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Finite Model Theory and Descriptive Complexity
, 2002
"... This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the ..."
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Cited by 27 (7 self)
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This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the
On the Power of Deterministic Transitive Closures
 INFORMATION AND COMPUTATION
, 1995
"... We show that transitive closure logic (FO + TC) is strictly more powerful than deterministic transitive closure logic (FO + DTC) on finite (unordered) structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every DTCquery is bounded and the ..."
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Cited by 11 (1 self)
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We show that transitive closure logic (FO + TC) is strictly more powerful than deterministic transitive closure logic (FO + DTC) on finite (unordered) structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every DTCquery is bounded and therefore first order expressible. On the other hand there are simple (FO + pos TC) queries on these classes that cannot be defined by first order formulae.
Some Aspects of Model Theory and Finite Structures
, 2002
"... this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures ..."
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Cited by 11 (0 self)
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this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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Cited by 2 (1 self)
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
On the FirstOrder Prefix Hierarchy
, 1998
"... We investigate the expressive power of fragments of firstorder logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence ' p ..."
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We investigate the expressive power of fragments of firstorder logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence ' p in prenex normal form with prefix p, over a single binary relation, such that for all sentences in prenex normal form, if is equivalent to ' p , then p can be embedded in the prefix of . This strengthens a theorem of Walkoe.
Game Representations of Complexity Classes
 Proc. Eur. Summer School on Logic, Language and Information (European Assoc. Logic, Language and Information
, 2001
"... Many descriptive and computational complexity classes have gametheoretic representations. These can be used to study the relation between different logics and complexity classes in finite model theory. ..."
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Many descriptive and computational complexity classes have gametheoretic representations. These can be used to study the relation between different logics and complexity classes in finite model theory.
Unifying Themes in Finite Model Theory
"... 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 discer ..."
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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 ” [21]. 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. As with most branches of mathematics, the growth of mathematical logic may be seen as fueled by its applications. The very birth of set theory was occasioned by Cantor’s investigations in real analysis, on subjects themselves motivated by developments in nineteenthcentury physics; and the study of subsets of the real line has remained the source of some of the deepest results of contemporary set theory. At the same time, model theory has matured through the development of ever deeper applications to algebra. The interplay between language and structure, characteristic of logic, may be discerned in all these developments. From the focus on definability hierarchies in descriptive set theory, to the classification of structures up to elementary equivalence in classical model theory, logic seeks order in the universe of mathematics through the medium of formal languages. As noted, finite model theory too has grown with its applications, in this instance not to analysis or algebra, but to combinatorics and computer science. Beginning with connections to automata theory, finite model theory has developed through a broader and broader range of applications to problems in graph theory, complexity theory, database theory, computeraided verification, and artificial intelligence. And though its applications have demanded
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"... Introduction Complexity Theory investigates the computational resources required to solve decision problems. A decision problem is usually given as a set of strings. Typical measures of complexity are time or space, that is the number of steps, or the amount of tape cells respectively, used by a Tur ..."
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Introduction Complexity Theory investigates the computational resources required to solve decision problems. A decision problem is usually given as a set of strings. Typical measures of complexity are time or space, that is the number of steps, or the amount of tape cells respectively, used by a Turing machine that accepts this set of strings. In many cases however, decision problems represent structural properties, say the property of a graph having a Hamiltonian cycle. Hence, it is natural to generalize the notion of a decision problem from sets of strings to classes of finite structures (it is a generalization, because a string can also be seen as a finite structure).