1 Automata-based presentations of infinite structures (original) (raw)

Finite Presentations of Infinite Structures: Automata and Interpretations

Theory of Computing Systems, 2004

We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations. These two ways of presenting a structure are related. Indeed, a structure is automatic if, and only if, it is first-order interpretable in an appropriate expansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefix order and equal length predicate. Similar results hold for ω-automatic structures and appropriate expansions of the real ordered group. We also discuss the relationship to automatic groups.

07441 Abstracts Collection--Algorithmic-Logical Theory of Infinite Structures}

Abstract From 28.10. to 02.11. 2007, the Dagstuhl Seminar 07441``Algorithmic-Logical Theory of Infinite Structures''was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. 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.

Computability and complexity properties of automatic structures and their applications

2008

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Continuing a tradition of studying computability in mathematics, we examine automatic structures, mathematical objects which can be represented by automata, and apply resulting observations to computer science. We measure the complexity of automatic structures via well-established concepts from model theory, topology, and set theory. We prove the following results. The ordinal height of any automatic well-founded partial order is bounded by ω ω. The ordinal heights of automatic well-founded relations are unbounded below ω CK 1 , the first uncomputable ordinal. For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ω CK 1 , ω CK 1 + 1. For any computable ordinal α, there is an automatic successor tree of Cantor-Bendixson rank α. Next, we study infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations have finite degree and can be described by finite automata over a one-letter alphabet. We investigate algorithmic properties of such graphs in terms of their finite presentations. In particular, we ask how hard it is to check whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms answering each of these questions. For a fixed input graph, the algorithm for infinite component membership works in constant time and reachability is decided uniformly by a single automaton. Hence, we improve on previous work, in which nonelementary or nonuniform algorithms were found. We turn our attention to automata techniques for deciding first-order logical theories. These techniques are useful in Integer Linear Programming and Mixed Integer Linear Programming, which in turn have applications in diverse areas of computer science and engineering. We extend known work to address the enumeration problem for linear programming solutions. Then, we apply a similar paradigm to give an automata theoretic decision procedure for the p-adic valued ring under addition and for formal Laurent series over a finite field with valuation and addition. BIOGRAPHICAL SKETCH Mor Minnes was born in Haifa, Israel in 1982. From an early age, Mor learned the importance of words and education: in the bath, Mor's parents (both students at the time) saw her and her sister typing intently on imaginary typewriters and proclaiming that they were working. In 1989, the nuclear family (now also including two brothers) moved to Vancouver, Canada for what was intended as a short visit for Mor's father to do a residency in psychiatry. Quickly, Mor became Mia, choosing a more suitable English name at the suggestion of her maternal grandmother. The family settled in Vancouver and Mia attended Prince of Wales Mini School and the International Baccalaureate Program at Sir Winston Churchill Secondary School. While in high school, Mia enjoyed both the sciences and the humanities. Not willing to give up either of these, Mia decided to pursue dual Bachelor's degrees at Queen's University in Kingston, Ontario. In 1999, Mia headed to snowy Kingston and enrolled in Applied Science (Mathematics & Engineering, Computing and Communications) and Philosophy. Mia had her first encounter with programming and fell in love with the power behind logical thinking. Considering graduate studies, logic was a natural fit with Mia's interests as it lay in the intersection of her favourite philosophy courses (philosophy of language and science) and engineering courses (computer architecture, programming, and math). Her best friend compiled a list of graduate programs in logic and Mia started browsing through their websites. When she discovered Anil Nerode's research outlining connections between automata, logic, and hybrid systems, she knew she had found what she wanted to work on. Mia graduated from Queen's and headed south to Ithaca, NY to begin graduate school in mathematics at Cornell University, along the way also earning a Master's degree in computer science. After five years of graduate school, Mia is looking forward to being a C.L.E. Moore Instructor at MIT next year.

1 Definability in classes of finite structures

2011

This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term ‘finite model theory’, as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject. The first theme c...

Approximating the Expressive Power of Logics in Finite Models

Lecture Notes in Computer Science, 2004

We present a probability logic (essentially a first order language extended with quantifiers that count the fraction of elements in a model that satisfy a first order formula) which, on the one hand, captures uniform circuit classes such as AC 0 and TC 0 over arithmetic models, namely, finite structures with linear order and arithmetic relations, and, on the other hand, their semantics, with respect to our arithmetic models, can be closely approximated by giving interpretations of their formulas on finite structures where all relations (including the order) are restricted to be "modular" (i.e. to act subject to an integer modulo). In order to give a precise measure of the proximity between satisfaction of a formula in an arithmetic model and satisfaction of the same formula in the "approximate" model, we define the approximate formulas and work on a notion of approximate truth. We also indicate how to enhance the expressive power of our probability logic in order to capture polynomial time decidable queries, There are various motivations for this work. As of today, there is not known logical description of any computational complexity class below NP which does not requires a built-in linear order. Also, it is widely recognized that many model theoretic techniques for showing definability in logics on finite structures become almost useless when order is present. Hence, if we want to obtain significant lower bound results in computational complexity via the logical description we ought to find ways of by-passing the ordering restriction. With this work we take steps towards understanding how well can we approximate, without a true order, the expressive power of logics that capture complexity classes on ordered structures.

Unifying Themes in Finite Model Theory

Texts in Theoretical Computer Science an EATCS Series, 2007

One of the fundamental insights of mathematical logic is that our understanding of mathematical phenomena is enriched by elevating the languages we use to describe mathematical structures to objects of explicit study. If mathematics is the science of pattern, then the media through which we discern patterns, as well as the structures in which we discern them, command our attention. It is this aspect of logic which is most prominent in model theory, "the branch of mathematical logic which deals with the relation between a formal language and its interpretations" . No wonder, then, that mathematical logic, in general, and finite model theory, the specialization of model theory to finite structures, in particular, should find manifold applications in computer science: from specifying programs to querying databases, computer science is rife with phenomena whose understanding requires close attention to the interaction between language and structure.