A boolean network is a map $ F:\{0,1\}^n \to \{0,1\}^n $ that defines a discrete dynamical system by the subsequent iterations of $ F $. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.
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Figure 3. Structure of a network $ G^{\tau,k} $ in which the connected component $ D $ has maximum delay $ \tau \geq 2 $ in every node and the connected component $ G $ has maximum delay $ \tau = 1 $ in every node. Additionally, the subset $ \partial $ contains every node in $ D $ whose have a neighbour in $ G $. (a).- The connected component $ G $ has an arbiratry topology. (b).- The connected component $ G $ is a bipartite graph. As it is shown in Proposition 1, the structure of $ G $ may condition the existance of different type of attractors for a conjunctive network
Figure 4. A scheme of the dynamics of a node $i \in \partial $ that is in state 0 in some time step t. Note that before the sequence of states Ï„; Ï„ âˆ' 1;...; 0 we donâ€™t know the state of i and we note it as? in order to illustrate this situation. As a consequence of the fact that the maximum delay Ï„ is even together with the fact that two nodes in state 0 can not be neighbours we have that there are no attractor different from the fixed points $\overrightarrow 1 $ and $\overrightarrow 0 $.
Figure 5. (a) A network $ G^{\tau,k} $ with maximum delay $ \tau = 3 $ and $ 10 $ nodes exhibiting an attractor of period $ 8 $. This network is defined by connecting the network $ D $ (squares and non-colored circles) to a component with maximum delay $ \tau = 1 $ in every vertex (gray nodes). The nodes $ 1 $ and $ 2 $ (in grey) are the only nodes in $ G^{\tau,k} $ with maximum delay $ \tau = 1 $. In addition, nodes represented by squares and non-coloured circles have maximum delay $ \tau = 3 $. Finally, nodes represented by squares are the nodes in the interface between the component $ D $ and $ G $ (we called this set $ \partial $ in the latter section), and nodes represented by non-coloured circles are the ones that have no neighbours in $ G $. (b) Configuration of an attractor with period $ 8 $ defined over the network showed in (a)
Figure 7. Structure of the $ j $-th block used in Proposition 5 to define a conjunctive network with firing memory and maximum delay values $ dt_i = \tau $ for every node $ i $ that admits attractors with period $ k(\tau+1) $. Every circle in the figure represents clock $ C(B_j)_{r,l} $ associated to a node $ r $ represented by a square. This gadget has $ \tau + 1 $ nodes and every node has $ \tau -1 $ clocks
Figure 10. Structure of a block used in Proposition 5 to define a conjunctive network with firing memory and maximum delay values $ dt_i = \tau $ for every node $ i $ that admits attractors with period $ k(\tau+1) $. Every circle in the figure represents clock associated to a node $ r $ represented by a square. This gadget has $ \tau + 1 $ nodes and every node has $ \tau -1 $ clocks
Figure 11. Interaction graph $ G $ associated to a conjunctive network with firing memory and maximum delay values $ dt_i = \tau $ for every node $ i \in V(G) $ that admits attractors with non polynomial period. Every component defines a local dynamics with period $ (\tau + 1) p_i $. Initial condition is defined verifying that there are no connected nodes in $ 0 $. Global period of the network is given by the product of prime numbers $ p_i $.
Figure 12. Interaction graph associated to a conjunctive network with firing memory and maximum delay vector $ dt_i = 2 $ for every node $ i $ that simulates an iterated monotone boolean circuit $ C $. Layers are made up by AND or OR gates exclusively, using the gadgets shown in Figure 13 and Figure 14 are alternately ordered
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Symmetric conjunctive network (
Symmetric AND (
Structure of a network
A scheme of the dynamics of a node
(a) A network
Attractors with period
Structure of the
Initial condition for the j-th block, j â‰¥ 2 used in Proposition 5 to define an attractor with period k(Ï„ + 1): For the first block we just define the state of the first node to 0 instead of Ï„.
Interaction graph G associated to a conjunctive network with firing memory and maximum delay dt_{i} = Ï„ for all
Structure of a block used in Proposition 5 to define a conjunctive network with firing memory and maximum delay values
Interaction graph
Interaction graph associated to a conjunctive network with firing memory and maximum delay vector
Gadget of AND gates used in the graph shown in Figure 12. Signals are transmitted and coded based on the block gadget
Gadget of OR gates used in the graph shown in Figure 12. Signals are transmitted and coded based on the block gadget
Iterations of the AND gate gadget. A
Iterations of the OR gate gadget. A