Reachability problems for sequential dynamical systems with threshold functions
Theoretical Computer Science - Mathematical foundations of computer science
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Abstract Consider a network of processors (sites) in which each site has finitely many neighbors. Each site has some transition function computing its next state from the states of the neighbors. These transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is r(x,t) then let us define the sequence r''(x,0),r''(x,1),... by taking the sequence r(x,0),r(x,1),... and deleting each repetition, i.e. each element equal to the preceding one. The system of transition functions is said to support asynchrony if the sequence r''(x,i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper gives a simple characterization of transition functions supporting asynchrony. The characterization says that it is equivalent to the following seemingly weaker commutativity condition: For any configuration, for any pair x,y of neighbors, if the updating would change both s(x) and s(y) then the result of updating first x and then y is be the same as the result of doing this in the reverse order.