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Fig. 1 | BMC Bioinformatics

Fig. 1

From: Reducing Boolean networks with backward equivalence

Fig. 1

Boolean backward equivalence shown on a simple example. (Top-left) BN with three variables denoted by \(x_1\), \(x_2\), and \(x_3\). (Bottom-left) The underlying STG. Each node is labelled by a vector that defines the state of each variable; a directed edge denotes a transition from a source state to a target state by a synchronous application of the update functions. States 110 and 111 form an attractor. (Top-right) Variables \(x_1\) and \(x_2\) can be shown to be BBE-equivalent by inspecting their update functions. If they have the same value in a state, i.e. \(x_1(t)=x_2(t)\), then they will be equivalent for all successor states since \(x_2(t+1) = x_1(t) \vee x_2(t) \vee \lnot x_3(t) =x_1(t) \vee x_1(t) \vee \lnot x_3(t)=x_1(t) \vee \lnot x_3(t)=x_1(t+1)\). Based on this, a reduced BN can be obtained by considering a representative variable for each block and rewriting the corresponding update functions in terms of those representatives (here the representative variable is denoted by \(x_{1,2}\)). (Bottom-right) The underlying STG agrees with the original one on all states that have equal values for variables in the same block (purple nodes in bottom-left panel). Instead, any other state (i.e. those where variables in the same BBE block have different value), is removed. The criteria for BBE only involve checks for the update functions of the original model, such that the generation of original STG can be circumvented

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