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

Weighted Automata and Logics on Hierarchical Structures and Graphs

Formal language theory, originally developed to model and study our natural spoken languages, is nowadays also put to use in many other fields. These include, but are not limited to, the definition and visualization of programming languages and the examination and verification of algorithms and systems. Formal languages are instrumental in proving the correct behavior of automated systems, e.g., to avoid that a flight guidance system navigates two airplanes too close to each other. This vast field of applications is built upon a very well investigated and coherent theoretical basis. It is the goal of this dissertation to add to this theoretical foundation and to explore ways to make formal languages and their models more expressive. More specifically, we are interested in models that are able to model quantitative features of the behavior of systems. To this end, we define and characterize weighted automata over structures with hierarchical information and over graphs. In particular, we study infinite nested words, operator precedence languages, and finite and infinite graphs. We show Büchi-like results connecting weighted automata and weighted monadic second order (MSO) logic for the respective classes of weighted languages over these structures. As special cases, we obtain Büchi-type equivalence results known from the recent literature for weighted automata and weighted logics on words, trees, pictures, and nested words. Establishing such a general result for graphs has been an open problem for weighted logics for some time. We conjecture that our techniques can be applied to derive similar equivalence results in other contexts like traces, texts, and distributed systems

Accepting networks of evolutionary picture processors

We extend the study of networks of evolutionary processors accepting words to a similar model, processing rectangular pictures. To this aim, we introduce accepting networks of evolutionary picture processors and investigate their computational power. We show that these networks can accept the complement of any local picture language as well as picture languages that are not recognizable. Some open problems regarding decidability issues and closure properties are finally discussed

Accepting networks of evolutionary picture processors

We extend the study of networks of evolutionary processors accepting words to a similar model, processing rectangular pictures. To this aim, we introduce accepting networks of evolutionary picture processors and investigate their computational power. We show that these networks can accept the complement of any local picture language as well as picture languages that are not recognizable. Some open problems regarding decidability issues and closure properties are finally discussed

Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing

DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick complementarity of DNA sequences, without external direction. Here, we propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. Moreover, we investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of {\it Smart Tiles}, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes

Two-dimensional grid grammars

Everyone has an intuitive understanding of the concept of an algorithm. It is generally agreed that an algorithm, or effective procedure, is a finite set of instructions which meet certain requirements. All instructions must be unambiguous and must not involve any element of chance. Each instruction must be executed in a finite amount of time. It is usually not required that algorithms halt. Instead, the set of instructions must be so specifically stated that any two people executing them would perform precisely the same operations. In this thesis we first introduce a class of mathematical machines which are used to define the concept of an algorithmic computation. These Turing machines [3], which operate on one-dimensional tapes, are defined in Chapter II. We consider machines as computational devices, function evaluators, and acceptor automata for syntactically correct input. A set of instructions, called the Wang programming language, which can be assembled in such a way as to simulate a given Turing machine, is presented in Chapter III. In Chapter IV we define grammars, which are systems for generating strings of symbols with certain structured properties. The Chomsky Hierarchy of classes of restricted grammars is outlined [1]. In Chapter V we introduce a class of machines which operate in a manner similar to Turing machines but on a two-dimensional tape. We refer to this class of machines as grid machines