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Describing Graphs: a FirstOrder Approach to Graph Canonization
, 1990
"... In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting ..."
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Cited by 57 (7 self)
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In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting quantifiers". We give efficient canonization algorithms for graphs characterized by Ck or Lk . It follows from known results that all trees and almost all graphs are characterized by C2 .
DynFO: A Parallel, Dynamic Complexity Class
 Journal of Computer and System Sciences
, 1994
"... Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of compu ..."
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Cited by 50 (4 self)
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Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic FirstOrder Logic (DynFO). This is the set of properties that can be maintained and queried in firstorder logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in DynFO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...
A FirstOrder Isomorphism Theorem
 SIAM JOURNAL ON COMPUTING
, 1993
"... We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds. ..."
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Cited by 24 (5 self)
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We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds.
Time, Hardware, and Uniformity
 In Complexity Theory Retrospective II
, 1997
"... We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of nonuniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of var ..."
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Cited by 16 (3 self)
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We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of nonuniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of variable bits, and type of numeric predicates respectively. A fairly simple picture arises in which the basic questions in complexity theory  solved and unsolved  can be understood as questions about tradeoffs among these three dimensions. 1 Introduction An initial presentation of complexity theory usually makes the implicit assumption that problems, and hence complexity classes, are linearly ordered by "difficulty ". In the Chomsky Hierarchy each new type of automaton can decide more languages, and the Time Hierarchy Theorem tells us adding more time allows a Turing machine to decide more languages. Indeed the word "complexity" is often used (e.g., in the study of algorithms) to mean "wo...
Reachability and the Power of Local Ordering
, 1994
"... The L ? = NL question remains one of the major unresolved problems in complexity theory. Both L and NL have logical characterizations as the sets of totally ordered () structures expressible in firstorder logic augmented with the appropriate Transitive Closure operator [I87]: (FO+DTC+ ) captur ..."
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Cited by 16 (4 self)
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The L ? = NL question remains one of the major unresolved problems in complexity theory. Both L and NL have logical characterizations as the sets of totally ordered () structures expressible in firstorder logic augmented with the appropriate Transitive Closure operator [I87]: (FO+DTC+ ) captures L and (FO+TC+ ) captures NL. On the other hand, in the absence of ordering, (FO + TC) is strictly more powerful than (FO +DTC) [GM92]. An apparently quite different "structured" model of logspace machines is the Jumping Automaton on Graphs (JAG), [CR80]. We show that the JAG model is intimately related to these logics on "oneway locally ordered" (1LO) structures. We argue that the usual JAG model is unreasonably weak and should be replaced, wherever possible, by the twoway JAG model, which we define. Furthermore, the language (FO + DTC + 2LO) over twoway locally ordered (2LO) graphs is more robust than even the twoway JAG model, and yet lower bounds remain accessible. We pro...
Descriptive Complexity: a Logician's Approach to Computation
 Notices of the American Mathematical Society
, 1995
"... this article is complete for NSPACE[log n].) ..."
McColm's conjecture
 In Proc. 9th IEEE Symp. on Logic in Computer Science
, 1994
"... Gregory McColm conjectured that positive elementary inductions are bounded in a class K of finite structures if every (FO + LFP) formula is equivalent to a firstorder formula in K. Here (FO + LFP) is the extension of firstorder logic with the least fixed point operator. We disprove the conjecture. ..."
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Cited by 9 (0 self)
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Gregory McColm conjectured that positive elementary inductions are bounded in a class K of finite structures if every (FO + LFP) formula is equivalent to a firstorder formula in K. Here (FO + LFP) is the extension of firstorder logic with the least fixed point operator. We disprove the conjecture. Our main results are two modeltheoretic constructions, one deterministic and the other randomized, each of which refutes McColm’s conjecture. 1
The Computational Complexity Column
, 1998
"... Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful too ..."
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Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], AmbosSpies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resourcebounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resourcebounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their