Joint Realizability of Monotone Boolean functions
aa r X i v : . [ m a t h . D S ] D ec JOINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS
PETER CRAWFORD-KAHRL, BREE CUMMINS, AND TOMAS GEDEON
Abstract.
The study of monotone Boolean functions (MBFs) has a long history. Weexplore a connection between MBFs and ordinary differential equation (ODE) models of generegulation, and, in particular, a problem of the realization of an MBF as a function describingthe state transition graph of an ODE. We formulate a problem of joint realizability of finitecollections of MBFs by establishing a connection between the parameterized dynamics of aclass of ODEs and a collection of MBFs. We pose a question of what collections of MBFs canbe realized by ODEs that belong to nested classes defined by increased algebraic complexityof their right-hand sides. As we progressively restrict the algebraic form of the ODE, weshow by a combination of theory and explicit examples that the class of jointly realizablefunctions strictly decreases. Our results impact the study of regulatory network dynamics,as well as the classical area of MBFs. We conclude with a series of potential extensions andconjectures. Introduction
The study of Boolean functions in general and monotone Boolean functions in particularhas a long history [5, 24, 11, 22, 32, 4, 1]. One area in which monotone Boolean functions(MBFs) have been used is in modeling the dynamics of gene regulatory networks. In thesemodels the (Boolean) state of each node i in the regulatory network is updated based on the(Boolean) states of the nodes j that that are connected by an edge from j to i . The mono-tonicity requirement on a Boolean function (Definition 2.3) reflects the fact that the edgesin gene regulatory networks are signed and thus the effect of one gene on another is alwayseither monotonically increasing (activating edge) or monotonically decreasing (repressingedge).An alternative class of network models uses continuous time dynamics of ordinary dif-ferential equations. We are interested in a particular class of such models with piecewiselinear right-hand sides [29, 16, 17, 13, 10, 3, 27]. For the most general of these models,which we call K -systems , the right-hand side is fully determined by a finite collection of con-stants K = ( K , . . . , K n ), where K i is a collection of constants that describes the activity ofnode i in the regulatory network. Each collection K i within K also satisfies a monotonicitycondition that reflects the monotone effect of the edges in the network.The main goal of this paper is to show that there is a close relationship between K -systemsand collections of monotone Boolean functions. In order to show this connection, we firstshow that to each K -system one can associate a state transition graph (STG), which is afinite directed graph that coarsely captures the progression of the trajectories of the K -system. There are a finite number of STGs, which permits an imposition of an equivalencerelation on the (infinite) set of K -systems, with an equivalence class denoted [ K ].Our first major result is the correspondence between the equivalence classes [ K ] andcollections of MBFs. For a fixed regulatory network with n nodes, each equivalence class[ K ] has the form [ K ] = ([ K ] , . . . , [ K n ]). Then each [ K i ] for a node i with m i input edges and b i output edges corresponds to a collection of b i monotone Boolean functions with m i inputs. Moreover, each such collection of b i MBFs, satisfying an additional condition thatthe truth sets are linearly ordered by inclusion, is associated to an equivalence class [ K i ].Using this result, the equivalence classes [ K ] are arranged into a parameter graph (PG)specific to the regulatory network under consideration. The edges of the PG are determinedby the adjacency of the collections of MBFs associated to each [ K ].Our next set of results examines the effect of imposing algebraic restrictions on the formof the right-hand side of the differential equations of the network model, which results inadditional structure on the set K . These additional algebraic restrictions decrease the sizeof the parameter graph. We ask which MBFs, and which collections of MBFs, are realizableas parameter nodes of the corresponding restricted parameter graphs.The classes of algebraic functions that we chose to examine are nested; the most restrictedand smallest class consists of linear functions, Σ, which is a subset of functions that canbe obtained as products of sums of individual variables, ΠΣ, and lastly sums of productsof sums, ΣΠΣ. These classes are all special cases of K -systems and therefore admit STGsand PGs. These algebraic restrictions are motivated by the software DSGRN [8, 9, 15, 14],which calculates the PGs and STGs for network models with the class of ΠΣ functions, andin principle can be extended to other classes of algebraic expressions.We show that the classes Σ, ΠΣ, and ΣΠΣ do impose constraints on pairs of MBFs thatcan be realized as parameter nodes of a PG. In fact, we show that the classes of pairs ofMBFs with three inputs that are realizable as linear functions are a strict subset of ΠΣ-jointly realizable pairs, which is in turn a strict subset of ΣΠΣ-jointly realizable pairs ofMBFs. We also show that there are pairs of MBFs for any n ≥ K -jointlyrealizable but are not ΣΠΣ realizable.These results show that the increased complexity of the algebraic expression provides aricher class of models as measured by the set of MBFs that can be realized in a PG. Atthe same time, the connection between differential equation models and collections of MBFsallows for the formulation of a host of interesting questions (see the Discussion) about what k -tuples of MBFs can be realized as nodes of parameter graphs of differential equation modelsas the complexity of the right-hand side varies.2. K -systems and Monotone Boolean Functions A regulatory network is a useful abstraction for organizing information about interactingunits. Nodes represent units and (directed) edges interaction between the nodes.
Definition 2.1. A regulatory network RN is a triple RN := { V, E, s } where • V is the set of vertices ; • E ⊂ V × V is a finite set of oriented edges , where ( i, j ) denotes the edge from i to j ; • s : E → { + , −} is the sign of the edge.We will generally use n = | V | . We denote S ( i ) to be the set of sources of node i and T ( i )the set of targets of node i : S ( i ) = { j ∈ V | ( j, i ) ∈ E } and T ( i ) = { j ∈ V | ( i, j ) ∈ E } . We split the set of sources into activating and repressing inputs as S ( i ) = S ( i ) + ∪ S ( i ) − where j ∈ S + ( i ) iff e = ( j, i ) ∈ E and s ( e ) = + OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 3 and j ∈ S − ( i ) iff e = ( j, i ) ∈ E and s ( e ) = − The interpretation of the signed edges comes from biology; a positive edge signifies up-regulation, where the rate of change of the target node concentration increases as the con-centration of the source node increases. A negative edge signifies down-regulation, wherethe rate of change of the target node concentration decreases as the concentration of thesource node increases. Inherent in this description is monotonicity of the rate of change ofthe target node with respect to changes in each of the source nodes [2, 14, 19, 21, 20, 12].One of the natural ways to associate dynamics to a network is using Boolean functions.Every node is assumed to be either OFF, corresponding to low concentration, representedby the state 0, or ON, corresponding to high concentration, represented by the state 1.
Definition 2.2.
Let B := { , } . We will use the notation B n := { , } n for the vertices ofa hypercube of dimension n . A Boolean function is a function f : B n → B .In examples, we will often write elements of B n as strings (e.g. 10010 ∈ B ). Definition 2.3. [6] A Boolean function f : B n → B is positive (resp. negative ) in x i if f | x i =0 ≤ f | x i =1 (resp. f | x i =0 ≥ f | x i =1 ), where f | x i =0 (resp. f | x i =1 ) denotes the value of f ( x , ..., x i − , , x i +1 , ..., x n ) (resp. f ( x , ..., x i − , , x i +1 , ..., x n )) for any Boolean values of x , . . . , x i − , x i +1 , . . . x n . We say that f is monotone in x i if it is either positive or negativein x i . f is monotone if it is monotone in x i for all i ∈ { , ..., n } .Positive and negative monotone Boolean functions (MBF) capture the effect of positiveand negative edges in the network RN , respectively. We will use the notation MBF ( n ) := { f : B n → B | f is monotone } MBF + ( n ) := { f : B n → B | f is positive in x i for all i ∈ { , ..., n }} The dynamics of the network with n nodes is described by iteration of f : B n → B n ,where f := ( f , f , . . . , f n ) is a collection of MBFs. Monotone Boolean function models arewidely used due to their simplicity, but matching their predictions to experimental valuesof continuous variables like concentration always poses a challenge. An effort to combinethe simplicity of Boolean maps with a continuous time description was initiated by [16,17, 30]. To explain this approach we extend the definition of regulatory network given inDefinition 2.1. Definition 2.4. A weighted regulatory network is a regulatory network RN with positive,real-valued weights assigned to each node, γ = ( γ , . . . , γ n )with n = | V | , and positive, real-valued weights assigned to each edge(1) θ i = { θ s i , θ s i , . . . , θ s bi i } where { s , s , . . . , s b i } = T ( i ) , with θ = S i θ i . We want to bring attention to the the indexing we use: θ ji is associated tothe edge from i to j , in the tradition of [8]. The node weights are called decay rates and theedge weights are called thresholds . We assume that for each node i , the b i thresholds in thecollection θ i are distinct. PETER CRAWFORD-KAHRL, BREE CUMMINS, AND TOMAS GEDEON θ = 3 θ = 1 . θ = 2 γ = 1 γ = 1 θ θ θ S + (1) = { } S − (1) = { } S + (2) = { } S − (2) = ∅ Figure 1.
An example weighted regulatory network. Here V = { , } . Weuse → to denote a positive (activating) edge, and ⊣ to denote a negative(inhibiting) edge. We also illustrate the sets of sources for each node.The idea of decay rates comes from biology and indicates how quickly a gene product willbreak down under natural cellular processes. A common model of enzymatic gene regulationis the sigmoidal Hill function model, which has a half-saturation value. These half-saturationvalues are sometimes treated as thresholds, here represented as weights on edges. An exampleweighted regulatory network is shown in Figure 1.2.1. K-systems.
The most general attempt to combine the simplicity of Boolean mapswith a continuous time description resulted in switching K -systems , consisting of a systemof differential equations on R n + . The “ K ” in K -system denotes a finite collection of realvalues that satisfy a monotonicity assumption (Definition 2.5) and are used to parameterizea system of ordinary differential equations (ODEs) with discontinuous right-hand sides.Given a weighted regulatory network, we associate to each node i a continuous non-negative variable x i ∈ R + , usually representing the concentration of a gene product. Wewrite the collection of gene concentrations as a vector x = ( x , . . . , x n ) ∈ R n + . The thresholds θ i divide the x i axis into b i + 1 intervals, where b i = | θ i | is the number of targets of node i .We enumerate these by the integers 0 , . . . , b i in ascending order. Then X = R n + \ { x i = θ ji | i ∈ , . . . , n, θ ji ∈ θ i } is an open rectangular grid where each component of X is an open domain. As we willsee next, the collection of real numbers K determines an ODE system defined on X whosesolutions are consistent with a discrete mapping between open domains in X and this discretemap can be interpreted as a collection of MBF s. The following definition of the K -systemgoes back to at least Thieffry and Romero [28]; we follow the exposition in [3]. Definition 2.5.
Recall the definition of a regulatory network RN in Definition 2.1, partic-ularly the nodes V and the sources S ± ( i ). Let(2) K := { K i,A,B ∈ R + | i ∈ V, A ⊂ S + ( i ) , B ⊂ S − ( i ) } be a collection of non-negative numbers that satisfies the monotonicity assumption : • For each i ∈ V , if A ( A ′ ⊂ S + ( i ) then K i,A,B ≤ K i,A ′ ,B for all B ⊂ S − ( i ) . • For each i ∈ V , if B ( B ′ ⊂ S − ( i ) then K i,A,B ≥ K i,A,B ′ for all A ⊂ S + ( i ) . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 5
Fixing a collection of constants K satisfying the monotonicity assumption, define a parameterassignment function for each continuous variable x i from the power sets of S + ( i ) and S − ( i )into the positive real numbers: k i : 2 S + ( i ) × S − ( i ) → R + , k i ( A, B ) := K i,A,B We write k = ( k , . . . , k n ) as the collection of these parameter assignment functions, one foreach component of the system.We continue the example from Figure 1 by listing an example assignment of numbers K that satisfy the monotonicity assumption. K , ∅ , = 0 . K , ∅ , ∅ = 0 . K , ∅ , ∅ = 0 . K , , ∅ = 0 . K , , = 5 K , , ∅ = 6Up to now,the construction of K has depended only on the structure of an unweightedregulatory network RN . We now take into account the weights associated to RN as inDefinition 2.4. The K-system ODE, that we describe next, will depend on the these weights.The dynamics of variables x i are affected by the incoming edges to node i in the regulatorynetwork RN . For each x j ∈ S ( i ), the value of x j is either above or below the threshold θ ij assigned to the edge from j to i in the weighted regulatory network. If x j ∈ S + ( i ) has anactivating effect, then x j > θ ij implies that x i will be produced at a greater rate than when x j < θ ij . The inequalities are swapped for a repressing effect, x j ∈ S − ( i ). With this in mind,we define the activity function for a node i and x ∈ X , as follows:(4) ζ i : X → S + ( i ) × S − ( i ) , ζ i ( x ) = ( A i , B i ) A i = { j ∈ S + ( i ) | x j > θ ij } B i = { j ∈ S − ( i ) | x j > θ ij } . The map ζ := ( ζ , . . . , ζ n ), defined on X , is constant on each open domain of X .The composition of the parameter assignment function with the activity function, k i ◦ ζ i ,assigns to a vector x ∈ X a scalar parameter k i ( A, B ) = k i ( ζ i ( x )) in the set K . Recalling thedecay rates γ from Definition 2.4, we are now in a position to define a differential equationparameterized by K and defined on X . Definition 2.6.
The system(5) ˙ x i = − γ i x i + k i ( ζ i ( x ))is called the K -system on X associated to the weighted regulatory network RN .Note that since ζ is constant on the open domains of X , the differential equation is linearin each such domain. On the boundaries of the domains, the system is undefined due to thediscontinuity in k i ( ζ i ( x )). However, we extend the system by continuity, whenever possible,from X to R n + . The assumption of the non-negativity of k guarantees that the non-negativeorthant R n + is positively invariant and the concentrations x i remain non-negative for all t ≥ PETER CRAWFORD-KAHRL, BREE CUMMINS, AND TOMAS GEDEON x x θ θ θ x k ◦ ζγ ( x ) (1 ,
1) (2 ,
1) (3 , ,
2) (2 ,
2) (3 , ζ ( x ) = ( ∅ , { } ) , ζ ( x ) = ( { } , ∅ ) k ◦ ζγ ( x ) = ( . , . κ ( . , .
4) = (1 , Figure 2.
Continuing the example from Figure 1 we construct the statetransition graph. Above left is X and right is D . For any x in the shadeddomain, the value of k ◦ ζγ ( x ) is constant and located in the lower left domain.This determines the value Φ K (2 ,
2) = (1 ,
1) denoted with a dashed arrow; see(8).2.1.1.
State transition graph.
Let D := n Y i =1 { , , . . . , b i } be a set of n integer sequences that will be referred to as states . Recall that b i is the numberof targets of the node i in RN , b i = | θ i | . We construct a function between the points of X and the states in D . Let κ i : X → { , , . . . , b i } be a map(6) κ i ( x ) = ℓ when x i ∈ ( θ ℓ ∗ i , θ ℓ +1 ∗ i )where superscript ℓ indicates the ℓ -th domain of the x i axis. Collecting the maps κ i in asingle map we define κ : X → D to be the index assignment map κ = ( κ , . . . , κ n ). Wesay that D indexes the open domains of X . By construction, the index assignment map isconstant on each domain in X . The map κ takes a vector x ∈ X and assigns it to the staterepresenting the domain of X in which x lies.The function k ◦ ζγ = (cid:18) k ◦ ζ γ , . . . , k n ◦ ζ n γ n (cid:19) is a map k ◦ ζγ : X → X . We define a discrete map Φ K : D → D on the set of states D byrequiring that(7) Φ K ◦ κ = κ ◦ k ◦ ζγ , OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 7 (1 ,
1) (2 ,
1) (3 , ,
2) (2 ,
2) (3 ,
2) (1 , ,
1) (2 ,
1) (3 , ,
2) (3 , Figure 3.
Left: The state transition graph is the asynchronous update dy-namics, and so does not allow the diagonal transition; instead, we replace thedashed arrow with arrows capturing one-step adjacency. Right: the completedSTG, where the process that was illustrated for state (2 ,
2) is repeated for eachstate.i.e. that the following diagram commutes(8)
X XD D k ◦ ζγ κ κ Φ K Note that the solution of (5) with initial condition x converges to the target point k ◦ ζγ ( x ).The map Φ K takes the state d = κ ( x ) and assigns it to the state Φ K ( d ) which contains thetarget point k ◦ ζγ ( x ). In this way, the map Φ K captures the behavior of solutions of (5). It isimportant to note that the convergence of the solution starting at x toward its target point k ◦ ζγ ( x ) is only valid while the solutions remain in the component of X containing x ; whenthey enter a neighboring domain, the target point will change.To capture this behavior, we define a state transition graph on states d ∈ D that coarselydescribes solutions of (5). It represents the asynchronous update dynamics for the discretevalued function Φ K . Definition 2.7 (State transition graph) . The state transition graph (STG) is a directedgraph with nodes D , where two nodes d, d ′ ∈ D are connected by a directed edge d → d ′ , ifand only if(1) either d = d ′ and Φ K ( d ) = d ; or(2) d and d ′ differ in exactly one component, say i , and d ′ i = d i + 1 and Φ Ki ( d ) > d i , or d ′ i = d i − Ki ( d ) < d i We construct the state transition graph of our example network in Figures 2 and 3.The number of maps Φ K for a given RN is finite. This induces an equivalence relation overall collections K satisfying the monotonicity condition in Definition 2.5 that are consistentwith the structure of RN . Definition 2.8.
For a given weighted regulatory network, we define an equivalence relationon the collection of all parameter sets K . We set K ∼ = K ′ ⇐⇒ Φ K ( d ) = Φ K ′ ( d ) for all d ∈ D. Notice that each equivalence class [ K ] has a component structure composed of n independentequivalence classes, [ K ] = ([ K ] , . . . , [ K n ]), one for each node i ∈ V in the regulatory network. PETER CRAWFORD-KAHRL, BREE CUMMINS, AND TOMAS GEDEON y y g g
00 0 001 0 010 1 111 1 1 y g ,
1) (2 ,
1) (3 , ,
2) (2 ,
2) (3 , y y g g
00 0 101 0 010 1 111 1 1 y g ,
1) (2 ,
1) (3 , ,
2) (2 ,
2) (3 , Figure 4.
Left: the collection of MBFs corresponding to our example net-work and choice of K in (3), and the associated state transition graph. Right:A collection of MBFs adjacent (in the parameter graph) to the collection onthe left. The single change is highlighted in gray. The corresponding statetransition graph is also pictured; differences caused by the shaded entry areshown as dashed edges .This is because the monotonicity assumption in Definition 2.5 applies independently to eachnode.2.2. Equivalence classes [ K ] are collections of MBFs. We now discuss the connectionbetween equivalence classes [ K ] and monotone Boolean functions. Each equivalence class[ K ] is uniquely associated to a collection of Q ni =1 b i MBFs, where b i = | T ( i ) | is the number oftargets of node i in RN . We label these MBFs g ki , one for each threshold θ ki in the weightedregulatory network, and construct them below.Let [ K ] = ([ K ] , . . . , [ K n ]) be an equivalence class, and consider an element K i ∈ [ K i ],where K i = { K i,A,B } A ⊂ S + ( i ) ,B ⊂ S − ( i ) is a collection of constants for node i in RN . Define afunction α i : B S ( i ) → S + ( i ) × S − ( i ) where α i ( ~y ) = ( A i , B i ) for A i = { j ∈ S ( i ) | y j = 1 } ∩ S + ( i ) , B i = { j ∈ S ( i ) | y j = 1 } ∩ S − ( i ) . Here we use the standard multi-index notation B S ( i ) = { y j y j . . . y j mi | j k ∈ S ( i ) , y j k ∈ B } ,i.e. elements of B S ( i ) are Boolean strings of length | S ( i ) | indexed by elements of S ( i ) inorder. As an example, if S ( i ) = { , , , } , then ~y = y y y y , where y i ∈ B .Let { θ s i , θ s i , . . . θ b i i } be the b i thresholds associated to the b i targets of node i in RN .With this assignment, we define positive Boolean functions g ki : B S ( i ) → B as g ki ( ~y ) = ( K i,α i ( ~y ) > θ ki γ k K i,α i ( ~y ) < θ ki γ k OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 9 and negative Boolean functions as g ki ( ~y ) = ( K i,α i ( ~y ) > θ ki γ k K i,α i ( ~y ) < θ ki γ k We observe that if j ∈ S + ( i ), then g ki will be positive in x j , and if j ∈ S − ( i ), then g ki willbe negative in x j . Therefore, any g ki constructed in this way will be a monotone Booleanfunction g ki ∈ MBF ( | S i | ). As we show next, the collection G = { g ki | i = 1 , . . . , n, k ∈ T ( i ) } is an equivalent representation of the equivalence class [ K ]. Proposition 2.9.
Fix a weighted regulatory network RN and thus sets S + ( i ) , S − ( i ) , and T ( i ) , as well as the weights γ i and θ ji for all i = 1 , . . . , n , j ∈ T ( i ) . Then Φ K = Φ K ′ ⇐⇒ G = G ′ . Proof.
Consider two different collections
K, K ′ associated to parameter assignment functions k, k ′ respectively. Note that the set of states D , the function κ : X → D and the function ζ : X → S + ( i ) × S − ( i ) are uniquely determined by the weighted regulatory network RN .Therefore, Φ K and Φ K ′ differ only in the functions k and k ′ :Φ K ◦ κ = κ ◦ k ◦ ζγ Φ K ′ ◦ κ = κ ◦ k ′ ◦ ζγ . Therefore it follows thatΦ K = Φ K ′ ⇐⇒ κ ◦ kγ = κ ◦ k ′ γ ⇐⇒ κ ◦ kγ ◦ α = κ ◦ k ′ γ ◦ α. This in turn leads to the equivalencies: ⇐⇒ κ ( K i,α i ( ~y ) /γ i ) = κ ( K ′ i,α i ( ~y ) /γ i ) ∀ i = 1 , . . . , n and ∀ ~y ∈ B S ( i ) ⇐⇒ K i,α i ( ~y ) /γ i , K ′ i,α i ( ~y ) /γ i ∈ ( θ ℓ ∗ i , θ ℓ +1 ∗ i ) for some ℓ ⇐⇒ K i,α i ( ~y ) , K ′ i,α i ( ~y ) ∈ ( θ ℓ ∗ i γ i , θ ℓ +1 ∗ i γ i ) for some ℓ ⇐⇒ (cid:0) K i,α i ( ~y ) > θ ji γ i ⇐⇒ K ′ i,α i ( ~y ) > θ ji γ i (cid:1) ∀ j ∈ T ( i ) ⇐⇒ (cid:0) g ji ∈ G ⇐⇒ g ji ∈ G ′ (cid:1) . (cid:3) For an example, compare the collection K in (3) for the weighted regulatory network inFigure 1 to the equivalent collection of three monotone Boolean functions in the left panelof Figure 4.2.3. Parameter Graph.
The fact that we can view equivalence classes [ K ] as collections ofmonotone Boolean functions allows us to organize the equivalence classes [ K ] into a graph,called the parameter graph (PG) , where each node is associated to an equivalence class. Let[ K ] and [ K ′ ] be different equivalence classes, with associated collections of MBFs G = { g ki } and G ′ = { g ′ ki } . The nodes [ K ] and [ K ′ ] will be connected by an edge, if, and only if, thereis i ∈ { , . . . , n } , k ∈ T ( i ), and ~y ∈ B S ( i ) such that g ki ( ~y ) = g ′ ki ( ~y ) g ki ( ~z ) = g ′ ki ( ~z ) ∀ ~z = ~y ×
20 node graph
Figure 5.
Continuing the example, we show the parameter graph for thenetwork in Figure 1. The parameter graph takes the form of a product graph,with one factor for each node in RN . The factor associated to node 1, whichhas two inputs and two outputs in RN , is the 20 node graph on the left.It is isomorphic to the graph shown in Appendix B, Figure 9. The factorassociated to node 2, which has one input and one output in RN , is shown onright. Each parameter node in the factor on the right contains the associatedmonotone Boolean function, where gray shading means g = 0, and similarlywhite shading implies g = 1. g ℓj = g ′ ℓj whenever j = i or ℓ = k. In other words, there is exactly one MBF whose value differs on one input. An example ofa single adjacency is shown in Figure 4 and the parameter graph for our running example isshown in Figure 5.2.4.
Representative networks.
It will be convenient to consider a subset of weightedregulatory networks RN with γ i = 1 for all i . It turns out that the class of weightedregulatory networks with this property exhibits the same parameter graphs with the samecollection of state transition graphs as the collection of weighted graphs with general positivedecay rates γ = ( γ , . . . , γ n ).To see this, fix a set K and its parameter assignment function k from Definition 2.5.Consider a weighted regulatory network RN with the collections of sources S + ( i ), S − ( i ),targets T ( i ), decay rates γ = ( γ , . . . , γ n ), and thresholds { θ ji } . Consider a network d RN with the same collection of sources S + ( i ), S − ( i ) and targets T ( i ), but with all decay ratesset to ˆ γ i = 1 and the thresholds set to { ˆ θ ji = γ i θ ji } .The threshold assignment induces a bijection x ˆ x with ˆ x = γx , from X to ˆ X :ˆ X = R n + \ { ˆ x i = ˆ θ ji | i = 1 , . . . , n, j ∈ T ( i ) } . The key observation is that(9) ˆ x i ∈ (cid:0) γ i θ ℓ ∗ i , γ i θ ℓ +1 ∗ i (cid:1) ⇔ x i = ˆ x i γ i ∈ (cid:0) θ ℓ ∗ i , θ ℓ +1 ∗ i (cid:1) . This allows us to conclude that the activity functions ζ and ˆ ζ (defined in (4)) satisfy ˆ ζ (ˆ x ) = ζ (ˆ x/γ ) = ζ ( x ), which leads to θ ji < k i ◦ ζγ i ( x ) < θ si for some j, s ∈ T ( i ), if and only if ˆ θ ji < k i ◦ ˆ ζ (ˆ x ) < ˆ θ si . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 11
In other words, the following diagram commutes:
X X ˆ X ˆ X k ◦ ζγ γx γxk ◦ ˆ ζ Since the underlying network topology is the same between two weighted networks RN and d RN , the discrete states of the state transition graph are the same D = ˆ D . Using (9)again, we conclude that the index assignment functions κ and ˆ κ from (6) satisfy the following:ˆ κ (ˆ x ) = ℓ ⇔ κ ( x ) = ℓ. Recalling that Φ K ◦ κ = κ ◦ ( k ◦ ζ ) /γ from (7), we see thatˆΦ K ◦ ˆ κ (ˆ x ) = Φ K ◦ κ ( x ) . This means that the state transition graphs are identical under K applied to RN and d RN .We conclude that by considering the restricted class of weighted regulatory networks with γ i = 1 for all i we will recover the same set of state transition graphs as the general system.Therefore we we will assume γ i = 1 from now on, and we will write g ki as g ki ( ~y ) = ( K i,α i ( ~y ) > θ ki K i,α i ( ~y ) < θ ki or(10) g ki ( ~y ) = ( K i,α i ( ~y ) > θ ki K i,α i ( ~y ) < θ ki . Differential equations from monotone Boolean functions.
In Definition 2.6 weassociated an ordinary differential equation to a weighted regulatory network RN . A moreexplicit way to do so is due to [16, 17, 30]. Again consider a weighted regulatory network RN with nodes i ∈ V summarizing regulatory activity for continuous variables x i . Assumethat regulation of x i by its regulatory input x j switches abruptly at the real-valued threshold θ ij from RN , written as one of two maps σ + ij ( x j ) = (cid:26) x j > θ ij x j < θ ij σ − ij ( x j ) = (cid:26) x j > θ ij x j < θ ij whenever j ∈ S ( i ) is a source of node i . We write σ i = ( σ is , . . . , σ is mi ), where s j ∈ S ( i ), m i = | S ( i ) | , σ ik = σ + ik whenever k ∈ S + ( i ), and σ ik = σ − ik whenever k ∈ S − ( i ). In otherwords, σ + models an activating input and σ − a repressing input.We shall again assume that any two thresholds θ ji and θ ki are distinct for variable x i . Alsoas before, these thresholds { θ ij | i = 1 , . . . , n, j ∈ S ( i ) } divide R n + into an open rectangulargrid X . In addition, assume that for very node i ∈ V there is an associated Boolean function f i : B S ( i ) → B that converts inputs of the node i to the new state of node i . Then we considerthe following system of ODEs on X :(11) ˙ x i = − x i + f i ( σ i ( x )) Algebraic switching systems
The system of ODEs (11) has no continuous parameters. In order to introduce suchparameters and allow comparison with K-systems we parameterize both the domain and therange of each function f i : B n → B . To parameterize the domain we replace in the definitionof σ ij the Boolean values 0 < L ij < U ij .To capture the sign along the network edges, we again consider two types of σ ij functions σ + ij ( x j ) = (cid:26) U ij x j > θ ij L ij x j < θ ij σ − ij ( x j ) = (cid:26) L ij x j > θ ij U ij x j < θ ij . We introduce the interaction function Λ i as a real-valued replacement for the function f i .All interaction functions will be algebraic expressions over the real numbers using additionand multiplication. In this notation, Equation (11) reads˙ x i = − x i + Λ i ( σ i ( x )) , and we refer to it as a switching system , as in [8, 15].For every x ∈ X , the composition Λ i ( σ i ( x )) assigns a real number that is a combination ofnumbers { L ij or U ij | j ∈ S ( i ) } , where for each j only one of L ij or U ij enters the functionΛ i . This combination is constant on each domain in X . For monotone functions Λ i , theimage of Λ i ( σ i ( x )) is a set K that satisfies the monotonicity assumption in Definition 2.5.Therefore the switching system with monotone Λ i for all i is a K-system (5) and thereforegives rise to a parameter graph and to a state transition graph for each parameter node.In this paper we consider three basic algebraic forms of functions Λ i . The set of linear (Σ)Λ functions is given by Σ ( n ) := ( Λ i : R n + → R + | V ⊆ { , . . . n } , Λ i ( z , . . . , z n ) = X i ∈ V z i ) . The products of sums (ΠΣ) Λ functions are ΠΣ ( n ) := ( Λ i : R n + → R + | Λ i ( z , . . . , z n ) = Y W k X i ∈ W k z i !) , where the sets W k partition S ( i ). The sums of products of sums (ΣΠΣ) Λ functions are(12) ΣΠΣ ( n ) := Λ i : R n + → R + | Λ i ( z , . . . , z n ) = X W k Y V k,j X i ∈ V k,j z i , where the disjoint union of V k,j is W k . As before, the sets W k partition S ( i ), and the sets V k,j partition the set W k . Observe that these classes of functions contain progressively morefunctions, i.e. for n ≥ Σ ( n ) ( ΠΣ ( n ) ( ΣΠΣ ( n ) . The restriction of the class of functions Λ to a product of sums (ΠΣ) goes back to atleast Snoussi [27], and was used extensively in the development of DSGRN (Dynamic Sig-natures Generated by Regulatory Networks) [8, 15, 9, 7, 33]. The main contribution of theDSGRN approach is the definition and explicit construction of a parameter graph (Section2.3) in terms of inequalities in the input combinations of { L ij , U ij | j ∈ S ( i ) } and thresholds { θ ki | k ∈ T ( i ) } given a collection of nonlinearities in ΠΣ. OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 13
As we have shown in Section 2.2, each component [ K i ] of the parameter node [ K ] =([ K ] , . . . , [ K n ]) is equivalent to a collection of monotone Boolean functions, one for eachedge in the regulatory network. When a node i has a single target, then there is onlyone MBF associated to node i , namely g ji , where j is the sole target of i . In the case ofmultiple targets, | T ( i ) | >
1, there is a collection of | T ( i ) | MBFs for node i . We consider asingle component [ K i ], first where node i has a single target and second where node i hasmore than one target. We ask which such singletons or collections of monotone Booleanfunctions can be associated to a parameter node in the parameter graphs of K -, Σ-, ΠΣ-and ΣΠΣ-systems. Definition 3.1.
We say that a monotone Boolean function h : B n → B is ∗ -realizable , where ∗ can stand for K , Σ, ΠΣ or ΣΠΣ, if there exists a regulatory network RN with a node j with a single target ℓ and weight θ ℓj and a parameter node [ K ] = ([ K ] , . . . , [ K n ]) for the ∗ -system, such that the Boolean map g ℓj that corresponds to [ K j ] is h .We say that a k -tuple of monotone Boolean functions h , h , . . . , h k : B n → B is ∗ -jointlyrealizable if there exists a regulatory network RN with a node j with k targets ℓ , . . . , ℓ k and weights { θ ℓ i j } and a parameter node [ K ] = ([ K ] , . . . , [ K n ]) for the ∗ -system such thatthe collection of Boolean maps g ℓ j , g ℓ j , . . . , g ℓ k j that corresponds to [ K j ] are the maps h , h , . . . , h k respectively. Remark . We note that Σ-realizability is a special case of ΠΣ-realizability, which is inturn a special case of ΣΠΣ-realizability, which is in turn a special case of K -realizability.These observations rely on the fact that Σ ⊂ ΠΣ ⊂ ΣΠΣ and that the images of anymonotone Λ functions give rise to K -systems.The definition of realizability 3.1 uses arbitrary functions h ∈ MBF . In some cases, it willbe convenient to assume g ki ∈ MBF + ( m i ), instead of the weaker condition g ki ∈ MBF ( m i ),where recall m i = | S ( i ) | is the number of sources of i . This is achieved via a coordinatechange. For each j ∈ S ( i ), define the function β ( i ) j : B → B as(13) β ( i ) j ( b ) = ( b if j ∈ S + ( i )1 − b if j ∈ S − ( i )Then define β ( i ) : B m i → B m i , β ( i ) = ( β ( i )1 , . . . , β ( i ) m i )component-wise. We observe that g ki ◦ β ( i ) ∈ MBF + ( m i ) and that β ( i ) is an involution i.e. β ( i ) ◦ β ( i ) = Id. We will use the notation β for a function where we do not specify the networknode identity i .Using the coordinate change β , we can see that h : B m i → B is ∗ -realizable if and onlyif, f := h ◦ β ∈ MBF + ( m i ) is a positive Boolean function and is also ∗ -realizable, via thecollection K ′ defined as K ′ i,A ′ , ∅ = K i,A,B where A ′ = A ∪ B, and network RN ′ which is the same as RN except that all edges are now activating. Likewise,if h , h , . . . , h k : B n → B is ∗ -jointly realizable if, and only if, f = h ◦ β (1) , . . . , f k := h k ◦ β ( k ) ∈ MBF + ( n ) are positive Boolean functions and are also ∗ -jointly realizable. Thereforeit is sufficient to consider in proofs only positive Boolean functions. The central question that we pose in this paper is to ask how much restriction the algebraicforms Σ, ΠΣ, and ΣΠΣ impose on the richness of the potential dynamics of the switchingsystem. We will interpret the number of k -tuples of MBFs that can be represented in theparameter graph as the richness of that particular class of switching systems. This ques-tion generalizes and extends a classical problem of determining when a monotone Booleanfunction is a threshold function. Definition 3.3. [6] A Boolean function f : B n → B , is called a threshold function (or a linearly separable function ) if there exist real numbers a , . . . , a n ∈ R and a threshold θ ∈ R such that for all ~y = ( y , . . . , y n ) ∈ B n ,(14) f ( ~x ) = ( P nj =1 a j y j > θ . The ( n + 1)-tuple ( a , a , . . . , a n , θ ) is called a (separating) structure of f .As we will see later in Lemma 4.21, any monotone Boolean function f is a thresholdfunction if and only if f is Σ-realizable, i.e. representable in the parameter graph of aΣ system. Framed in terms of threshold functions, determining which f functions are Σ-realizable is then equivalent to determining which MBFs are indeed threshold functions.This is a classical problem in the Boolean literature. Paull [24] showed that monotonicityis a necessary condition for a Boolean function to be a threshold function. As shown byChow [4] and Elgot [11], a Boolean function is a threshold function if, and only if, it is assumable , where assumable was defined by Winder [32]. An algorithm for determiningwhether a Boolean function is a threshold function was given by Peled and Simeone [25].Their algorithm will produce a , . . . , a n in the sense of Definition 3.3 if the Boolean functionis indeed a threshold function. An algorithm for producing additional linearly separableBoolean functions and further characterization of threshold functions was given in Rao andZhang [26]. The number of threshold functions for n ≤ n = 9 by Gruzling [18].Note that f is a threshold function if the set of points in B n ⊂ R n at which f attains value1 is linearly separable from those points where f attains value 0. Following this connection,Pantovic et al. [23], Zunic [34], and Wang and Williams [31] all examined partitions ofsets of points with surfaces that are not necessarily hyperplanes. This is intimately relatedto the questions of ΠΣ- and ΣΠΣ-realizability, i.e. which monotone Boolean functions arerepresentable in a parameter graph of a ΠΣ- vs. ΣΠΣ- systems.The parameter graph node [ K ] = ([ K ] , . . . , [ K n ]) represents n collections of monotoneBoolean functions, where each [ K j ] corresponds to b j = | T ( j ) | MBFs, the number of targetsof j in RN . Not every collection of b j monotone Boolean functions is ∗ -jointly realizablefor the algebraic classes we consider. We introduce the idea of considering multiple Booleanfunctions simultaneously and asking which of them are *-jointly realizable, i.e. realizable ina node in the parameter graph of a K , Σ, ΠΣ or ΣΠΣ system.4. Realizability Results
The main results are summarized in Table 1, where we consider pairs of ∗ -jointly realizableMBFs. The equality sign between two categories indicates that whenever a pair of MBFswith given set of inputs n (row index) is realizable in one category, it is also realizable inthe other category. The strict subset relationship shows that any pair of functions realizable OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 15 n ∗ -Joint Realizability of f ≺ g ( ΠΣ ( ΣΠΣ = K ≥ ( ΠΣ ( ΣΠΣ ( K Table 1.
Summary of the results. The n is the number of inputs for eachof the pair of MBF, where f ≺ g means that the truth set of f is a subset oftruth set of g (Definition 4.5). The (in)equalities express realizability relationsamong categories of functions (see text). Row 1 and 2 are a result of Propo-sition 4.22. Row 3 is a result of Subsections 4.4.1, 4.4.2, and 4.5. Row 4 is aresult of Subsection 4.4.3.in the smaller category is also realizable in the larger category, and, furthermore, there is apair of Boolean functions f ≺ g that is realizable in the larger category that is not realizablein the smaller category.4.1. K -realizability. The main goal of this section is to prove the following two results: (1)any monotone Boolean function is K -realizable, and (2) any k -tuple f ≺ . . . ≺ f k of MBFsis K -jointly realizable. Therefore K -systems are general enough to represent any collectionof monotone Boolean functions. Definition 4.1.
For a Boolean function f : B n → B , we define the truth set of f as True ( f ) := { ~y ∈ B n | f ( ~y ) = 1 } . Similarly we define the false set of f as False ( f ) := { ~y ∈ B n | f ( ~y ) = 0 } . For U ⊆ B n any subset, we will also use the notation True ( f ) | U := { ~y ∈ U | f ( ~y ) = 1 } , False ( f ) | U := { ~y ∈ U | f ( ~y ) = 0 } . We start our discussion of K -realizability and K -joint realizability by proving two resultsrelating K -realizability to the existence of what we call a realizing function . Definition 4.2.
The positive monotonicity assumption for a function R : B n → R + is thefollowing: for all j ∈ { , . . . , n } , for all ~y ∈ B n with y j = 0 R ( ~y ) ≤ R ( ~y + ˆ e j ) . Theorem 4.3. (1) f ∈ MBF + ( n ) is K -realizable if, and only if, there exist • a weighted regulatory network RN with a node i with only one target j and aweight θ ji and • a function R ( i ) such that R ( i ) : B n → R + satisfies the positive monotonicityassumption and f ( ~y ) = ( if R ( i ) ( ~y ) > θ ji if R ( i ) ( ~y ) < θ ji . (2) A k -tuple of MBFs f , f , . . . , f k ∈ MBF + ( n ) is K -jointly realizable if, and only if,there exist • a weighted regulatory network RN with a node i with k targets ℓ , . . . , ℓ k andweights { θ ℓ j i } and • a function R ( i ) such that R ( i ) : B n → R + satisfies the positive monotonicityassumption and for all j ∈ { , . . . , k } , f j can be expressed as (15) f j ( ~y ) = ( if R ( i ) ( ~y ) > θ ℓ j i if R ( i ) ( ~y ) < θ ℓ j i . Proof.
Using Definition 3.1, Equation (10), and setting R ( i ) ( ~y ) ≡ K i,α i ( ~y ) , ~y ∈ B n , the theorem follows. It remains to note that the positive monotonicity assumption on R ( i ) induces the (positive) monotonicity condition on K i from Definition 2.5. Likewise, when f , . . . , f k ∈ MBF + ( n ), then S ( i ) = S + ( i ) and K i satisfying the monotonicity condition inDefinition 2.5 implies that R ( i ) must satisfy the positive monotonicity condition. (cid:3) Definition 4.4.
If Theorem 4.3 is satisfied, then the pair ( R ( i ) , RN ) is called a realizingfunction and realizing network for f , . . . , f k , respectively.Theorem 4.3 shows that K -systems can be thought of as arising from monotone Booleanfunctions via realizing functions R ( i ) , one for each node i in a realizing network RN . In thefollowing, we will restrict our focus to a single node in RN and drop the superscript. Definition 4.5.
For two Boolean functions f, g : B n → B , we say f implies g and write f ≺ g if, and only if, True ( f ) ⊆ True ( g ).Now we prove the main result of this section, namely that all k -tuples of MBFs that arelinearly ordered f ≺ f ≺ . . . ≺ f k are K -(jointly) realizable for all k ≥ Theorem 4.6. (1) f ∈ MBF + ( n ) if and only if f is K -realizable.(2) A collection f , . . . , f b ∈ MBF + ( n ) of monotone Boolean functions has a linear order f ≺ f ≺ . . . ≺ f b if and only if it is K -jointly realizable.Proof. Since realizability is a special case of joint realizability and since a single MBF triviallyhas an order, it is sufficient to prove the second point.( ⇒ ) Let R ( ~y ) := P bj =1 f j ( ~y ). R satisfies the positive monotonicity assumption of Defini-tion 4.2, since if y i = 0 for some ~y , then R ( ~y ) ≤ R ( ~y + ˆ e i ) by summation and the positivity of f j . Now for each j ∈ { , . . . , b } , let θ j = b − j + . Suppose ~y ∈ True ( f j ). Then ~y ∈ True ( f k )for k = j, . . . , b , since True ( f j ) ⊆ True ( f k ) by the ≺ relationship. So R ( ~y ) = b − j + 1 andwe have θ j = b − j + 1 / < R ( ~y ) as desired. Then the function R and thresholds θ , . . . , θ b satisfy the assumptions of Theorem 4.3.( ⇐ ) Given the b thresholds, establish the indexing θ b < · · · < θ using the order of R .Then for any i < j , we have θ i > θ j and ( θ i , ∞ ) ⊆ ( θ j , ∞ ). Given the realizing function R ,define a collection of b positive monotone Boolean functions by True ( f i ) = R − ( θ i , ∞ ) . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 17
Then by construction f i ( ~y ) = ( R ( ~y ) > θ i R ( ~y ) < θ i . Moreover, if i < j , we have
True ( f i ) ⊂ True ( f j ), implying f i ≺ f j . (cid:3) ∗ -Realizability. In this section we discuss technical points needed later for Σ, ΠΣ,and ΣΠΣ realizability.Since Σ, ΠΣ, and ΣΠΣ realizability are based on Λ functions, Λ : R n + → R + , we need toconsider a restricted class of realizing functions of the form R := Λ ◦ φ, where φ : B n → R n + component-wise monotonically encodes a Boolean vector into a realvalued vector, i.e. φ = ( φ , . . . , φ n ), where φ i : B → R + , φ i (0) < φ i (1) and Λ is an algebraicfunction that belongs to one of the classes Σ ( n ) ( ΠΣ ( n ) ( ΣΠΣ ( n ).The following Lemma is a direct consequence of Theorem 4.3, the definition of the classesof algebraic functions Σ ( n ), ΠΣ ( n ) and ΣΠΣ ( n ), and the previously made observation thatall switching systems are K -systems. Lemma 4.7.
In the following, ∗ could be Σ , ΠΣ , or ΣΠΣ . A function f ∈ MBF + (n) is ∗ -realizable if, and only if, there exist a realizing network RN and realizing function R = Λ ◦ φ ,where(1) the ∗ -interaction function Λ : R n → R belongs to the class Λ ∈ ΣΠΣ ( n ) , Λ ∈ ΠΣ ( n ) ,or Λ ∈ Σ ( n ) if ∗ = ΣΠΣ , ∗ = ΠΣ , or ∗ = Σ respectively; and(2) for each i ∈ { , . . . , n } , the function φ i : B → R + satisfies φ i (0) < φ i (1) .Similarly, a k -tuple of MBFs f , f , . . . , f k ∈ MBF + ( n ) is ∗ -jointly realizable if and onlyif there exist a realizing network RN and a realizing function R = Λ ◦ φ for Λ a ∗ -interactionfunction from (1) and φ a map satisfying (2). The general question of which k -tuples of Boolean functions are ∗ -jointly realizable seemsvery difficult and is likely connected to fundamental problems in algebraic geometry. Wefocus here on some initial results for k = 2 and will consider pairs of Boolean functionswith different numbers of inputs. We start with definitions and results that enumerateconsequences of joint realizability of f ≺ g on relationships between True and
False sets onsubsets of the space of Boolean inputs.In the following, and many times throughout the rest of the manuscript, it will be usefulto view B n as a hypercube embedded in R n + with side lengths of 1. This gives rise toa geometrical structure of B n , where if ~y = ( y , . . . , y i − , , y i +1 , . . . , y n ), then ~y + ˆ e i =( y , . . . , y i − , , y i +1 , . . . , y n ), where ˆ e i is the standard i -th basis vector in R n . This definesthe geometrical idea of floor and ceiling in the i -th direction of the hypercube B n . When f ≺ g , there are relationships between the True and
False sets on the floors and ceilings inall directions.
Definition 4.8.
We define the ceiling (of B n ) in the i -th normal direction as the set C i := { ( y , y , . . . , y n ) ∈ B n | y i = 1 } and similarly we define the floor (of B n ) in the i -th normal direction as the set F i := { ( y , y , . . . , y n ) ∈ B n | y i = 0 } . Notice that C i and F i are both hypercubes of dimension ( n − B n = C i ∪ F i , andthat C i = F i + ˆ e i .Next we define the idea of a collapse , in which a floor and ceiling of B n are consideredobjects embedded in the hypercube B n − . Definition 4.9.
For a given i ∈ { , . . . , n } , we define the i -th collapse as the function Col i : B n → B n − which removes the i -th coordinate, defined as Col i (( y , . . . , y n )) := ( y , . . . , y i − , y i +1 , . . . , y n )Then for any subset U ⊂ B n we have Col i ( U ) = { ( y , . . . , y i − , y i +1 , . . . , y n ) ∈ B n − | ( y , . . . , y n ) ∈ U } . Using the notions of floor, ceiling, and collapse, we move through a series of results thatare critical to future proofs involving ∗ -joint realizability for Λ function classes. Lemma 4.10. If f ∈ MBF + ( n ) , then for all i ∈ { , . . . , n } , Col i ( True ( f ) | F i ) ⊆ Col i ( True ( f ) | C i ) Proof.
Observe that the hypercube can be viewed as a distributive lattice via the relation ≤ on the corners of the hypercube by ~y ≤ ~z ∈ B n ⇐⇒ ( y i = 1 ⇒ z i = 1) for all i = 1 , . . . , n. Notice that since any f ∈ MBF + ( n ) is positive monotone, True ( f ) is an upperset of B n viewed as a lattice. Therefore, for any ~y ∈ True ( f ) | F i , we have ~y + ˆ e i ∈ True ( f ) | C i . Underthe collapse operation, we have Col i ( ~y ) = Col i ( ~y + ˆ e i ), completing the proof. (cid:3) Proposition 4.11. If f, g ∈ MBF + ( n ) and f ≺ g , then for all i ∈ { , . . . , n } , (16) Col i ( True ( f ) | F i ) ⊆ Col i ( True ( g ) | F i ) and Col i ( True ( f ) | C i ) ⊆ Col i ( True ( g ) | C i ) . Proof.
Recall that f ≺ g implies True ( f ) ⊆ True ( g ), which implies that True ( f ) | F i ⊆ True ( g ) | F i and that True ( f ) | C i ⊆ True ( g ) | C i . Since in both cases, the collapse operationoccurs over the same set, we have (16) as desired. (cid:3) Proposition 4.12. If f, g ∈ MBF + ( n ) and f ≺ g , then for all i ∈ { , . . . , n } , Col i ( True ( f ) | F i ) ⊆ Col i ( True ( g ) | C i ) Proof.
By Lemma 4.10 we see have
Col i ( True ( f ) | F i ) ⊆ Col i ( True ( f ) | C i ), and by Proposi-tion 4.11 we have Col i ( True ( f ) | C i ) ⊆ Col i ( True ( g ) | C i ), which completes the proof. (cid:3) The following two technical results for special forms of Λ functions are used extensively inthe coming sections. Proofs for Theorem 4.14 and Theorem 4.15 are found in Appendix A.
Definition 4.13.
Let Λ be a ∗ -interaction function. We say z i is a factor (of Λ ) if thereis another map Λ ′ that does not depend on z i such that Λ = z i Λ ′ . Similarly we say z i is a simple term (of Λ ) if we can write Λ as Λ = z i + Λ ′ . Notice if Λ is a Σ-interaction function,for all i ∈ { , . . . , n } , z i is a simple term. Theorem 4.14.
Let ∗ be Σ , ΠΣ , or ΣΠΣ . Let f, g ∈ MBF + ( n ) , with f ≺ g , be ∗ -jointlyrealizable. Let (Λ ◦ φ, RN ) ∗ -jointly realize ( f, g ) . For each ℓ ∈ { , . . . , n } , if z ℓ is a factoror a simple term of Λ , then Col ℓ ( True ( f ) | C ℓ ) ⊆ Col ℓ ( True ( g ) | F ℓ ) , or (17) Col ℓ ( True ( f ) | C ℓ ) ⊇ Col ℓ ( True ( g ) | F ℓ )(18) OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 19
Theorem 4.15.
Let ∗ be Σ , ΠΣ , or ΣΠΣ . Let n > . Let f, g ∈ MBF + ( n ) be ∗ -jointlyrealizable MBFs on B n with f ≺ g . Let ℓ ∈ { , . . . , n } and U ∈ { F ℓ , C ℓ } . Thena) the functions f ′ U , g ′ U ∈ MBF + ( n − defined by True ( f ′ U ) = Col ℓ ( True ( f ) (cid:12)(cid:12) U ) and True ( g ′ U ) = Col ℓ ( True ( g ) (cid:12)(cid:12) U ) are ∗ -jointly realizable MBFs on B n − , andb) there is a single ∗ -interaction function Λ ′ along with maps φ C ℓ , φ F ℓ and weightedregulatory networks RN C ℓ and RN F ℓ such that (Λ ′ ◦ φ F ℓ , RN F ℓ ) ∗ -jointly realizes ( f ′ F ℓ , g ′ F ℓ ) and (Λ ′ ◦ φ C ℓ , RN C ℓ ) ∗ -jointly realizes ( f ′ C ℓ , g ′ C ℓ ) . Joint realizability in B n and realizability in B n +1 . We will show that there isa bijection η between pairs ( f, g ) ∈ MBF + ( n ) × MBF + ( n ) satisfying f ≺ g and h ∈ MBF + ( n + 1). We use η to relate the ∗ -joint realizability of a pair f ≺ g and the ∗ -realizability of single function η ( f, g ) ∈ MBF + ( n + 1). We will use this fact at the end ofthe section to prove rows 1 and 2 of Table 1. Definition 4.16.
Define the map η : { ( f, g ) | f, g ∈ MBF + ( n ) and f ≺ g } → MBF + ( n + 1)by h = η ( f, g ), where for ~y ∈ B n +1 , h ( ~y ) = ( f ( y , . . . , y n ) if y n +1 = 0 g ( y , . . . , y n ) if y n +1 = 1 . Lemma 4.17.
The map η is a bijection onto MBF + ( n + 1) .Proof. First we describe the range of η . Let f, g ∈ MBF + ( n ) with f ≺ g . By definition f and g describe the floor and ceiling of η ( f, g ) = h in the ( n + 1)-th direction,(19) True ( f ) = Col n +1 (cid:16) True ( h ) (cid:12)(cid:12) F n +1 (cid:17) and True ( g ) = Col n +1 (cid:16) True ( h ) (cid:12)(cid:12) C n +1 (cid:17) . Then the positive monotonicity of f and g induce positive monotonicity on the floor andceiling of h , and f ≺ g gives positive monotonicity in the ( n + 1)-th direction. So the rangeof η is contained in MBF + ( n + 1).It is clear that η is injective. Indeed, if η ( f, g ) = η ( f ′ , g ′ ) then it follows immediately fromthe definition that f = f ′ and g = g ′ .To show that η is surjective, consider h ∈ MBF + ( n + 1) and define f and g by setting (19)to be true. Since h satisfies positive monotonicity on its floor and ceiling, f, g ∈ MBF + ( n ).Also, since Col n +1 (cid:16) True ( h ) (cid:12)(cid:12) F n +1 (cid:17) ⊆ Col n +1 (cid:16) True ( h ) (cid:12)(cid:12) C n +1 (cid:17) , by Lemma 4.10, we have that f ≺ g . We have then constructed the desired pair ( f, g ) with f ≺ g such that η ( f, g ) = h is well-defined. (cid:3) As a consequence of this result, note that η − ( h ) = ( f, g ) is well defined. Theorem 4.18.
Let n ≥ . Suppose f ≺ g is ∗ -jointly realizable, where f, g ∈ MBF + ( n ) .Then h = η ( f, g ) is ∗ -realizable. Proof.
Suppose f ≺ g is ∗ -jointly realizable. Then by Lemma 4.7 there exists (Λ ◦ φ, RN )that ∗ -jointly realizes f and g . Moreover, the proof of Theorem 4.6 tells us that the thresholdsin RN associated to these functions satisfy θ g < θ f . We seek to construct (Λ ′ ◦ φ ′ , RN ′ ) that ∗ -realizes h . To build RN ′ , we take RN and add a source to the node under considerationfrom any other node in the network. It remains to discover the weight, θ ′ of that edge. Case 1: ( ∗ = Σ or ∗ = ΣΠΣ ). By the assumption since f ≺ g is ∗ -jointly realizable, wehave f ( y , . . . , y n ) = ( φ ( y ) , . . . φ n ( y n )) > θ f φ ( y ) , . . . φ n ( y n )) < θ f g ( y , . . . , y n ) = ( φ ( y ) , . . . φ n ( y n )) > θ g φ ( y ) , . . . φ n ( y n )) < θ g For ~y ∈ B n +1 , we assign Λ ′ ◦ φ ′ = Λ ◦ φ + φ ′ n +1 for φ ′ = ( φ , . . . , φ n , φ ′ n +1 ) with some choiceof φ ′ n +1 . We know f ( y , . . . , y n ) = h ( y , . . . , y n ,
0) and g ( y , . . . , y n ) = h ( y , . . . , y n , φ ′ n +1 and θ ′ we choose must satisfy f ( y , . . . , y n ) = h ( y , . . . , y n ,
0) = ( φ ( y ) , . . . φ n ( y n )) + φ ′ n +1 (0) > θ ′ φ ( y ) , . . . φ n ( y n )) + φ ′ n +1 (0) < θ ′ and g ( y , . . . , y n ) = h ( y , . . . , y n ,
1) = ( φ ( y ) , . . . φ n ( y n )) + φ ′ n +1 (1) > θ ′ φ ( y ) , . . . φ n ( y n )) + φ ′ n +1 (1) < θ ′ Consider the assignment φ ′ n +1 (0) = ǫ , φ ′ n +1 (1) = θ f + ǫ − θ g , and θ ′ := θ f + ǫ , where ǫ is anysufficiently small real number 0 < ǫ < θ f − θ g . It is easy to check that with this assignment, h ( y , . . . , y n ,
0) = 1 if and only if f ( y , . . . , y n ) = 1 and that h ( y , . . . , y n ,
1) = 1 if and onlyif g ( y , . . . , y n ) = 1. This completes the construction of (Λ ′ ◦ φ ′ , RN ′ ) that ∗ -realizes h . Case 2: ( ∗ = ΠΣ ) The proof proceeds analogously with Case 1, where the only differenceis replacement of a simple term by a factor in Λ ′ . It is easy to verify that the followingassignments ∗ -realize h : φ ′ = ( φ , . . . , φ n , φ ′ n +1 ),Λ ′ ◦ φ ′ ( y , . . . , y n , y n +1 ) = φ ′ n +1 ( y n +1 ) · (Λ ◦ φ ( y , . . . , y n )) ,θ ′ = θ f , φ ′ n +1 (0) = 1, φ n +1 (1) = θ f /θ g . (cid:3) We do not know if the converse of Theorem 4.18 is true in general. However, with anadditional constraint we obtain the following theorem.
Theorem 4.19.
Let n ≥ . Suppose (Λ ◦ φ, RN ) ∗ -realizes h ∈ MBF ( n + 1) . If z i is afactor or simple term of Λ , then f, g ∈ MBF + ( n ) defined by True ( f ) = Col i ( True ( h ) | F i ) , True ( g ) = Col i ( True ( h ) | C i ) are ∗ -jointly realizable.Proof. Without loss of generality assume i = n + 1. Let θ be the threshold associated to therealization of h . Case 1: ( z n +1 is a factor) Then Λ = z n +1 Λ ′ and from Lemma 4.7 h ( y , . . . , y n +1 ) = ( φ n +1 ( y n +1 )Λ ′ ( φ ( y ) , . . . φ n ( y n )) > θ φ n +1 ( y n +1 )Λ ′ ( φ ( y ) , . . . φ n ( y n )) < θ OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 21
If we restrict our attention to f , dividing by φ n +1 (0), the above equation gives us f ( y , . . . , y n ) = h ( y , . . . , y n ,
0) = ( ′ ( φ ( y ) , . . . φ n ( y n )) > θ/φ n +1 (0)0 if Λ ′ ( φ ( y ) , . . . φ n ( y n )) < θ/φ n +1 (0) . Restricting our attention to g we see that g ( x , . . . , x n ) = h ( x , . . . , x n ,
1) = ( ′ ( φ ( x ) , . . . φ n ( x n )) > θ/φ n +1 (1)0 if Λ ′ ( φ ( x ) , . . . φ n ( x n )) < θ/φ n +1 (1) . Construct RN ′ by removing the source edge associated to n + 1 and adding one target edgeto the node under consideration. Assign to one target edge the weight θ f = θ/φ n +1 (0) andassign θ g = θ/φ n +1 (1) to the other. After setting φ ′ = ( φ , . . . , φ n ), we obtain (Λ ′ ◦ φ ′ , RN ′ )that ∗ -jointly realizes ( f, g ). Case 2: ( z n +1 is a simple term) The argument for this case is similar but instead ofdividing by φ n +1 ( y n +1 ), we will subtract. Specifically, since y n +1 is a simple term, h ( y , . . . , y n +1 ) = ( φ n +1 ( y n +1 ) + Λ ′ ( φ ( y ) , . . . φ n ( y n )) > θ φ n +1 ( y n +1 ) + Λ ′ ( φ ( y ) , . . . φ n ( y n )) < θ and so f ( y , . . . , y n ) = h ( y , . . . , y n ,
0) = ( ′ ( φ ( y ) , . . . φ n ( y n )) > θ − φ n +1 (0)0 if Λ ′ ( φ ( y ) , . . . φ n ( y n )) < θ − φ n +1 (0) g ( y , . . . , y n ) = h ( y , . . . , y n ,
1) = ( ′ ( φ ( y ) , . . . φ n ( y n )) > θ − φ n +1 (1)0 if Λ ′ ( φ ( y ) , . . . φ n ( y n )) < θ − φ n +1 (1)Construct RN ′ as before with threshold assignments θ f = max { , θ − φ n +1 (0) } , θ g =max { , θ − φ n +1 (1) } , and a further perturbation by small enough ǫ > θ f = θ g . Thentuple (Λ ′ ◦ φ ′ , RN ′ ) ∗ -jointly realizes ( f, g ). (cid:3) The following Corollary is an immediate result of Theorems 4.18 and 4.19. It states that,in the Σ class of functions, joint realizability of a pair ( f, g ) in dimension n is equivalent tothe realizability of η ( f, g ) in dimension n + 1, since every term in Λ is simple. Corollary 4.20.
Let n ≥ . Suppose f ≺ g and let h = η ( f, g ) . Then ( f, g ) is Σ -jointlyrealizable if and only if h is Σ -realizable. As promised, we now show the equivalence of threshold (linearly separable) functions andΣ-realizability, see Definition 3.3.
Lemma 4.21.
Let f ∈ MBF + ( n ) .(1) If f is Σ -realizable then f is a threshold function.(2) If f is a threshold function with separating structure ( a , . . . , a n , θ ′ ) such that a , . . . , a n ≥ and θ ′ > − n , then f is Σ -realizable.Proof. (1) Suppose f is Σ-realizable, and let (Λ ◦ φ, RN ) Σ-realize f . We construct a , . . . , a n and θ ′ as in the sense of Definition 3.3 as follows: set a i = φ i (1) − φ i (0), and let θ ′ =max { , θ − ( φ (0) + · · · + φ n (0)) } .(2) Now suppose f is a threshold function with separating structure ( a , . . . , a n , θ ′ ) suchthat a , . . . , a n ≥ θ ′ > − n . Set Λ = z + · · · + z n . Set φ i (0) = 1, φ i (1) = 1 + a i , and θ = θ ′ + n , to obtain the desired (Λ ◦ φ, θ ). (cid:3) The following proposition, together with Remark 3.2, proves the first two rows of Table 1.
Proposition 4.22.
Assume f ≺ g ∈ MBF + ( n ) with n = 1 or n = 2 . Then the pair ( f, g ) is Σ -realizable.Proof. By simple enumeration, one can check that, for n = 1 , ,
3, all functions in
MBF + ( n )are threshold functions, and admit separating structures with a , . . . , a n , θ >
0. Via Lemma 4.21,these functions are Σ-realizable. This fact, when combined with Corollary 4.20, proves thefirst two rows of Table 1. (cid:3)
Strict subset relations in Table 1.
This section contains a series of examples il-lustrating the differences between Σ, ΠΣ, and ΣΠΣ realizability, proving some of the strictsubset results in Table 1. We will use Theorems 4.14 and 4.15 extensively.The idea behind all of the examples is to show that there exists a pair of ∗ -jointly realizablefunctions ( f, g ) that are not ∗ ′ -jointly realizable, where ∗ ′ is a more restrictive class than ∗ .The proofs are inductive, with different base case constructions and very similar inductivesteps. The methodology for the induction is to take an ( f, g ) ∗ -jointly realizable, but not ∗ ′ -jointly realizable, pair in B n and to set ( f, g ) to be the floors of new ( ˜ f , ˜ g ) functions in B n +1 .It then remains to construct ceilings that ensure ˜ f , ˜ g ∈ MBF + ( n + 1). For any ~y ∈ C n +1 ,we choose to set ˜ f ( ~y ) = ˜ g ( ~y ) = 1. By this choice, True ( ˜ f ) ⊇ C n +1 , True (˜ g ) ⊇ C n +1 , which ensures that ˜ f , ˜ g ∈ MBF + ( n + 1). By the contrapositive of Theorem 4.15 (a), thisconstruction ensures that ( ˜ f , ˜ g ) are not ∗ ′ -jointly realizable. We then show that ( ˜ f , ˜ g ) are ∗ -jointly realizable.000001 010 100110101011 111 y y y φ ( y y y ) Λ( φ ( y y y ))000 (1,1,1) 2001 (1,1,2) 4100 (4,1,1) 5010 (1,4,1) 5110 (4,4,1) 8101 (4,1,2) 10011 (1,4,2) 10111 (4,4,2) 16 Figure 6.
Left: An example pair f, g : B → B with f ≺ g . Dark grey is False ( f ) ∩ False ( g ), light grey is False ( f ) ∩ True ( g ), and white is True ( f ) ∩ True ( g ).Nodes are labels with y y y . The pair ( f, g ) are ΠΣ-jointly realizable, butnot Σ-jointly realizable. Right: A table of values proving ( f, g ) are ΠΣ-jointlyrealizable. Here Λ = ( z + z ) z and θ = 4 . θ = 9. The coloring in thetable column is consistent with vertex coloring on the left.4.4.1. Σ -jointly realizable ( ΠΣ -jointly realizable for n ≥ . In this section we prove thefirst strict inclusion in the third and fourth rows of Table 1.
Lemma 4.23.
Let n ≥ . There exists a pair f ≺ g ∈ MBF + ( n ) such that ( f, g ) is not Σ -jointly realizable, but is ΠΣ -jointly realizable. OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 23
Proof.
We first construct an explicit pair ( f, g ) for n = 3. Consider the pair f ≺ g of MBFsdepicted on the left of Figure 6. We use Theorem 4.14 to show that ( f, g ) is not Σ-jointlyrealizable, and provide an explicit (Λ ◦ φ, RN ) that ΠΣ-jointly realizes ( f, g ). Choose any RN with a node with three sources and two targets, with threshold values to be determined.First, we illustrate the use of Theorem 4.14. Observe that True ( f ) | C = { , } , andso Col ( True ( f ) | C ) = { , } . Similarly True ( g ) | F = { , } , so Col ( True ( g ) | F ) = { , } . Therefore, we can see that Col ( True ( f ) | C ) * Col ( True ( g ) | F ) and Col ( True ( f ) | C ) + Col ( True ( g ) | F ) . By the contrapositive of Theorem 4.14, we see that if (Λ ◦ φ, RN ) ∗ -jointly realizes ( f, g ),then Λ cannot have a simple term or factor z . Therefore, ( f, g ) is not Σ-jointly realizable,as any Σ-interaction function Λ has every variable as a simple term.However, ( f, g ) is ΠΣ-jointly realizable. To see this, set Λ = ( z + z ) z , φ (0) = φ (0) = φ (0) = 1, φ (1) = 4, φ (1) = 4 . φ (1) = 2, and θ = 4 . θ = 9. The results of such anassignment are displayed in the table in Figure 6.We now prove the inductive step. Let n ≥
3. Assume there exists f, g : B n → B such that( f, g ) is not Σ-jointly realizable, but is ΠΣ-jointly realizable. Let(Λ ◦ φ, RN ) with thresholds θ and θ ΠΣ-jointly realize ( f, g ). Now over B n +1 define˜ f ( y . . . y n +1 ) := ( f ( y . . . y n ) y n +1 = 01 y n +1 = 1˜ g ( y . . . y n +1 ) := ( g ( y . . . y n ) y n +1 = 01 y n +1 = 1Observe that Col n +1 (cid:16) True ( ˜ f ) | F n +1 (cid:17) = True ( f ) , Col n +1 (cid:0) True (˜ g ) | F n +1 (cid:1) = True ( g ) , in other words the floor of ˜ f has the same truth set as f and the floor of ˜ g has the same truthset as g . Since ( f, g ) are not Σ-jointly realizable, the contrapositive of Theorem 4.15 (a) tellsus that ( ˜ f , ˜ g ) are not Σ-jointly realizable. Let m = min { Λ( φ ( B n )) } and define ˜ φ n +1 (0) = 1and ˜ φ n +1 (1) = max { , C } , where C is large enough such that mC > max { θ , θ } . Define˜Λ = z n +1 Λ and ˜ φ := ( φ , . . . , φ n , ˜ φ n +1 ). Then ˜Λ is a valid ΠΣ-interaction function, and( ˜Λ ◦ ˜ φ, RN ) ΠΣ-jointly realizes ( ˜ f , ˜ g ), completing the proof. (cid:3) -jointly realizable ( ΣΠΣ -jointly realizable for n ≥ . In this section we prove thesecond strict inclusion in the third and fourth rows of Table 1.
Lemma 4.24.
Let n ≥ . There exists a pair f ≺ g ∈ MBF + ( n ) such that ( f, g ) is not ΠΣ -jointly realizable, but is ΣΠΣ -jointly realizable.Proof.
Again, we construct an explicit pair ( f, g ) for n = 3. Consider the pair f ≺ g shownin Figure 7. We will show that ( f, g ) is not ΠΣ-jointly realizable. Suppose, by way ofcontradiction, that (Λ ◦ φ, RN ) ΠΣ-jointly realizes ( f, g ) for some RN with a node withthree sources and two targets.From Lemma 4.7, we can see that the only allowable ΠΣ-interaction function for n = 3are z + z + z , ( z + z ) z , ( z + z ) z , ( z + z ) z , z z z y y y φ ( y y y ) Λ( φ ( y y y ))000 (1,1,1) 2100 (3,1,1) 4010 (1,3.1,1) 4.1001 (1,1,4) 5101 (3,1,4) 7011 (1,3.1,4) 7.1110 (3,3.1,1) 10.3111 (3,3.1,4) 13.3 Figure 7.
An example pair f, g : B → B with f ≺ g . Dark grey is False ( f ) ∩ False ( g ), light grey is False ( f ) ∩ True ( g ), and white is True ( f ) ∩ True ( g ). Nodesare labels with y y y . The pair ( f, g ) is ΣΠΣ-jointly realizable, but not ΠΣ-jointly realizable. Right: A table of values proving ( f, g ) are ΣΠΣ-jointlyrealizable. Here Λ = z z + z and and θ = 4 . θ = 9. The coloring in therightmost column is consistent with vertex coloring on the left.For f ≺ g in Figure 7, observe that(20) Col ( True ( f ) | C ) * Col ( True ( g ) | F ) and Col ( True ( f ) | C ) + Col ( True ( g ) | F ) , and(21) Col ( True ( f ) | C ) * Col ( True ( g ) | F ) and Col ( True ( f ) | C ) + Col ( True ( g ) | F ) . By Theorem 4.14, we see that z and z cannot be simple terms or factors of Λ. Thisconstraint implies that Λ = ( z + z ) z . The following inequality argument will show thatthis choice of Λ is also impossible. To reduce notation, we will write φ i (0) = ℓ i and φ i (1) = u i .We have the following relations from Figure 7:Λ( φ (100)) < Λ( φ (001)) (dark grey < light grey)Λ( φ (010)) < Λ( φ (001)) (dark grey < light grey) −− Λ( φ (101)) < Λ( φ (110)) (light grey < white)Λ( φ (011)) < Λ( φ (110)) (light grey < white)Written in the language of ℓ and u this means( u + ℓ ) ℓ < ( ℓ + ℓ ) u ( ℓ + u ) ℓ < ( ℓ + ℓ ) u ( u + ℓ ) u < ( u + u ) ℓ ( ℓ + u ) u < ( u + u ) ℓ . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 25
We consider first and fourth equation; the second and third together lead to similar contra-diction. First equation: ( u + ℓ ) ℓ < ( ℓ + ℓ ) u u ℓ + ℓ ℓ < ℓ u + ℓ u u ℓ − ℓ u < ℓ ( u − ℓ )Fourth equation: ( ℓ + u ) u < ( u + u ) ℓ ℓ u + u u < u ℓ + u ℓ u ( u − ℓ ) < u ℓ − ℓ u Comparing the last line in each equation block we get u ( u − ℓ ) < u ℓ − ℓ u < ℓ ( u − ℓ ) , which, after cancellation, gives u < ℓ . Therefore φ (0) > φ (1), contradicting Lemma 4.7, so Λ = ( z + z ) z is also impossible.Therefore, ( f, g ) is not ΠΣ-jointly realizable.However, the pair ( f, g ) is ΣΠΣ-jointly realizable. To see this, let Λ = z z + z , let φ (0) = φ (0) = φ (0) = 1, φ (1) = 3, φ (1) = 3 .
1, and φ (1) = 4, and let θ = 4 .
5, and θ = 9. Such an assignment is displayed in the table in Figure 7.We now prove the inductive step. Let n ≥
3. Assume there exists f, g : B n → B suchthat ( f, g ) is not ΠΣ-jointly realizable, but is ΣΠΣ-jointly realizable. Let (Λ ◦ φ, RN ) withthresholds θ and θ ΣΠΣ-jointly realize ( f, g ). Define˜ f ( y . . . y n +1 ) := ( f ( y . . . y n ) y n +1 = 01 y n +1 = 1˜ g ( y . . . y n +1 ) := ( g ( y . . . y n ) y n +1 = 01 y n +1 = 1As in the proof of Lemma 4.23, observe that Col n +1 (cid:16) True ( ˜ f ) | F n +1 (cid:17) = True ( f ) , Col n +1 (cid:0) True (˜ g ) | F n +1 (cid:1) = True ( g ) , in other words the floor of ˜ f has the same truth set as f and the floor of ˜ g has the same truthset as g . Since ( f, g ) are not ΠΣ-jointly realizable, the contrapositive of Theorem 4.15 (a) tellsus that ( ˜ f , ˜ g ) are not ΠΣ-jointly realizable. Let m = min { Λ( φ ( B n )) } . Define ˜ φ n +1 (0) := 1and ˜ φ n +1 (0) := max { , C } , where C is large enough such that m + C > max { θ , θ } . Define˜Λ := z n +1 + Λ and ˜ φ := ( φ , . . . , φ n , ˜ φ n +1 ). Then ˜Λ is a valid ΣΠΣ-interaction function, and( ˜Λ ◦ ˜ φ, RN ) ΣΠΣ-jointly realizes ( ˜ f , ˜ g ), completing the proof. (cid:3) Figure 8.
An example pair f, g : B → B with f ≺ g . Dark grey is False ( f ) ∩ False ( g ), light grey is False ( f ) ∩ True ( g ), and white is True ( f ) ∩ True ( g ). Nodesare labels with y y y y . The pair ( f, g ) is K -jointly realizable, but not ΣΠΣ-jointly realizable. Visually, the inner cube is the floor in the fourth direction,and the outer cube is the ceiling in the fourth direction.4.4.3. ΣΠΣ -jointly realizable ( K -jointly realizable for n ≥ . In this section we prove thelast strict inclusion in the fourth row of Table 1.
Lemma 4.25.
Let n ≥ . There exists a pair f ≺ g ∈ MBF + ( n ) such that ( f, g ) is not ΣΠΣ -jointly realizable, but ( f, g ) is K -jointly realizable.Proof. Recall that any pair f ≺ g ∈ MBF + ( n ) is K -jointly realizable by Theorem 4.6. Itremains to construct an example that is not ΣΠΣ-jointly realizable and apply induction. Weconstruct an explicit pair for n = 4. Consider the pair ( f, g ) in Figure 8. Observe that thefloor (inner cube) has the same truth set as the cube in Figure 7. In Lemma 4.24 we showedthat this pair can be realized by the ΣΠΣ-interaction function Λ = z z + z . It turns outthis is the only valid ΣΠΣ-interaction function that can realize the pair. To see this, we firstlist all possible ΣΠΣ-interaction functions for n = 3, which are z + z + z ( z + z ) z ( z + z ) z ( z + z ) z z z z z z + z z z + z z z + z Via Equations (20) and (21) we can rule out all but z z + z and ( z + z ) z . In addition,Lemma 4.24 showed that Λ = ( z + z ) z also does not work. Therefore, we can make thefollowing claim: if (Λ ◦ φ, RN ) ΣΠΣ-jointly realizes the pair of MBFs from Figure 8, thenΛ = z z + z . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 27
Suppose that f ≺ g ∈ MBF + (4) in Figure 8 are ΣΠΣ-jointly realizable. Then by Theo-rem 4.15 the floor (inner cube) and ceiling (outer cube) pairs ( f ′ F , g ′ F ) and ( f ′ C , g ′ C ) definedrigorously in Theorem 4.15 are ΣΠΣ-jointly realizable, and there is a single ΣΠΣ-interactionfunction Λ along with maps φ C , φ F , realizing networks RN ′ C , RN ′ F , and thresholds θ C , , θ C , , θ F , , θ F , such that (Λ ◦ φ F , RN ′ F ) ΣΠΣ-jointly realizes ( f ′ F , g ′ F ) and (Λ ◦ φ C , RN ′ C ) ΣΠΣ-jointly realizes ( f ′ C , g ′ C ). By our above claim, we know Λ = z z + z . However,consider ( f ′ C , g ′ C ). By inspection, we see that Col ( True ( f ′ C ) | C ) * Col ( True ( g ′ C ) | F ) Col ( True ( f ′ C ) | C ) + Col True ( g ′ C ) | F )By Theorem 4.14 we know that if (Λ ◦ φ C , RN ′ C ) ΣΠΣ-jointly realizes ( f ′ C , g ′ C ), then z cannot be a factor or simple term of Λ. This contradicts our claim that Λ = z z + z , andso ( f, g ) are not ΣΠΣ-jointly realizable.We now prove the inductive step. Let n ≥
4. Assume there exists f, g : B n → B such that( f, g ) is not ΣΠΣ-jointly realizable, but is K -jointly realizable. Define˜ f ( y . . . y n +1 ) := ( f ( y . . . x n ) y n +1 = 01 y n +1 = 1˜ g ( y . . . y n +1 ) := ( g ( y . . . y n ) y n +1 = 01 y n +1 = 1Observe that ˜ f ′ , ˜ g ′ ∈ MBF + ( n + 1) such that ˜ f ′ ≺ ˜ g ′ . Since ˜ f ′ F n +1 = f and ˜ g ′ F n +1 = g , byTheorem 4.15 we know ( ˜ f , ˜ g ) is not ΣΠΣ-jointly realizable. By Theorem 4.6, we know allpairs f ≺ g ∈ MBF + ( n + 1) are always K -jointly realizable. (cid:3) -joint realizability = K -joint realizability for n = 3 . This is the final resultremaining to be proven in Table 1. To find the total number of pairs ( f, g ) such that f, g ∈ MBF + (3) and f ≺ g , we use the bijection given in Definition 4.16. The numberof K -realizable pairs ( f, g ) where f ≺ g ∈ MBF + (3) is the same as (cid:12)(cid:12) MBF + (4) (cid:12)(cid:12) . In [5] itwas found that (cid:12)(cid:12) MBF + (4) (cid:12)(cid:12) = 168. We used the software DSGRN to find that 150 MBFsin MBF + (4) are Σ-realizable. The software was also used to computationally check thatthere are exactly 150 pairs f ≺ g ∈ MBF + (3) that are Σ-jointly realizable. We explicitlyconstructed the remaining 18 pairs ( f i ≺ g i ) , i = 1 , . . .
18 that are provably not Σ-jointrealizable by applying Theorem 4.14 to at least one direction in each case. These pairs areall presented in Appendix Table 2 with the direction that allows application of Theorem 4.14indicated. Finally, in Appendix Table 3, we provide specific realizing functions Λ ◦ φ andrealizing network thresholds θ , θ that ΣΠΣ-jointly realize all 18 pairs.5. Discussion
In this work we linked two classes of dynamical systems, one a continuous time ordinarydifferential equation (ODE) model and the other a discrete time monotone Boolean function(MBF) model. Both of these classes have been used to model dynamics of gene regulatorynetworks. We show that a very general class of ODE models with a discontinuous righthand side admits an equivalence relation, such that all ODEs in an equivalence class sharethe approximate description of dynamics in terms of the identical state transition graphSTG. We then showed that each equivalence class corresponds to a collection of MBFs.
The collections of MBFs can be arranged into a parameter graph where edges between thecollections indicate a one-step change in one of the MBFs.After establishing the equivalence between collections of MBFs and equivalence classes of K systems of ODEs, we pose the question about what restrictions, if any, the algebraic formof the right-hand side of the ODE imposes on k -tuples of MBFs that correspond to realizableequivalence classes of ODEs.We show that the classes of pairs of MBFs with three inputs that are realizable as linearfunctions are a strict subset of ΠΣ-jointly realizable pairs, which is in turn a strict subset ofΣΠΣ-jointly realizable pairs of MBFs. We also show that there are pairs of MBFs with any n ≥ K -jointly realizable, but are not ΣΠΣ realizable. To summarize,theincreased complexity of the algebraic expression provides a richer class of models as measuredby the set of MBFs that can be realized in a parameter graph.Our work opens up many interesting questions about the joint realizability of collectionsof MBFs. We will briefly discuss two potential sets of questions. First, we defined an infinitenested set of classes of nonlinearities. While we only discussed the first three Σ ( ΠΣ ( ΣΠΣ , adding alternating products and sums creates larger and larger classes of functions.Do our results extend in this direction? In other words, are there pairs of monotone Booleanfunctions that are realizable in a parameter graph via class s + 1, but are not realizable inclass s ? Furthermore, is it possible that there are pairs of MBFs that are not realizable inany of the infinite progression of algebraic classes with alternating products and sums, butare K -jointly realizable?The second class of questions generalizes pairs of monotone Boolean functions to k -tuples,Our results derive some constraints for pairs of MBFs, which are then used to prove themain results about differences in ∗ -joint realizability. While these results apply pairwise toany k -tuple of MBFs with f ≺ f ≺ . . . ≺ f k , to rule out realizability of some tuples, wedo not know if there any additional constraints that arise from considering, say, triples offunctions f ≺ g ≺ h , or k -tuples of MBFs.By providing a link between a class of discontinuous differential equations and the collec-tion of k -tuples of MBFs, this paper provides an opening to a class of new problems in thefield of monotone Boolean functions. Appendix A. Proofs of Theorem 4.14 and 4.15.
Proof of Theorem 4.14.
We will proceed by contradiction. Suppose z ℓ is a factor or simpleterm and suppose the negation of Equations (17) and (18). The negation of (17) is thereexists some point ~y ∈ F ℓ such that g ( ~y ) = 0 and f ( ~y + ˆ e ℓ ) = 1. The negation of (18) impliesthere exists some point ~w ∈ F ℓ such that g ( ~w ) = 1 and f ( ~w + ˆ e ℓ ) = 0. Let θ f and θ g bethe thresholds from RN associated to the two functions respectively. From the definition of ∗ -jointly realizable, we know g ( ~y ) = 0 ⇐⇒ Λ( φ ( ~y )) < θ g (22) f ( ~y + ˆ e ℓ ) = 1 ⇐⇒ Λ( φ ( ~y + ˆ e ℓ )) > θ f (23) g ( ~w ) = 1 ⇐⇒ Λ( φ ( ~w )) > θ g (24) f ( ~w + ˆ e ℓ ) = 0 ⇐⇒ Λ( φ ( ~w + ˆ e ℓ )) < θ f (25)We now proceed by cases. OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 29
Case 1: ( z ℓ is a factor) Recall from the definition of factor that there exists a functionΛ ′ that does not depend on z ℓ such that Λ = z ℓ Λ ′ . We have z ℓ = φ ℓ ( v ℓ ) for any ~v ∈ B n .Taking ~v ∈ F ℓ we have Λ( φ ( ~v )) = φ ℓ (0)Λ ′ ( φ ′ ( Col ℓ ( ~v )))Λ( φ ( ~v + ˆ e ℓ )) = φ ℓ (1)Λ ′ ( φ ′ ( Col ℓ ( ~v ))) . We used the collapse operation because Λ ′ is independent of z ℓ and we took φ ′ ( Col ℓ ( ~v )) =( φ ( v ) , . . . , φ ℓ − ( v ℓ − ) , φ ℓ +1 ( v ℓ +1 ) , . . . , φ n ( v n )). Lastly, we used Col ℓ ( ~v ) = Col ℓ ( ~v + ˆ e ℓ ). Weconclude that for any ~v ∈ F ℓ φ ℓ (1) φ ℓ (0) Λ( φ ( ~v )) = Λ( φ ( ~v + ˆ e ℓ )) . From (23) and (25), we may write φ ℓ (1) φ ℓ (0) Λ( φ ( ~y )) > θ f ⇐⇒ Λ( φ ( ~y )) > φ ℓ (0) φ ℓ (1) θ f φ ℓ (1) φ ℓ (0) Λ( φ ( ~w )) < θ f ⇐⇒ Λ( φ ( ~w )) < φ ℓ (0) φ ℓ (1) θ f and combining with (22) and (24) we obtain θ g > φ ℓ (0) φ ℓ (1) θ f and θ g < φ ℓ (0) φ ℓ (1) θ f a clear contradiction. Case 2: ( z ℓ is a simple term) Similar to Case 1, the key fact is, if z ℓ is a simple term,we know for any ~v ∈ F ℓ Λ( φ ( ~v )) + ( φ ℓ (1) − φ ℓ (0)) = Λ( φ ( ~v + ˆ e ℓ )) , We then make a similar argument as before. Equations (23) and (25) giveΛ( φ ( ~y )) + ( φ ℓ (1) − φ ℓ (0)) > θ ⇐⇒ Λ( φ ( ~y )) > θ − ( φ ℓ (1) − φ ℓ (0))Λ( φ ( ~w )) + ( φ ℓ (1) − φ ℓ (0)) < θ ⇐⇒ Λ( φ ( ~w )) < θ − ( φ ℓ (1) − φ ℓ (0))and combining with (22) and (24) we obtain θ g > θ f − ( φ ℓ (1) − φ ℓ (0)) and θ g < θ f − ( φ ℓ (1) − φ ℓ (0))which is our desired contradiction. (cid:3) Proof of Theorem 4.15.
Let f, g : B n → B , with f ≺ g , be ∗ -jointly realizable MBFs, and let(Λ ◦ φ, RN ) jointly ∗ -realize ( f, g ). Let θ f and θ g be the thresholds associated to the tworealizations of these functions.We seek to construct a ∗ -interaction function Λ ′ : R n − → R + , a map φ ′ : B n − → R n − ,and a weighted regulatory network RN ′ U with thresholds θ ′ f and θ ′ g so that (Λ ′ ◦ φ ′ , RN ′ U ) ∗ -jointly realizes f ′ U , g ′ U : B n − → B . In doing so we will prove a). In each of the followingcases, the construction of Λ ′ does not depend on whether U = F ℓ or U = C ℓ , and so b) willfollow immediately.Recall that we are going to collapse over the ℓ th dimension. In the following proof, wewill be considering a node in RN that has an incoming edge from node ℓ and two targets,one of which is associated to the MBF f and the other to g . It will be useful to define the graph RN ′ to be the network RN without the edge from ℓ to the node under consideration. RN ′ is an intermediate step to the construction of RN ′ U .Using the observations in Remark 3.2, we note that regardless of the specific value of ∗ , wecan always assume that Λ ∈ ΣΠΣ defined in (12). Therefore there exist sets W , . . . , W L ,where W , . . . , W L partitions the set { , . . . , n } , and for each k ∈ { , . . . , L } , there exists M k ∈ N so that the sets V k, , . . . , V k,M k partition the set W k , such thatΛ = X W k Y V k,j X i ∈ V k,j z i There is exactly one set in the partition, call it W p , such that ℓ ∈ W p . Furthermore, thereis exactly one V p, ∗ that contains ℓ , call it V p,q . We now proceed by cases. Define the map δ : { , . . . , n } \ { ℓ } → { , . . . , n − } as δ ( j ) = ( j if j < ℓj − j > ℓ We will use the map δ to construct the interaction function Λ ′ . If ∗ is ΣΠΣ, we need toconsider Cases 1, 2, and 3. However, if ∗ is ΠΣ, then W = { , . . . , n } , and we only needto consider Cases 2 and 3, and if ∗ is Σ, then W = V , = { , . . . , n } , and we only need toconsider Case 3. Case 1: ( W p = { ℓ } ) In this case the map Λ has the formΛ = z ℓ + X W k k = p Y V k,j X i ∈ V k,j z i where z i = φ i ( y i ) for ~y ∈ B n . We construct the interaction function Λ ′ asΛ ′ = X W k k = p Y V k,j X i ∈ V k,j z δ ( i ) and define z δ ( i ) = φ ′ i ( y i ) = φ δ − ( i ) ( y i ). We then set φ ′ = ( φ ′ , . . . , φ ′ n − ). This constructionensures, for any ~y = ( y , . . . , y n ) ∈ B n ,Λ( φ ( ~y )) = Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − )) + φ ℓ ( y ℓ )If U = F ℓ , then for all ~y = ( y , . . . , y n ) ∈ U , we know y ℓ = 0. Therefore,Λ( φ ( ~y )) = Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − )) + φ ℓ (0) . Set θ ′ f = max { , θ f − φ ℓ (0) } and θ ′ g = max { , θ g − φ ℓ (0) } . It is possible that θ ′ f = θ ′ g atthis point. However, since Λ ◦ φ takes on finitely many values, we can always perturb onethreshold by a small enough ǫ to guarantee our inequalities still hold. After this potentialperturbation, replace θ f and θ g in RN ′ with θ ′ f and θ ′ g to complete the construction of RN ′ F ℓ .This construction means thatΛ( φ ( y , . . . , y n )) ≶ θ f ⇐⇒ Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − )) ≶ θ ′ f and likewise for θ g and θ ′ g . Therefore (Λ ′ ◦ φ ′ , RN ′ F ℓ ) ∗ -jointly realizes f ′ F ℓ and g ′ F ℓ . OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 31
Similarly, if U = C ℓ , then for all ~y = ( y , . . . , y n ) ∈ U , we haveΛ( φ ( ~y )) = Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − )) + φ ℓ (1) . We set θ ′ f = max { , θ f − φ ℓ (1) } and θ ′ g = max { , θ g − φ ℓ (1) } , perturbed by small enough ǫ >
0, if necessary, to replace θ f and θ g in RN ′ and complete the construction of RN ′ C ℓ .Then (Λ ′ ◦ φ ′ , RN ′ C ℓ ) ∗ -jointly realizes f ′ C ℓ and g ′ C ℓ . Case 2: ( W p \ { ℓ } 6 = ∅ and V p,q = { ℓ } ) The interaction function Λ ′ without the ℓ thelement is Λ ′ = X W k Y V k,j j = q X i ∈ V k,j z δ ( i ) In this case we know that W p is a partition of at minimum size two. Pick exactly one t ∈ W p \ { q } . Construct z δ ( j ) = φ ′ j ( y j ) as follows: if U = F i , then for ~y ∈ B n φ ′ j ( y j ) = ( φ δ − ( j ) ( y j ) φ ℓ (0) if j ∈ V p,t φ δ − ( j ) ( y j ) otherwise . However, if U = C i , then φ ′ j ( y j ) = ( φ δ − ( j ) ( y j ) φ ℓ (1) if j ∈ V p,t φ δ − ( j ) ( y j ) otherwise . We have ensured for any ~y = ( y , . . . , y n ) ∈ U ,Λ( φ ( ~y )) = Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − ))Setting θ ′ f = θ f and θ ′ g = θ g obtains the desired result; i.e., RN ′ U = RN ′ and (Λ ′ ◦ φ ′ , RN ′ ) ∗ -jointly realizes f ′ U , g ′ U . Case 3: ( W p \ { ℓ } 6 = ∅ and V p,q \ { ℓ } 6 = ∅ ) In this case the original interaction functionΛ takes the form Λ = X W k Y V k,j ( k,j ) =( p,q ) X i ∈ V k,j z i + Y V p,q X i ∈ V p,q z i . The interaction function Λ ′ is constructed asΛ ′ = X W k Y V k,j ( k,j ) =( p,q ) X i ∈ V k,j z δ ( i ) + Y V p,q X i ∈ V p,q i = ℓ z δ ( i ) . Pick exactly one element t ∈ V p,q \ { ℓ } , and construct φ ′ j as follows: if U = F ℓ , then φ ′ j ( y j ) := ( φ δ − ( j ) ( y j ) + φ ℓ (0) if j = δ ( t ) φ δ − ( j ) ( y j ) otherwise . However, if U = C ℓ , then φ ′ j ( y j ) := ( φ δ − ( j ) ( y j ) + φ ℓ (1) if j = δ ( t ) φ δ − ( j ) ( y j ) otherwise . We then set φ ′ = ( φ ′ , . . . , φ ′ n − ). This construction ensures, for any ~y = ( y , . . . , y n ) ∈ U ,Λ( φ ( ~y )) = Λ ′ ( φ ′ ( y δ − (1) , . . . , y δ − ( n − ))and so by setting RN ′ U = RN ′ as in Case 2, we obtain the desired result. (cid:3) Appendix B. Supporting tables and figures
Table 2 lists explicitly all pairs f ≺ g ∈ MBF + ( ) of Boolean functions with three inputsthat are not Σ-jointly realizable. These correspond to non-threshold monotone Booleanfunctions in MBF + (4) with 4 inputs. In each case the direction that allows us to useTheorem 4.14 to rule out Σ-joint realizability is indicated in the last column.Input000 001 010 100 110 101 011 111 Direction(s) f ≺ g
00 01 00 01 11 01 01 11 y f ≺ g
00 00 01 01 01 01 11 11 y f ≺ g
00 01 01 00 01 11 01 11 y f ≺ g
00 01 01 00 11 01 01 11 y f ≺ g
00 01 00 01 01 01 11 11 y f ≺ g
00 00 01 01 01 11 01 11 y f ≺ g
00 01 00 00 11 01 01 11 y , y f ≺ g
00 00 00 01 01 01 11 11 y , y f ≺ g
00 00 01 00 01 11 01 11 y , y f ≺ g
00 01 00 01 11 01 11 11 y , y f ≺ g
00 00 01 01 01 11 11 11 y , y f ≺ g
00 01 01 00 11 11 01 11 y , y f ≺ g
00 01 00 00 11 01 11 11 y f ≺ g
00 00 00 01 01 11 11 11 y f ≺ g
00 00 01 00 11 11 01 11 y f ≺ g
00 00 00 01 11 01 11 11 y f ≺ g
00 00 01 00 01 11 11 11 y f ≺ g
00 01 00 00 11 11 01 11 y Table 2.
All 18 non Σ-jointly realizable pairs ( f, g ) for n = 3 such that f ≺ g . The column for input ~y shows the pair of values of f i ( y ) g i ( y ). Forexample, column 001 has 01 in the first row. This means that f (001) = 0 and g (001) = 1. The direction(s) that allow the use of Theorem 4.14 to rule outΣ-joint realizability are indicated in the last column.In Table 3 we show explicitly the form of the interaction function Λ ∈ ΣΠΣ , the values φ (1) = ( φ (1) , φ (1) , φ (1)), and the value of the thresholds θ g , θ f that ΣΠΣ-jointly realizeall pairs f i ≺ g i given in Table 2. In all cases, we set φ (0) = φ (0) = φ (0) = 1. This list, OINT REALIZABILITY OF MONOTONE BOOLEAN FUNCTIONS 33 together with 150 pairs that are Σ-jointly realizable, exhausts all pairs f ≺ g of functions in MBF + (3) and proves that for n = 3 every such pair is ΣΠΣ-joint realizable.Figure 9 shows all pairs f ≺ g ∈ MBF + (2). It is also the factor of the parameter graphassociated to node 1 in the network in Figure 1, after transforming the Boolean functions tobe positive under the map β given in (13). i Λ φ (111) θ g θ f z z + z (3 , ,
3) 3.5 6.52 z + z z (3 , ,
2) 3.5 6.53 z + z z (2 , ,
3) 3.5 6.54 z z + z (2 , ,
3) 3.5 6.55 z + z z (3 , ,
3) 3.5 6.56 z + z z (3 , ,
2) 3.5 6.57 z z + z (3 , ,
4) 4.5 88 z + z z (4 , ,
3) 4.5 89 z + z z (3 , ,
3) 4.5 810 z ( z + z ) (4 , ,
4) 4.5 911 z ( z + z ) (4 , ,
2) 4.5 912 z ( z + z ) (2 , ,
4) 4.5 913 z z + z (2 , ,
4) 4.5 6.514 z + z z (4 , ,
3) 4.5 6.515 z + z z (3 , ,
2) 4.5 6.516 z ( z + z ) (4 , ,
3) 4.5 7.517 z ( z + z ) (3 , ,
2) 4.5 7.518 z ( z + z ) (2 , ,
4) 4.5 7.5
Table 3.
Example (Λ ◦ φ, θ , θ ) that ΣΠΣ-jointly realize all pairs f i ≺ g i given in Table 2. For all rows, φ (0) = φ (0) = φ (0) = 1.
00 0110 1100 0110 1100 0110 1100 0110 11 00 0110 1100 0110 1100 0110 11 00 0110 1100 0110 11 00 0110 11 00 0110 1100 0110 11 00 0110 11 00 0110 1100 0110 11 00 0110 11 00 0110 1100 0110 11 00 0110 1100 0110 11
Figure 9.
All 20 pairs of functions f ≺ g ∈ MBF + (2). Each node inthe above graph represents a pair of functions, where dark grey is False ( f ) ∩ False ( g ), light grey is False ( f ) ∩ True ( g ), and white is True ( f ) ∩ True ( g ). EFERENCES 35
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