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**1 - 6**of**6**### Opponent

, 2014

"... A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of su ..."

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A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of subshifts are called cellular automata, and we call the set of cellular automata on a subshift its endomorphism monoid. It is known that the set of all sequences (the full shift) allows cellular automata with complex dynamical and com-putational properties. We are interested in subshifts that do not support such cellular automata. In particular, we study countable subshifts, min-imal subshifts and subshifts with additional universal algebraic structure that cellular automata need to respect, and investigate certain criteria of ‘simplicity ’ of the endomorphism monoid, for each of them. In the case of countable subshifts, we concentrate on countable sofic shifts, that is, countable subshifts defined by a finite state automaton. We

### Automated Discharging Arguments for Density Problems in Grids

, 2015

"... Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. How-ever, the minimum density of an identifying code in the hexagonal grid remains open, with an uppe ..."

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Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. How-ever, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of 37 ≈ 0.428571 and a lower bound of 512 ≈ 0.416666. We present a new framework for pro-ducing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of 23 55 ≈ 0.41818 on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, triangular, and pentagonal grids. We also present a new method to find matching upper bounds. ∗Pages 1-10 of this PDF contain the extended abstract, with some figures, tables, and proofs appearing in appendices (pp.

### Adaptive Identification in Torii in Triangular Grids

, 2012

"... Adaptive identification consists in asking the questions one after the other, allowing one to choose the next question according to the answers received so far, and its goal is to identify a (posible) faulty vertex in a graph. One can view adaptive identification also as a game, with the first playe ..."

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Adaptive identification consists in asking the questions one after the other, allowing one to choose the next question according to the answers received so far, and its goal is to identify a (posible) faulty vertex in a graph. One can view adaptive identification also as a game, with the first player secretly choosing a vertex to be faulty, or no vertex at all, and the second player trying to locate the faulty vertex by asking questions of the type “is there a faulty vertex in the ball B(v) center at some vertex v? ” for vertices in graph G. The goal of the first player is to maximize the number of needed queries and the goal of the second player is to minimize this number. In this paper we study adaptive identification in torii in the triangular lattice.

### Tolerant identification with Euclidean balls

, 2012

"... The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z 2 where one sensor can check its neighbours within Euclidean distance r. We construct ..."

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The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z 2 where one sensor can check its neighbours within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r.