Graph subshifts

We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the question whether such local conditions can enforce a specific support graph, and thus relate the model to classical symbolic dynamics. We prove that the subshifts that contain only infinite graphs are either aperiodic, or feature no residual finiteness of their period group, yielding non-trivial examples as well as two natural undecidability theorems.Comment: 13 pages, 4 figure

Ogden's lemma for random permitting and forbidding context picture languages and table-driven context-free picture languages

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, February 16, 2015.Random context picture grammars are used to generate pictures through successive refinement. There are three important subclasses of random context picture grammars, namely random permitting context picture grammars, random forbidding context picture grammars and table-driven context-free picture grammars. These grammars generate the random permitting context picture languages, random forbidding context picture languages and table-driven context-free picture languages, respectively. Theorems exist which provide necessary conditions that have to be satisfied by a language before it can be classified under a particular subclass. Some of these theorems include the pumping and shrinking lemmas, which have been developed for random permitting context picture languages and random forbidding context picture languages respectively. Two characterization theorems were developed for the table-driven context-free picture languages. This dissertation examines these existing theorems for picture languages, i.e., the pumping and shrinking lemmas and the two characterisation theorems, and attempts to prove theorems, which will provide an alternative to the existing theorems and thus provide new tools for identifying languages that do not belong to the various classes. This will be done by adapting Ogden’s idea of marking parts of a word which was done for the string case. Our theorems essentially involve marking parts of a picture such that the pumping operation increases the number of marked symbols and the shrinking operation reduces it

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Compensatory Contempt: Plaintiff\u27s Remedy When Defendant Violates an Injunction

None availabl

Enforcing the Hidden U.S. Equal Rights Law

Since 1945 the law of the United States has required the United States government to take action to promote universal observance of human rights for all without distinction as to sex. This equal rights for women law is part of the supreme law of the land, to be faithfully executed by the President and the Administration, to be enforced by the federal courts and by the courts of the several states, to be implemented by Congress, and to be obeyed by industry, reported by the media, and relied on and obeyed by the people in their daily lives. Busy practitioners representing women whose equal rights have been denied will save time and increase their effectiveness by making use of this hidden law. This is an opportune moment to rediscover and enforce it. The needs of women are becoming more acute in the face of increasing attacks on women\u27s advancement. At the same time the cold war is ending, and the peace dividend must make it possible to fund some reforms. This broad equal rights law was adopted as two articles in a multilateral treaty that is better known for establishing the United Nations as the international organization to keep and promote peace. Like all treaties, on signing by the President and passage by the U.S. Senate by a two-thirds vote, it became part of the supreme law of the land to be enforced by the courts, under the U.S. Constitution

Enforcing the Hidden U.S. Equal Rights Law

Since 1945 the law of the United States has required the United States government to take action to promote universal observance of human rights for all without distinction as to sex. This equal rights for women law is part of the supreme law of the land, to be faithfully executed by the President and the Administration, to be enforced by the federal courts and by the courts of the several states, to be implemented by Congress, and to be obeyed by industry, reported by the media, and relied on and obeyed by the people in their daily lives. Busy practitioners representing women whose equal rights have been denied will save time and increase their effectiveness by making use of this hidden law. This is an opportune moment to rediscover and enforce it. The needs of women are becoming more acute in the face of increasing attacks on women\u27s advancement. At the same time the cold war is ending, and the peace dividend must make it possible to fund some reforms. This broad equal rights law was adopted as two articles in a multilateral treaty that is better known for establishing the United Nations as the international organization to keep and promote peace. Like all treaties, on signing by the President and passage by the U.S. Senate by a two-thirds vote, it became part of the supreme law of the land to be enforced by the courts, under the U.S. Constitution