Simulation of automata networks
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An automata network is a finite graph where each node holds a state from some finite alphabet and is equipped with an update function that changes its state according to the configuration of neighboring states. More concisely, it is given by a finite map f:Q^n→ Q^n. They are studied using various update modes: sequential (one node update at a time), asynchronous (any subset of nodes at a time) or synchronous (all nodes simultaneously). In this paper we study how some (sets of) automata networks can be simulated by some other (set of) automata networks with prescribed update mode or interaction graph. For instance, it is known [Cameron-Fairbairn-Gadouleau-2014] that for any alphabet Q and network size n (with finite exceptions) there is a universal automaton network f such that any permutation of Q^n is a composition of sequential updates of f. However there is no such universal f for the whole set of transformations. Our contributions are the following. For non-Boolean alphabets and for any network size, there are intrinsically non-sequential transformations (that can not be obtained as composition of sequential updates of some network). Moreover there is no universal automaton network that can produce all non-bijective functions via compositions of asynchronous updates. On the other hand, we show that there are universal automata networks for sequential updates if one allows to use a larger alphabet and then use either projection onto or restriction to the original alphabet. We also characterize the set of functions that are generated by non-bijective sequential updates. Following [Tchuente-1986], we characterize the interaction graphs D whose semigroup of transformations is the full semigroup of transformations on Q^n, and we show that they are the same if we force either sequential updates only, or all asynchronous updates.
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