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60
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bound ..."
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Cited by 38 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Local Properties of Query Languages
"... predeterminedportionoftheinput.Examplesincludeallrelationalcalculusqueries. everyrelationalcalculus(rstorder)queryislocal,thegeneralresultsprovedforlocalqueriescan manyeasyinexpressibilityproofsforlocalqueries.Wethenconsideracloselyrelatedproperty, namely,theboundeddegreeproperty.Itdescribestheoutp ..."
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Cited by 34 (23 self)
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predeterminedportionoftheinput.Examplesincludeallrelationalcalculusqueries. everyrelationalcalculus(rstorder)queryislocal,thegeneralresultsprovedforlocalqueriescan manyeasyinexpressibilityproofsforlocalqueries.Wethenconsideracloselyrelatedproperty, namely,theboundeddegreeproperty.Itdescribestheoutputsoflocalqueriesonstructuresthat locallylook\simple."Everyquerythatislocalisshowntohavetheboundeddegreeproperty.Since Westartbyprovingageneralresultdescribingoutputsoflocalqueries.Thisresultleadsto toapplythanEhrenfeuchtFrassegames.Wealsoshowthatsomegeneralizationsofthebounded degreepropertythatwereconjecturedtohold,failforrelationalcalculus. beviewedas\otheshelf"strategiesforprovinginexpressibilityresults,whichareofteneasier maintenanceofviews,andshowthatSQLandrelationalcalculusareincapableofmaintainingthe gregates,whichisessentiallyplainSQL,hastheboundeddegreeproperty,thusansweringaques tionthathasbeenopenforseveralyears.Consequently,rstorderquerieswithHartigorRescher quantiersalsohavetheboundeddegreeproperty.Finally,weapplyourresultstoincremental Wethenprovethatthelanguageobtainedfromrelationalcalculusbyaddinggroupingandag
Notions of Locality and Their Logical Characterizations Over Finite Models
, 1997
"... Many known tools for proving expressibility bounds for firstorder logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with senten ..."
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Cited by 33 (18 self)
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Many known tools for proving expressibility bounds for firstorder logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the firstorder case. We use them to derive expressibility bounds for firstorder logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. 1 Introduction It is well known that firstorder logic has limited expressive power. Typically, inexpressibility proofs are based on either a compactness argument, or EhrenfeuchtFraiss'e games. In ...
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 31 (8 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
Logics with Aggregate Operators
 Journal of the ACM
"... We study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, a ..."
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Cited by 28 (15 self)
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We study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, aggregates are not adequately captured by the existing logical formalisms. Consequently, all previous approaches to analyzing the expressive power of aggregation were only capable of producing partial results, depending on the allowed class of aggregate and arithmetic operations. We consider a powerful counting logic, and extend it with the set of all aggregate operators. We show that the resulting logic satis es analogs of Hanf's and Gaifman's theorems, meaning that it can only express local properties. We consider a database query language that expresses all the standard aggregates found in commercial query languages, and show how it can be translated into the aggregate logic, thereby pro...
Directed planar reachability is in unambiguous logspace
 In Proceedings of IEEE Conference on Computational Complexity CCC
, 2007
"... We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1. ..."
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Cited by 25 (6 self)
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We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.
Relational queries over interpreted structures
 Journal of the ACM
, 2000
"... We rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries. We identify parts of the classical theory that go through `as before ' when interpreted structure is present, parts that go through only for ..."
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Cited by 24 (13 self)
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We rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries. We identify parts of the classical theory that go through `as before ' when interpreted structure is present, parts that go through only for classes of nicelybehaved structures, and parts that only arise in the interpreted case. The rst category includes a number of results on language equivalence and expressive power characterizations for the activedomain semantics for a variety of logics. Under this semantics, quanti ers range over elements of a relational database. The main kind of results we prove here are generic collapse results: for generic queries, adding operations beyond order, does not give us extra power. The second category includes results on the natural semantics, under which quantiers range over the entire interpreted structure. We prove, for a variety of structures, naturalactive collapse results, showing that using unrestricted quantication does not give us any extra power. Moreover, for a variety of structures, including the real eld, we give a set of algorithms for eliminating unbounded quantications in favor of bounded ones. Furthermore, we extend these collapse results to a new class of higherorder logics that mix unbounded and bounded quantication. We give a set of normal forms for these logics, under special conditions on the interpreted structures. As a byproduct, we obtain an elementary proof of the fact that parity test is not denable in the relational calculus with polynomial inequality constraints. We also give examples of structures with nice modeltheoretic properties over which the naturalactive collapse fails. 1
The directed planar reachability problem
 In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science
, 2005
"... Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the ..."
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Cited by 23 (9 self)
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Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previouslystudied subclass of planar graphs known as grid graphs. We show that the directed planar stconnectivity problem reduces to the reachability problem for directed grid graphs. A special case of the gridgraph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1
Verifiable properties of database transactions
 Information and Computation
, 1998
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On the forms of locality over finite models
 Logic in Computer Science
, 1997
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