Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a "stabilizable" discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n 2 log n) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n 2 log n) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n 2 log n).
Keywords: Δ 0 α categorical structure, structure that is not relatively Δ 0 α categorical, field.In [3,6,7], it was proved that for each computable ordinal α, there is a structure that is Δ 0 α categorical but not relatively Δ 0 α categorical. The original examples were not familiar algebraic kinds of structures. In [11], it was shown that for α = 1, there are further examples in several familiar classes of structures, including rings and 2-step nilpotent groups. Similar example for all computable successor ordinals were constructed in [12]. In the present paper, this result is extended to computable limit ordinals. We know of an example of an algebraic field that is computably categorical but not relatively computably categorical. Here we show that for each computable limit ordinal α > ω, there is a field which is Δ 0 α categorical but not relatively Δ 0 α categorical. Examples on dimension and complexity of relations are given.
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