Formal Languages in Dynamical Systems

We treat here the interrelation between formal languages and those dynamical systems that can be described by cellular automata (CA). There is a well-known injective map which identifies any CA-invariant subshift with a central formal language. However, in the special case of a symbolic dynamics, i.e. where the CA is just the shift map, one gets a stronger result: the identification map can be extended to a functor between the categories of symbolic dynamics and formal languages. This functor additionally maps topological conjugacies between subshifts to empty-string-limited generalized sequential machines between languages. If the periodic points form a dense set, a case which arises in a commonly used notion of chaotic dynamics, then an even more natural map to assign a formal language to a subshift is offered. This map extends to a functor, too. The Chomsky hierarchy measuring the complexity of formal languages can be transferred via either of these functors from formal languages to symbolic dynamics and proves to be a conjugacy invariant there. In this way it acquires a dynamical meaning. After reviewing some results of the complexity of CA-invariant subshifts, special attention is given to a new kind of invariant subshift: the trapped set, which originates from the theory of chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe

Strong fairness and ultra metrics

AbstractWe answer an open question of Costa and Hennessy and present a characterization of the infinite fair computations in finite labeled transition systems—without any structure of the states—as cluster points in metric spaces. This technique is applied to reduce the logical complexity of several known fairness concepts from Π03 to Π02 and from Σ11 to Π03, respectively

Baire and automata

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science

Topological Conjugacies Between Cellular Automata

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant

Digital architecture and difference: a theory of ethical transpositions towards nomadic embodiments in digital architecture

This thesis contributes to histories and theories of digital architecture of the past two decades, as it questions the narratives of its novelty. The main argument this thesis puts forward is that a plethora of methodologies, displacing the centrality of the architect from the architectural design process, has folded into the discipline in the process of its rewriting along digital protocols. These steer architecture onto a post-human path. However, while the redefinition of the practice unfolds, it does so epistemically only without redefining the new subject of architecture emerging from these processes, which therefore remains anchored to humanist-modern definitions. This unaccounted-for position, I argue, prevents novelty from emerging. Simultaneously, the thesis unfolds a creative approach – while drawing on nomadic, critical theory concepts, there surfaces an alternative genealogy already underpinning digital methodologies that enable a reconceptualization of novelty framed with difference to be articulated through nomadic digital embodiment. Regarding the first claim, I turn to the narratives as well as to the mechanisms of digital discourse emerging in two modes of production – mathematical and biological – in exploration of the ways perceptions of novelty are articulated: a) through close readings of its narratives as they consolidate into digital architectural theory (Carpo 2011; Lynn 2003, 2012; Terzidis 2006; Migayrou 2004, 2009); b) through an analysis of the two digital methodologies that support these narratives – parametric architecture and biodigital architecture. In parallel, this thesis draws on twentieth-century critical theory and twenty-firstcentury nomadic feminist theory to rethink two thematic topics: difference and subjectivity. Specifically, these are Gilles Deleuze’s non-essentialist, nonrepresentational philosophy of difference (1968, 1980, 1988) and Rosi Braidotti’s nomadic feminist reconceptualization of post-human, nonunitary subjectivity (2006, 2011, 2015). Nomadic feminist theory also informs my methodology. I draw on Rosi Braidotti’s cartographing and transposing (2006, 2011) because they engender a non-dualist approach to research itself that is dynamic and affirmative, insisting on grounding techniques – grounding in subject positions that are nevertheless post-human and nonunitary. This leads to a redefinition of novel digital practices with ethical ones

Layered Cylindrical Algebraic Decomposition

In this report the idea of a Layered CAD is introduced: atruncation of a CAD to cells of dimension higher than a prescribedvalue. Limiting to full-dimensional cells has already beeninvestigated in the literature, but including more levels is shown toalso be beneficial for applications. Alongside a direct algorithm, arecursive algorithm is provided. A related topological property isdefined and related to robot motion planning. The distribution of celldimensions in a CAD is investigated and layered CAD ideas are combinedwith other research. All research is fully implemented within a freelyavailable Maple package, and all results are corroborated withexperimental results