Subshifts, MSO Logic, and Collapsing Hierarchies

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014, published by Springe

Quasiperiodicity and non-computability in tilings

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the fixed point construction; we improve this general technique and make it enforce the property of local regularity of tilings needed for quasiperiodicity. We prove also a stronger result: any effectively closed set can be recursively transformed into a tile set so that the Turing degrees of the resulted tilings consists exactly of the upper cone based on the Turing degrees of the later.Comment: v3: the version accepted to MFCS 201

Return times, recurrence densities and entropy for actions of some discrete amenable groups

Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the initial $k$-block within larger $n$-blocks approaches $2^{-hk}$, uniformly in $n>s^k$, as $k$ tends to infinity. Again, similar results hold for processes with other indexing groups.Comment: To appear in Journal d'Analyse Mathematiqu

On the density of periodic configurations in strongly irreducible subshifts

Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense

Governing through representatives of the community: A case study on farmer organizations in rural Australia

The concept ‘governing through the community’ has been used frequently to interpret the neoliberal policy embraced by Australian governments since the 1990s. Yet explanation is still inadequate of how ‘governing through the community’ is conducted in practice, particularly the specific mechanisms that regulate interaction among government agencies, groups seeking to represent the community and individuals in the community. In this study, we find that ‘governing through the community’ is actually ‘governing through representatives of the community’ because it is the representatives that make the community visible and governable. This observation is based on a case study of three kinds of farmer organizations, in two states of Australia, who see their role as serving the community and are regarded by outsiders as representatives of the community at least on certain issues. An understanding of the interaction among different stakeholders within and outside of the community is developed through three themes of ‘paperwork’, ‘data’ and ‘price’ that were used by locals from Landcare groups, grower groups and farmer cooperatives, respectively, to articulate how they experience the mechanisms through which their interactions are regulated. This paper concludes that these groups can claim to represent some residents within a defined geographical area, rather than any exact definition of ‘the community’ and that this is a sufficient claim to enable these groups to participate in the process of ‘governing through the community’. The tensions between government agencies, community representatives and community members threaten the legitimacy of the community representatives as intermediaries. Government agencies do try to contribute to reduce these tensions by strengthening the legitimacy of community representatives through various policy and project mechanisms. However, while the stated aim of ‘governing through the community’ is often focused on producing a ‘flourishing rural community’ through improving democratic modes of representation, this study demonstrates that it is only part of the community, namely the ‘targeted customers’ of the farmer organizations, that is potentially reachable to ‘the state’