Primitive groups synchronize non-uniform maps of extreme ranks

Journal of the Australian Mathematical Society, 2003

Let G be a permutation group on a set Ω with no fixed point in Ω. If for each subset Г of Ω the size |Гg - Г| is bounded, for g ∈ G, we define the movement of g as the max|Гg − Г| over all subsets Г of Ω. In particular, if all non-identity elements of G have the same movement, then we say that G has constant movement. In this paper we will first give some families of groups with constant movement. We then classify all transitive permutation groups with a given constant movement m on a set of maximum size.

Theoretical Computer Science, 2007

Reversible cellular automata (RCA) are models of massively parallel computation that preserve information. We generalize these systems by introducing the class of ωω bijective finite automata. It consists of those finite automata where for any bi-infinite word there exists a unique path labelled by that word. These systems are strictly included in the class of local automata. Although the synchronization delay of an n-state local automaton is known to be Θ(n 2 ) in the worst case, we prove that in the case of ωω bijective finite automata the synchronization delay is at most n − 1. Based on this we prove that for a one-dimensional n-state RCA where the neighborhood consists of m consecutive cells, the neighbourhood of the inverse automaton consists of at most n m−1 − (m − 1) cells. Similar bounds are obtained also in [E. Czeizler, J. Kari, A tight linear bound on the neighborhood of inverse cellular automata, in: Proceedings of ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 410-420] but here the result comes as a direct consequence of the more general result. We also construct examples of RCA with large inverse neighbourhoods proving that the upper bounds provided here are the best possible in the case m = 2.

Groups St Andrews 2013, 2015

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.

Lecture Notes in Computer Science, 2002

In spite of its simple formulation, the problem about the synchronization of a finite deterministic automaton is not yet properly understood. The present paper investigates this and related problems within the general framework of a composition theory for functions over a finite domain N with n elements. The notion of depth introduced in this connection is a good indication of the complexity of a given function, namely, the complexity with respect to the length of composition sequences in terms of functions belonging to a basic set. Our results show that the depth may vary considerably with the target function. We also establish criteria about the reachability of some target functions, notably constants. Properties of n such as primality or being a power of 2 turn out to be important, independently of the semantic interpretation. Most of the questions about depth, as well as about the comparison of different notions of depth, remain open. Our results show that the study of functions of several variables may shed light also to the case where all functions considered are unary.

Information and Computation, 1990

Mathematische Annalen, 2012

Every automaton group naturally acts on the space X ω of infinite sequences over some alphabet X. For every w ∈ X ω we consider the Schreier graph Γ w of the action of the group on the orbit of w. We prove that for a large class of automaton groups all Schreier graphs Γ w have subexponential growth bounded above by n (log n) m with some constant m. In particular, this holds for all groups generated by automata with polynomial activity growth (in terms of S. Sidki), confirming a conjecture of V. Nekrashevych. We present applications to ω-periodic graphs and Hanoi graphs.

Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, 1990

The diameter of a group G with respect to a set S of generators is the maximum over g E G of the length of the shortest word in S U S-' representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent, to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if IS1 is allowed to exceed (logIGI)'+C (c > 0), it plays a heavy role when 1. 51 = O(1). In particular, every nonabelian finite simple group has a set of 5 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index.-Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((nlnn)'/2(1 + o(1))) upper bound. In the average case, the upper bound improves to exp((lnn)'(l + o(1))). As a first step toward extending this result to simple groups other than A,, , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for A,. In the limited spa.ce of this article, we try to illuminate some of the basic underlying techniques.

Lecture Notes in Computer Science, 2016

We have improved an algorithm generating synchronizing automata with a large length of the shortest reset words. This has been done by refining some known results concerning bounds on the reset length. Our improvements make possible to consider a number of conjectures and open questions concerning synchronizing automata, checking them for automata with a small number of states and discussing the results. In particular, we have verified the Černý conjecture for all binary automata with at most 12 states, and all ternary automata with at most 8 states.

Glasgow Mathematical Journal, 1991

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2006

Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. Then we define the movement of G as, m := move(G) := sup Γ {|Γ g \ Γ||g ∈ G}. Let p be a prime, p ≥ 5, and let move(G) = m. We show that if G is not a 2-group and p is the least odd prime dividing |G|, then n := |Ω| ≤ 4m − p. Moreover for an infinite family of groups the maximum bound n = 4m − p is attained.