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Definability by constantdepth polynomialsize circuits
 Information and Control
, 1986
"... A function of boolean arguments is symmetric if its value depends solely on the number of l's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constantdepth polynomialsize sequences of boolean circuits, and discus ..."
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A function of boolean arguments is symmetric if its value depends solely on the number of l's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constantdepth polynomialsize sequences of boolean circuits, and discuss the complete characterization. (We treat both uniform and nonuniform sequences of circuits.) Our results imply that these circuits can compute functions that are not definable in firstorder logic. In the second part of the paper we generalize from circuits computing symmetric functions to circuits recognizing firstorder structures. By imposing fairly natural restrictions we develop a circuit model with precisely the power of firstorder logic: a class of structures is firstorder definable if and only if it can be recognized by a constantdepth polynomialtime sequence of such circuits. © 1986 Academic Press, Inc.
Boolean algebras, Tarski invariants and index sets
 JOURNAL OF FORMAL LOGIC
, 2006
"... Tarski defined a way of assigning to each boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from N, such that two boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a boolean algebra, there is a computa ..."
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Tarski defined a way of assigning to each boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from N, such that two boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this paper we analyze the complexity of the question “Does B have invariant x?”. For each x ∈ In we define a complexity class Γx, that could be either Σn, Πn, Σn ∧ Πn, or Πω+1 depending on x, and prove that the set of indices for computable boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in [Sel90] and [Sel91]. According to [Sel03] similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras.
What is a Structure Theory
 Bulletin of the London Mathematical Society
, 1987
"... into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consis ..."
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into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups
Three lectures on automatic structures, in:
 Proceedings of Logic Colloquium 2007,
, 2008
"... Preface This paper grew out of three tutorial lectures on automatic structures given at the Logic Colloquium 2007 in Wroc law (Poland). The paper will follow the outline of the tutorial lectures, supplementing some material along the way. We discuss variants of automatic structures related to sever ..."
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Preface This paper grew out of three tutorial lectures on automatic structures given at the Logic Colloquium 2007 in Wroc law (Poland). The paper will follow the outline of the tutorial lectures, supplementing some material along the way. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Büchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Büchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Finite automata are the most commonly known in the general computer science community. Such an automaton reads finite input strings from left to right, making state transitions along the way. Depending on its last state after processing a given string, the automaton either accepts or rejects the input string. Automatic structures are mathematical objects which can be represented by (word, tree, Büchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research. We begin with some motivation and history for studying automatic structures. We introduce definitions of automatic structures, present examples, and discuss decidability and definability theorems. Next, we concentrate on finding natural isomorphism invariants for classes of automatic structures. These classes include wellfounded partial orders, Boolean algebras, linear orders, trees, and finitely generated groups. Finally, we address the issue of complexity for automatic structures. In order to measure complexity of automatic structures we involve topology (via the Borel hierarchy), model theory (Scott ranks), computability theory (Σ 1 1 completeness), and computational complexity (the class P). This paper consists of three sections based on the tutorial lectures. The first lecture provides motivating questions and historical context, formal definitions of the different types of automata involved, examples of automatic structures, and decidability and definability results about automatic structures. The second lecture begins by introducing tree and Rabin automatic structures. We then outline techniques for proving nonautomaticity of sets and structures and study