Sequentialization and procedural complexity in automata networks

Bridoux, Florian

doi:10.1016/j.tcs.2021.11.001

Cited by 1 publication

(10 citation statements)

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“…First, assume that at least two instructions of Y , say F (1) and F (2) , are singular. We claim that no assignment g = (a → b), with a i = b i , i = 1, 2, can be computed using only instructions in Y .…”

Section: Complete Transformations Of Minimal Sizementioning

confidence: 99%

“…Let u, v ∈ A n , u 1 = v 1 , be such that (u)F (1) = (v)F (1) . For any g ∈ Sing(A n ), we may write g = π • F (1) • h, where h ∈ Tran(A n ) and π ∈ F (2) , . .…”

Section: Complete Transformations Of Minimal Sizementioning

confidence: 99%

“…All that is left to prove is that those transformations, acting on A, can be simulated by f . Firstly, it is easy to check that F (2,1) simulates the cycle c = (0, 1 . .…”

Section: Proof Letmentioning

confidence: 99%

“…. , g (n) generating Sym(A n ) such that there exists z ∈ A n fixed by g (1) but not by g (2) . We then denote the assignment instruction (x…”

Section: Proof Letmentioning

confidence: 99%

“…For reminder, we apply functions from the left to the right. Thus, the image of x by g is obtained by applying on x, first h (1) , then h (2) , . .…”

Section: Introductionmentioning

confidence: 99%

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Complete simulation of automata networks

Bridoux

Castillo-Ramirez

Gadouleau

2020

Journal of Computer and System Sciences

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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 goal is to better understand the computational power of sequential and asynchronous update modes, with a particular focus on non-bijective functions. 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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Section: Complete Transformations Of Minimal Sizementioning

confidence: 99%

“…Let u, v ∈ A n , u 1 = v 1 , be such that (u)F (1) = (v)F (1) . For any g ∈ Sing(A n ), we may write g = π • F (1) • h, where h ∈ Tran(A n ) and π ∈ F (2) , . .…”

Section: Complete Transformations Of Minimal Sizementioning

confidence: 99%

“…All that is left to prove is that those transformations, acting on A, can be simulated by f . Firstly, it is easy to check that F (2,1) simulates the cycle c = (0, 1 . .…”

Section: Proof Letmentioning

confidence: 99%

“…. , g (n) generating Sym(A n ) such that there exists z ∈ A n fixed by g (1) but not by g (2) . We then denote the assignment instruction (x…”

Section: Proof Letmentioning

confidence: 99%

“…For reminder, we apply functions from the left to the right. Thus, the image of x by g is obtained by applying on x, first h (1) , then h (2) , . .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations