The model-theoretic complexity of automatic linear orders

Automatic structures are—possibly infinite—structures which are finitely presentable by means of finite automata on strings or trees. Largely motivated by the fact that their first-order theories are uniformly decidable, automatic structures gained a lot of attention in the "logic in computer science" community during the last fifteen years. This thesis studies the model-theoretic complexity of automatic linear orders in terms of two complexity measures: the finite-condensation rank and the Ramsey degree. The former is an ordinal which indicates how far a linear order is away from being dense. The corresponding main results establish optimal upper bounds on this rank with respect to several notions of automaticity. The Ramsey degree measures the model-theoretic complexity of well-orders by means of the partition relations studied in combinatorial set theory. This concept is investigated in a purely set-theoretic setting as well as in the context of automatic structures.Auch im Buchhandel erhältlich: The model-theoretic complexity of automatic linear orders / Martin Huschenbett Ilmenau : Univ.-Verl. Ilmenau, 2016. - xiii, 228 Seiten ISBN 978-3-86360-127-0 Preis (Druckausgabe): 16,50

Model Theoretic Complexity of Automatic Structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS 4978 pp 514-52

Comparisons of Animal “Smarts” Using the First Four Stages of the Model of Hierarchical Complexity

The Model of Hierarchical Complexity is a behavioral model of development and evolution of the complexity of behavior. It is based on task analysis. Tasks are ordered in terms of their hierarchical complexity, which is an ordinal scale that measures difficulty. The hierarchical difficulty of tasks is categorized as the order of hierarchical complexity. Successful performance on a task is called the behavioral stage. This model can be applied to non-human animals, and humans. Using data from some of the simplest animals and also somewhat more complex ones, this analysis describes the four lowest behavioral stages and illustrate them using the behaviors of a range of simple organisms. For example, Stage 1 tasks, and performance on them, are addressed with automatic unconditioned responses. Behavior at this Stage includes sensing, tropisms, habituation and, other automatic behaviors. Single cell organisms operate at this Stage. Stage 2 tasks include these earlier behaviors, but also include respondent conditioning but not operant conditioning. Animals such as some simple invertebrates have shown respondent conditioning, but not operant conditioning. Stage 3 tasks coordinate three instances of these earlier tasks to make possible operant conditioning. These stage 3 performances are similar to those of some invertebrates and also insects. Stage 4 tasks organisms coordinate 2 or more circular sensory-motor task actions into a superordinate “concept”. This explanation of the early stages of the Model of Hierarchical Complexity may help future research in animal behavior, and comparative psychology.

Advances and applications of automata on words and trees : abstracts collection

From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

The First-Order Theory of Ground Tree Rewrite Graphs

We prove that the complexity of the uniform first-order theory of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower bound, we show that there is some fixed ground tree rewrite graph whose first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory.Comment: accepted for Logical Methods in Computer Scienc

Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow $\epsilon$-contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy.Comment: 12 pages Accepted for STACS 201

An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet

We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical characterizations. We also consider a slight extension of the Feferman-Vaught formalism which allows to express more relations between component values (such as equality), and prove related decidability results. From this result we get new classes of decidable logics for words over an infinite alphabet.Comment: 24 page

What's Decidable About Sequences?

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl