÝÑ p , , q . In this case, we need to shift the state of p y to 0 al-though the guard is satisfied with s x “
0. We thus need to characterize thesituations in which the transition necessarily shifts the value of a supportvariable to 1. This restricts the set of admissible encodings triggering theguard, outside of which the transition always shifts the support variablestate to 0.Let s y i ÝÑ s be an integer unitary stepwise transition with s y i “ l and s p y i q “ l such that | l ´ l | “
1, and p y i be the support of y i ( p y ik P p y i ),we denote by w ÝÑ w the asynchronous Boolean transition bisimilar to s y i ÝÑ s . Two cases where p y i,k “ p y i,k is shifted from 0 to 1 during the transition( i.e. , w p y i,k “ w p p y i,k q “ p y i,k remains as 1 ( i.e. , w p y i,k “ w p p y i,k q “ l such that p y ik is either shifted to or remainsat 1. The initial level l is determined from the target level l by consideringthat it is either l ´ , l or l ` p y ik is shifted from0 to 1 during the transition ( ψ is implicitly restricted to ψ : B p y i Ñ N y i ):Ψ Ñ p l , p y ik q “ t w p y i P dom ψ | D max p , l ´ q ď l ď min p l ` , L q : ψ p w p y i q “ l ^ w p y ik “ ^ ψ p w p y i r p y ik ÞÑ s q “ l u . Similarly, we define the set of codes for which a shift from 1 to 0 occurs:Ψ Ñ p l , p y ik q “ t w p y i P dom ψ | D max p , l ´ q ď l ď min p l ` , L q : ψ p w p y i q “ l ^ w p y ik “ ^ ψ p w p y i r p y ik ÞÑ s q “ l u . Finally, we define the set of codes where p y ik is 1 in both l and l :Ψ Ñ p l , p y ik q “ t w p y i P dom ψ | D max p , l ´ q ď l ď min p l ` , L q , D w p y ik P ψ ´ p l q : ψ p w p y i q “ l ^ w p y ik “ ^ w p y ik “ u . The set of Boolean states coding for level l , always reaching state 1 andnever a state 0 for p y ik in a transition to a code of level l , is defined as:Ψ ‹Ñ p l , p y ik q “ Ψ Ñ p l , p y ik q Y p Ψ Ñ p l , p y ik qz Ψ Ñ p l , p y ik qq . Note that the set difference in the previous equation is not necessarily empty.Indeed, there may exist a pair of states w p y i and w p y i , with ψ p w p y i q “ l and19 p w p y i q “ l , such that w p y ik “ w p y ik “
1, but for which ψ p w p y i r p y ik ÞÑ s q “ l alsoholds. We need to exclude such states w p y i from Ψ ‹Ñ p l , p y ik q , because theystill allow reaching a Boolean profile coding for l by setting p y ik to 0. Theresulting transition is bisimilar to an integer transition, and thus must bekept.The guard admissibility condition G of C l B is thus defined as the disjunc-tion of the minterms of the admissible codes: G Ψ ‹Ñ p l , p y ik q “ ł c P Ψ ‹Ñ p l , p y ik q minterm p c q . (13)In our running example, consider the levels that potentially reach level2 in a unitary stepwise transition (levels 1, 2, and 3). The final simplifiedformulas of the code admissibility conditions G Ψ ‹Ñ p , p y k q , ď k ď
3, for eachsupport variable are detailed in Table 2.From these conditions (Table 2), we deduce that the asynchronous tran-sitions from level 3 coded by p , , q to level 2 all set to 0 one of the Booleansupport variables. Indeed, the update of p y to 0 leads to p , , q and sim-ilarly for p y , p , , q and p y , p , , q that all represent a Summing code oflevel 2. G Ψ ‹Ñ p , p y q “ p y ^ y q _ p y ^ y q _ p y ^ y q G Ψ ‹Ñ p , p y q “ p y ^ y q _ p y ^ y q _ p y ^ y q G Ψ ‹Ñ p , p y q “ p y ^ y q _ p y ^ y q _ p y ^ y q Table 2: Guard admissibility condition for level 2 of the support variables of y . Boolean formula of a support variable.
The final formula f i,k for a sup-port variable p y i,k can be understood as the Boolean version of the guardsrestricted to the codings admissible for their triggering, defined as: f i,k “ ł ď l ď L ` C l B ^ G Ψ ‹Ñ p l, p y ik q ˘ . (14)The Boolean network gathers the formulas defined by (14) for each sup-port variable. For the running example (Figure 1), the final Boolean networkprovides a clean description of the formulas once simplified for the Summingcode (Figure 5) and the Gray code (Figure 6), that differ due to the codings.Theorem 4 demonstrates the correction of the conversion method.20 “ $’’&’’% p x “ p y _ p y _ p y p y “ p x p y “ p x p y “ p x p x p y p y p y ` ` ` ` ` ` p y p x p y p x p y p y p x p y p x p y p y p x p y p y p x p y p y p y p x p x p y p y p y p y p y p y p y p y p y p y p y p y Figure 5: A Boolean network bisimilar to the Multi-valued network in Figure 1, itsinteraction graph (right), and its asynchronous dynamics for the Summing code withoutthe self-loops (below). &% p x “ p y _ p y p y “ p p x ^ p y q _ p p y ^ p y q p y “ p p x ^ p y q _ p p x ^ p y q p x p y p y ˘ ` ` ` ˘˘
000 110 001010 011100 101111 p y p x p y p x p y p x p x p y p y p y Figure 6: A Boolean network bisimilar to the Multi-valued network in Figure 1, itsinteraction graph (below), and its asynchronous dynamics for the Gray code without theself-loops (right).
Bisimilar reflexive reduction.
Under the asynchronous mode, some supportvariables may maintain their value inducing self-loops that are not bisimilarto any transition in the integer dynamics. In the running example, shifting y from 2 to 3 is bisimilar to p , , q p y ÝÑ p , , q , which modifies the value of p y . However, any of the Boolean variables may be updated in asynchronousdynamics leading to two self loops on p y and p y for maintaining the stateof these variables at 1. Obviously, these self-loops are not bisimilar to theinteger state transition since the variation of the integer state from 2 to3 is carried out by one transition only. No alternatives preventing theseadditional self-loops in the Boolean network are possible since any one ofthe Boolean variables may be updated, but the state must not change for p y and p y . This situation explains why our method operates on reflexivereductions of the networks, effectively discarding these extra self-loops. Alsonotice that this transformation can only be performed if the integer level0 is coded by the 0 Boolean profile, meaning that the behaviours of theMulti-valued and Boolean networks match when no guards are satisfied.The reflexive reduction of a network N is denoted N ‰ .22 heorem 4. Given a neighbourhood preserving Boolean coding ψ such that is coded by the Boolean profile, the inference by (14) from a Multi-valuedunitary stepwise network N “ x g, Y, Y y produces a Boolean network B “x f, p Y , p Y y such that the reflexive reductions of both networks are bisimilar: N ‰ „ ψ B ‰ .Extension to other modes. The method can be applied to any local-to-support mode but the admissible region may be reduced compared to theasynchronous mode. This reduction is caused by the decrease of the updatecapacity allowed by the mode. Hence, by Definition 1, this implies select-ing the appropriate codes for always reaching a code supporting an unitarystepwise transition for each update according to the mode components. Forthe running example, by using the parallel local-to-support mode with theSumming code, M “ tt x u , t p y , p y , p y uu , all the Boolean variables of p y areupdated jointly allowing to reach a single code instead of reaching the dif-ferent codes of the same integer by separate updates of variables. Thus, thecoding is reduced to Van Ham coding.Moreover, the trajectories starting from a state located outside the ad-missible region always end in the admissible region and no supplementaryequilibra are thus created (Proposition 4). This property generally holdsfor any coding that partially covers the Boolean state space.In conclusion, the domain of ψ may thus be reduced for a surjective de-coding function such as the Summing code without altering the asymptoticdynamics, but remains unchanged for a bijective decoding. Proposition 4.
Let f be an evolution function defined according to (14)from a Multi-valued unitary stepwise network N “ x g, Y, Y y . Let B “x f, p Y , M y be the corresponding Boolean network with M a local-to-supportmode, and ψ a decoding function such that N ‰ „ ψ B ‰ . All the states of theBoolean space eventually reach a state in the admissible region: @ w P B p Y , D w P dom ψ : w ÝÑ ‹ f w . Complexity of the algorithm.
Assume that the Multi-valued network has n variables reaching at most level L , the upper bound on the number ofregulators for a variable is r , the maximal number of support variables is m , and the maximal bound of code variants is c . Then the complexity ofthe Boolean guard is in O p nL r rcm q and the complexity of computing the23uard admissibility condition is in O p nLcm q . Thus the complexity of thealgorithm is dominated by the complexity of computing the Boolean guard.Accordingly, the computation time mainly depends on the maximal leveland the number of regulators. The algorithm is efficient in practice sincethe maximal level and the number of regulators often remain tractable onreal biological network models.
7. Conclusion
Bisimulation is at the core of the Multi-valued-to-Boolean network con-version process with the coding as a parameter. Although bisimulationpreserves reachability, more features are expected to be preserved in orderto directly perform analysis on the Boolean network. These features pertainto the nature of the equilibria and to the regulation of the variables. Wepropose a method whose outcome is a Boolean network explicitly defined byits Boolean formulas, with equilibria of the same nature as in the bisimilarMulti-valued network. The interaction graph of the Multi-valued networkcan be recovered from this Boolean network for any coding. In particular,we show that the Gray code providing the shortest Boolean representationhas the same properties as the Summing code, usually considered standardfor this conversion.Such automatic conversion sketches a pipeline where the Multi-valuednetwork becomes an input specification for modelling only, while the bulkof the analysis is performed on the Boolean network. Such pipeline suggeststhat the Boolean framework is central and sufficient for biological networkmodelling, thus calling to focus theoretical efforts on this framework sincethe results will benefit to both categories of discrete models via this pipeline.A perspective research direction would concern the study of bisimula-tion between Boolean networks. As bisimulation formally represents a formof behavioural equivalence, we could investigate the global properties offamilies of bisimilar Boolean networks in order to discover general rulesgoverning their behaviour. 24 ppendix
Proposition 1.
The global Boolean transition relation is the union of the lo-cal transition relations that are bisimilar to the local Multi-valued relations.As the union of bisimilar relations is bisimilar to the union relation, we de-duce that the global Boolean relation is bisimilar to the global Multi-valuedrelation.
Theorem 1.
We prove that Property (7) is met if and only if N „ ψ B . Wefirst prove the implication and next the reciprocal. Before, we prove thefollowing property for any mode component m P M used in the proofs: ´ m “ p p y i z m Y ´ p y i q (T1) Proof. ´ m = p Y z m by definition of ´ m ;= p p y i Y ´ p y i qz m as ´ p y i “ p Y z p y i by definition;= p y i z m Y ´ p y i z m = p p y i z m Y ´ p y i q since m Ď p y i by definition of thelocal-to-support mode, meaning that ´ p y i z m “ ´ p y i . p ùñ q Assume that Property (7) is met for the local-to-support mode M , i.e. , @ m P M : m Ď p y i ^ ψ p f m p w q Y w p y i z m q “ g i ˝ ψ p w q . • N simulates B . Let w m ÝÑ f w , m P M , be a transition in the model of B such that w, w P dom ψ . We define the transition ψ p w q ÝÑ ψ p w q by application of ψ on w and w . We have: ψ p w q “ ψ p f m p w q Y w ´ m q by definition of a transition(Section 2); “ ψ p f m p w q Y w p y i z m Y´ p y i q by (T1); “ ψ p f m p w q Y w p y i z m Y w ´ p y i q from (5) “ ψ p f m p w q Y w p y i z m q Y ψ p w ´ p y i q from (5) and (6); “ g i ˝ ψ p w q Y ψ p w ´ p y i q from (7), true byhypothesis.Set s “ ψ p w q and s “ ψ p w q . Then s ´ y i “ ψ p w ´ p y i q , because w ´ p y i isthe Boolean encoding of the rest of the state s ´ y i . We finally have: ψ p w q ÝÑ ψ p w q “ s ÝÑ g i p s q Y s ´ y i “ s y i ÝÑ g i s , which defines a transition of ÝÑ g i with the asynchronous mode y i .25 B simulates N . Let s y i ÝÑ g s be a transition in the model of N . As ψ : B p Y Ñ N Y is surjective, there exist two Boolean states w, w P B p Y such that: ψ p w q “ s and ψ p w q “ s . We prove that we can select w in the preimage of s so that a transition w m ÝÑ f w exists in themodel of the Boolean network B .Firstly, let us characterize s based on w . s “ g i p s q Y s ´ y i by definition of transition(Section 2); “ g i ˝ ψ p w q Y s ´ y i as ψ p w q “ s by hypothesis; “ ψ p f m p w q Y w p y i z m q Y s ´ y i from (7), true d by hypothesis; “ ψ p f m p w q Y w p y i z m q Y ψ p w y ´ y i q from (6) and ψ p w q “ s ; “ ψ p f m p w q Y w p y i z m q Y ψ p w ´ p y i q by definition of the support (5); “ ψ p f m p w q Y w p y i z m q Y w ´ p y i q from (6); “ ψ p f m p w q Y w p y i z m Y´ p y i by definition of the support (5); “ ψ p f m p w q Y w ´ m q by (T1).Thus, we conclude that ψ p w q “ s implies that w “ f m p w q Y w p y i z m .Hence, by definition of a transition, we have w m ÝÑ f w , meaning that B simulates N .In conclusion, if Property 7 is verified then networks N and B are bisimilarwith respect to ψ . p ðù q Assume that N „ ψ B . Hence, for all transitions w m ÝÑ f w such that w, w P dom ψ , there exist s, s P N Y such that s ÝÑ g i s and s “ ψ p w q , s “ ψ p w q .From the bisimulation, we deduce that: s “ ψ p w q by hypothesis; “ ψ p f m Y w ´ m q by definition of w m ÝÑ f w ; “ ψ p f m Y w p y i z m Y´ p y i q by (T1); “ ψ p f m Y w p y i z m Y w ´ p y i q from (5); “ ψ p f m Y w p y i z m q Y ψ p w ´ p y i q by (5), (6); “ ψ p f m Y w p y i z m q Y s ´ y i as s “ ψ p w q by hypothesis.From the definition of a transition, we deduce the following: s “ g i p s q Y s ´ y i as s ÝÑ g i s by hypothesis; “ g i ˝ ψ p w q Y s ´ y i as s “ ψ p w q by the bisimulationhypothesis.26s ´ y i X y i “ H because ´ y i “ Y z y i , we can simplify the equation byremoving s ´ y i in both part, leading to: ψ p f m Y w p y i z m q “ g i ˝ ψ p w q for all w P dom ψ, which defines Property (7). Lemma 1.
That admissibility for modes is reflexive and symmetric followsdirectly from Definition 3. To show transitivity of admissibility for modes,consider three arbitrary modes M , M , M Ď p Y , such that both adm ψ p M , M q and adm ψ p M , M q (with respect to the functional bisimulation B „ ψ N ).We can show that clause (1) of Definition 3 is satisfied for modes M and M in the following way: adm ψ p M , M q ^ adm ψ p M , M qùñ ` @ m P M , D m P M : adm ψ p m , m q ˘ ^ ` @ m P M , D m P M : adm ψ p m , m q ˘ Definition 3 (1) ùñ @ m P M , D m P M , D m P M : adm ψ p m , m q ^ adm ψ p m , m qùñ @ m P M , D m P M : adm ψ p m , m q , where the last transition is done by the symmetricity and transitivity ofadmissibility for modalities. Showing that clause (2) of Definition 3 is satis-fied for M and M can be done symmetrically, which implies adm ψ p M , M q and the transitivity of admissibility for modes. Theorem 2.
Consider the Boolean network B “ x f, p Y , M y and the Multi-valued network N “ x g i , Y, y i y , related by the bisimulation B „ ψ N . Pickan M -admissible mode M Ď p Y and consider the Boolean network B “x f, p Y , M y . We will show that B bisimulates N , B „ ψ N , by directlychecking clauses (1) and (2) of the definition of bisimulation (Definition 1). Clause (1) (forward simulation):
Take two Boolean states w, w P dom ψ such that w m ÝÑ f w for some m P M . We can then write the followingdeduction: w m ÝÑ f w ùñ D m P M , D w P dom ψw m ÝÑ f w ^ ψ p w q “ ψ p w q M is M -admissible ùñ D m P M , D w P dom ψψ p w q µ p m q ÝÑ g ψ p w q ^ ψ p w q “ ψ p w q B „ ψ N ùñ D m P M , D w P dom ψψ p w q µ p m q ÝÑ g ψ p w q . N is only allowed to update one variable, y i , µ p m q canonly be equal to t y i u . Clause (2) (backward simulation):
Take any two integer states s, s P N Y and an arbitrary Boolean state w P B X . Since N is only allowed to update y i , we can carry out the following deduction: ψ p w q “ s ^ s y i ÝÑ g s ùñ D w P B X , D m P M : µ p m q “t y i u ^ ψ p w q “ s ^ w m ÝÑ f w B „ ψ N ùñ D w P B X , D m P M , D w P B X , D m P M : µ p m q “ t y i u ^ ψ p w q “ s ^ w m ÝÑ f w ^ ψ p w q “ ψ p w q ^ w m ÝÑ f w M is M -admissible ùñ D w P B X , D m P M : ψ p w q “ s ^ w m ÝÑ f w . The two previous paragraphs show that the clauses of the definition ofbisimulation (Definition 1) are satisfied for the Boolean network B , runningunder mode M , and for the Multi-valued network N , meaning that B „ ψ N . The associated function mapping the modalities of B to those of N isthe unique total function 2 p Y Ñ t y i u (i.e., the same as for the bisimulation B „ ψ N ). Proposition 2.
By definition of a transition (Section 2), we have: @ w, w P B p Y , @ m P M : w m ÝÑ w ùñ d p w, w q ď | m | , As by hypothesis, @ y i P Y, @ s y i P N p y i , @ v, v P ψ ´ p s y i q : v ‰ v ùñ γ ă d p v, v q , where γ stands here for the greatest cardinality of M components, γ “ max t| m | , m P M u , we deduce that: d p w, w q ď | m | ď γ ă d p v, v q , meaning that the transitioncannot be achieved between codes of the same integer by hypothesis, thusleading to: @ w, w P B p Y , @ m P M : w ‰ w ^ w m ÝÑ w ùñ ψ p w q ‰ ψ p w q , As w “ f m p w q Y w ´ m by definition of a transition, this statement is equiv-alent to (9), concluding that the equilibrium stability is preserved.28 emma 2. let N be an asynchronous Multi-valued network bisimulating anasynchronous Boolean network B with: migs p N q “ x Y, , σ y , and bigs p B q “ x p Y , , σ B y , as their respectivesigned interaction graphs; let M p Y Ď p Y be a set of Boolean variables com-plying to (11), we prove Statement (4.1) by considering that Statement (4.2)holds.First we demonstrate two properties (L2.a) and (L2.b) used in the proof: @ w, w P dom ψ, @ p y ik P M p Y : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik ùñ ψ p w q ď ψ p w q . (L2.a) Proof.
Assume that: @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik for p y ik P M p Y .Two cases occur:1. w p y ik “ w p y ik : in this case w “ w leading to ψ p w q “ ψ p w q since ψ is afunction, thus satisfying ψ p w q ď ψ p w q .2. w p y ik ă w p y ik : as only two values are possible, 0 or 1, we deduce that w p y ik “
1. Hence, w can be defined as w “ w r p y ik ÞÑ s . As p y ik P M p Y by hypothesis, we conclude from (11) that ψ p w q ď ψ p w r p y ik ÞÑ s q . Thisinequality is equivalent to ψ p w q ď ψ p w q . @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik ùñ ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q . (L2.b) Proof.
Assume that: @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik . As w ´ p y i Ď w ´ p y ik and since p y ik P p y i , we have: w ´ p y ik “ w p y i ùñ w ´ p y i “ w p y i , thus implying that: @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y i “ w p y i . Hence, from Property (L2.a) applied to w p y i , w p y i , we deduce that: @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y i “ w p y i ùñ ψ p w p y i q ď ψ p w p y i q , Moreover, as ψ is a function defined on supports, we have: @ w, w P dom ψ : w ´ p y i “ w p y i ùñ ψ p w ´ p y i q “ ψ p w p y i q In conclusion, the following statement holds: ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q . ow we prove that Statement 4.1 is satisfied. The proof is given for positiveinteraction. p ùñ q
By definition (3), a positive interaction, p y ik ` p y jr is definedas: @ w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik ùñ f j,r p w q ď f j,r p w q . From (L2.b), we can rewrite this statement as: @ w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q^ ψ p w ´ p y i q “ ψ p w p y i q ùñ f j,r p w q ď f j,r p w q . Let v “ f j,r p w q Y w ´ x y jr and v “ f j,r p w q Y w x y jr , as f j,r p w q ď f j,r p w q byhypothesis, we conclude that: ψ p v q ď ψ p v q from (L2.a), thus leading to: @ w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q ùñ ψ p f j,r p w q Y w ´ x y jr q ď ψ p f j,r p w q Y w x y jr q . As N and B are bisimilar, Property (7) holds. By application of thisproperty we have: ψ p f j,r p w q Y w ´ x y jr q “ g j ˝ ψ p w q and similarly for w . Thuswe deduce that: @ w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q ùñ g j ˝ ψ p w q ď g j ˝ ψ p w q . Finally, as codom ψ “ N Y by definition, we can rewrite the previous state-ment as follows by setting, s “ ψ p w q , s “ ψ p w q : @ s, s P N Y : s i ď s i ^ s ´ i “ s i ùñ g j p s q ď g j p s q , that defines the positive interaction on migs (N): y i ` y j . p ðù q Assume that an interaction y i ` y j exists and there existtwo Boolean variables p y ik P M p Y X p y i and p y jr P M p Y X p y j with no positiveinteractions between these variables, i.e. , p y ik σ p y jr ùñ σ ‰ ` . We givea proof for the case σ “ ´ ; the proof for σ “ D w, w P dom ψ : w p y ik ď w p y ik ^ w ´ p y ik “ w p y ik ^ f j,r p w q ą f j,r p w q . From Property (L2.b), we can rewrite the previous statement as: D w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q ^ f j,r p w q ą f j,r p w q . v “ f j,r p w q Y w ´ x y jr and v “ f j,r p w q Y w x y jr , as f j,r p w q ą f j,r p w q byhypothesis, we conclude that: ψ p v q ą ψ p v q from (L2.a), thus leading to: @ w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q ^ ψ p w ´ p y i q “ ψ p w p y i q ùñ ψ p f j,r p w q Y w ´ x y jr q ą ψ p f j,r p w q Y w x y jr q . As N and B are bisimilar, Property (7) holds. By application of thisproperty we have: ψ p f j,r p w q Y w ´ x y jr q “ g j ˝ ψ p w q and similarly for w . Thuswe have: D w, w P dom ψ : ψ p w p y i q ď ψ p w p y i q^ ψ p w ´ p y i q “ ψ p w p y i q^ g j ˝ ψ p w q ą g j ˝ ψ p w q . As codom ψ “ N Y by definition, we can rewrite the previous statement asfollows by setting, s “ ψ p w q , s “ ψ p w q : D s, s P N Y : s i ď s i ^ s ´ i “ s i ^ g j p s q ą g j p s q , that contradicts the existence of a positive interaction y i ` y j , which isfalse by hypothesis.The proof for negative interaction follows the same scheme. Thus, weconclude that Statement 4.1 is satisfied. Theorem 3.
We prove that (11) holds for a set of Boolean variables belong-ing to a support p y i .Let p y ik be a Boolean variable of this set, two cases occur: either w p y ik “ w p y ik “
1. For the latter, w is left untouched by substitution leading to ψ p w q “ ψ p w r p y ik ÞÑ s q since ψ is a function, thus fulfilling (11). Hence, weaddress the case when w p y ik “ Summing code.
The following property holds when w p y ik “ ÿ p y ij P p y i z p y ik w p y ij “ ÿ p y ij P p y i w p y ij , thus, we have: ψ p w r p y ik ÞÑ s q “ ÿ p y ij P p y i z p y ik w p y ij ` “ ÿ p y ij P p y i w p y ij ` “ ψ p w q ` . We conclude that: ψ p w q ă ψ p w r p y ik ÞÑ s .31 an Ham code. Van Ham code is a sub-code of the Summing code, thuscomplying to its results.
Gray code.
Let p y i be a Boolean variable carrying the most significant digit,we separate p y i from the other variables in the definition of ψ : ψ p w q “ | p y i | ÿ k “ | p y i |´ k . k à j “ w p y ij “ | p y i |´ .w p y i ` | p y i | ÿ k “ | p y i |´ k . k à j “ w p y ij . Hence, when w p y i “
0, we deduce that ψ p w r p y i ÞÑ s q “ | p y i |´ ` ψ p w q ,leading to ψ p w q ă ψ p w r p y i ÞÑ s q .Thus we conclude that p Y is the set of markers for the Summing and VanHam code, while t p y i | y i P Y u are the markers for Gray code by applicationof Lemma 2. Theorem 4.
We first show that the computation of the Boolean function f defined by (14) is correct with respect to the integer function g and theasynchronous mode (A). Next (B), we examine the satisfaction of Prop-erty (7). Finally we demonstrate the bisimulation of the reflexive reductionfor both networks (C). A) The construction of f is correct. Let s y i ÝÑ s be a Multi-valued transition, such that s y i “ g i p s q “ l and s y j “ s y j for all 1 ď j ď n, j ‰ i , by definition of the asynchronousdynamics. We have: max p l ´ , q ď s y i ď min p l ` , L i q since the evolutionis unitary stepwise. There exist two Boolean states w, w P B p Y such that ψ p w q “ s as codom ψ “ N Y . We check that for all p y ik P p y i if f i,k p w q “ w p y ik then ψ p w q “ s and w ´ p y ik “ w p y ik , thus proving the correction of f i,k . Thefact that w ´ p y ik “ w p y ik is a direct consequence an asynchronous transitionupdating one variable only.Two cases are considered qualifying whether s y i ‰ s y i “
0. Foreach, we examine whether the target state of the support variable p y ik is 0or 1. Let us consider the following cases:1. s y i ‰
0: By definition of a Multi-valued network (1) C l p s q is necessarysatisfied as l “ s y i ‰
0. Let R p y i q be the set of regulators of y i ,we have: s R p y i q P C R p y i q ,l . Hence, we deduce that w p y i satisfies theBoolean version of the condition, C B l , by construction of the Booleancondition (12). Now we examine, the possible target states of thesupport variable p y ik , w p y ik : 32 w p y ik “
1: in this case w p y i belongs to Ψ ‹Ñ p l , p y ik q by definition,meaning that w admissible for the guard. Thus we have: f i,k p w q “ C B l p w q ^ C Ψ ‹Ñ p l , p y ik q p w q “ . • w p y ik “
0: in this case w p y i does not belong to Ψ ‹Ñ p l , p y ik q bydefinition meaning that w is not admissible for the guard. Thuswe have: f i,k p w q “ C B l p w q ^ C Ψ ‹Ñ p l , p y ik q p w q “ . In both cases, f i,k provides the expected result.2. s y i “
0: By definition of the Multi-valued dynamics (1), no guards aresatisfied. The conjunction of the guards for all levels is unsatisfiable,thus by definition of the part related to the guard in f i,k (12), wededuce that f i,k p w q “ w p y ik “ k . f returns the appropriate result regarding a pair w, w encoding the pair s, s . If s ‰ s then there exists a Boolean support variable p y ik , ď k ď | p y i | such that: ψ p f i,k p w q Y w ´ p y ik q “ s , corresponding to the following condition: f i,k p w q ‰ w p y ik . Otherwise ( s “ s ) any index k satisfies ψ p f i,k p w q Y w ´ p y ik q “ s . Notice that this part is not sufficient for proving bisimulation, since wemay have f i,j p w q “ w p y ij “ w p y ij by definition of the asynchronous dynamics,thus also leading to a transition w p y ij ÝÑ w by definition. This transition doesnot simulate the transition s y i ÝÑ s when s ‰ s , motivating the proof of thebisimulation restricted to the reflexive reduction. However a Multi-valuedself-loop ( s “ s ) is simulated by a self-loop in the Boolean network byconstruction of f . B) Property (7) is satisfied.
Let s y i ÝÑ s be a Multi-valued asynchronous transition such that s ‰ s ,then there exist w, w P B p Y such that ψ p w q “ s, ψ p w q “ s , by constructionof f (A). Moreover, we have: ψ p w q ‰ ψ p w q , leading to w ‰ w , as ψ is afunction. Thus, a Boolean support variable p y ik verifies that w p y ik ‰ w p y ik ,as g and f are neighbourhood preserving (8). In this case, we have: w “ f i,k p w q Y w ´ p y ik from (A). Thus, we have the following equalities:33 = ψ p w q by definition of ψ ;= ψ p f i,k p w q Y w ´ p y ik q by construction of f (A);= ψ p f i,k Y w p y i z p y ik Y´ p y i q from (T1);= ψ p f i,k Y w p y i z p y ik Y w ´ p y i q from (5);= ψ p f i,k Y w p y i z p y ik qY ψ p w ´ p y i q by (5), (6);= ψ p f i,k Y w p y i z p y ik q Y s ´ y i as s “ ψ p w q .As s “ g i p s q Y s ´ y i by definition of a transition, we deduce by simplifi-cation of s ´ y i that: g i p s q “ g i ˝ ψ p w q “ ψ p f i,k Y w p y i z p y ik q , concluding thatProperty (7) holds. C) Bisimulation between reflexive reductions.
It follows from (B), that wecan set that Property (7) holds whenever s y i ‰ g i p s q . As the asynchronousmode is local-to-support, and we always have s y i ÝÑ s ùñ s y i ‰ s y i “ g i p s q by reflexive reduction, we conclude by application of Theorem 1 that thereflexive reduction of the Multi-valued and Boolean networks are bisimilarwith respect to ψ . Proposition 4.
We denote: m a Boolean state with 0 for all variables in m Ď p Y .If a state is in the admissible region then it always reaches states inthe admissible region, and only in the admissible region, by definition ofbisimulation.If the Boolean state w P B p Y is outside of the admissible region, w R dom ψ , then it is not accounted for by the computation of admissibility ofthe guard condition, by definition of Ψ ‹Ñ . Therefore we have: f m p w q “ m , @ m P M . Thus, all the trajectories starting from w successively cancel(set to 0) the states of the variables of m P M whenever the result of thecancellation leads to a state outside the admissible region; otherwise theproposition holds. As Ť m P M m “ p Y by definition of a mode, the cancellationprocess terminates at state p Y , which is always in the admissible region since p y i , @ y i P Y, is the sole code for the integer value 0, regardless of the variable y i . In conclusion, the trajectories starting from any w P B p Y eventually endup in a state in the admissible region dom ψ .34 eferences [1] Fadi A. Aloul, Arathi Ramani, Igor L. Markov, and Karem A. Sakallah. GenericILP versus specialized 0-1 ILP: An update. IEEE/ACM International Conferenceon Computer-Aided Design, Digest of Technical Papers , 1:450–457, 2002.[2] Gilles Didier, Elisabeth Remy, and Claudine Chaouiya. Mapping multivalued ontoBoolean dynamics.
Journal of Theoretical Biology , 270(1):177–184, 2011.[3] Adrien Faur´e and Shizuo Kaji. A circuit-preserving mapping from multilevel toBoolean dynamics.
Journal of Theoretical Biology , 440:71–79, 2018.[4] Leon Glass and Stuart A. Kauffman. The logical analysis of continuous, non-linearbiochemical control networks.
Journal of Theoretical Biology , 39(1):103–129, 1973.[5] S. A. Kauffman. Metabolic stability and epigenesis in randomly constructed geneticnets.
Journal of Theoretical Biology , 22(3):437–467, 1969.[6] Davide Sangiorgi.
Introduction to bisimulation and coinduction , volume 9781107003.2011.[7] Denis Thieffry and Ren´e Thomas. Dynamical behaviour of biological regulatorynetworks-II. Immunity control in bacteriophage lambda.
Bulletin of MathematicalBiology , 57(2):277–297, 1995.[8] Rene Thomas, Denis Thieffry, and Marcelle Kaufmann. Behaviour of BiologicalBiological Role of Feedback Loops and Practical Use of the Concept of the Loop-Characteristic.
Bulletin of mathematical biology , 57(2):247–276, 1995.[9] Elisa Tonello. On the conversion of multivalued to Boolean dynamics.
DiscreteApplied Mathematics , 259:193–204, 2019.[10] Phillipe Van Ham. How to deal with variables with more than two levels. In Ren´eThomas, editor,
Kinetic Logic A Boolean Approach to the Analysis of ComplexRegulatory Systems , pages 326–343, Brussel, 1979. Springer Berlin Heidelberg., pages 326–343, Brussel, 1979. Springer Berlin Heidelberg.