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Relational Queries Computable in Polynomial Time
 Information and Control
, 1986
"... We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several ..."
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Cited by 271 (17 self)
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We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation. This proves that the fixed point query hierarchy suggested by Chandra and Harel collapses at the first fixed point level. It is also a general result showing that in finite model theory one application of fixed point suffices. Introduction and Summary Query languages for relational databases have received considerable attention. In 1972 Codd showed that two natural languages for queries  one algebraic and the other a version of first order predicate calculus  have identical powers of expressibility, [Cod72]. Query languages which are as expressive as Codd's Relational Calculus are sometimes called complete. This term is misleading however becau...
Languages That Capture Complexity Classes
 SIAM Journal of Computing
, 1987
"... this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first ..."
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Cited by 230 (21 self)
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this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first order language of graphs (respectively groups, binary strings, etc.) together with a least fixed point operator. As another example, a property is in logspace if and only if it is expressible in first order logic together with a deterministic transitive closure operator. The roots of our approach to complexity theory go back to 1974 when Fagin showed that the NP properties are exactly those expressible in second order existential sentences. It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy. We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time, respectively. The existence of such natural languages for each important complexity class sheds a new light on complexity theory. These languages reaffirm the importance of the complexity classes as much more than machine dependent issues. Furthermore a whole new approach is suggested. Upper bounds (algorithms) can be produced by expressing the property of interest in one of our languages. Lower bounds may be demonstrated by showing that such expression is impossible.
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by ..."
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Cited by 127 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
Multiparty protocols
 In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing
, 1983
"... process communication have been examined from a complexity pbint of view [SP, Y]. We study a new model, in which a collection of processes eo, "'', e~:~l ..."
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Cited by 111 (1 self)
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process communication have been examined from a complexity pbint of view [SP, Y]. We study a new model, in which a collection of processes eo, "'', e~:~l
merging, and sorting in parallel models of computation
 in “Proc. 14th Annual ACM Sympos. on Theory of Cornput
, 1982
"... A variety of models have been proposed for the study of synchronous parallel computation. These models are reviewed and some prototype problems are studied further. Two classes of models are recognized, fixed connection networks and models based on a shared memory. Routing and sorting are prototype ..."
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Cited by 105 (3 self)
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A variety of models have been proposed for the study of synchronous parallel computation. These models are reviewed and some prototype problems are studied further. Two classes of models are recognized, fixed connection networks and models based on a shared memory. Routing and sorting are prototype problems for the networks; in particular, they provide the basis for simulating the more powerful shared memory models. It is shown that a simple but important class of deterministic strategies (oblivious routing) is necessarily inefficient with respect to worst case analysis. Routing can be viewed as a special case of sorting, and the existence of an O(log n) sorting algorithm for some n processor fixed connection network has only recently been established by Ajtai, Komlos, and Szemeredi (“15th ACM Sympos. on Theory of Cornput., ” Boston, Mass., 1983, pp. l9). If the more powerful class of shared memory models is considered then it is possible to simply achieve an O(log n loglog n) sort via Valiant’s parallel merging algorithm, which it is shown can be implemented on certain models. Within a spectrum of shared memory models, it is shown that loglogn is asymptotically optimal for n processors to merge two sorted lists containing n elements. 0 1985 Academic Press, Inc.
Lower Bounds for the Size of Circuits of Bounded Depth in Basis
, 1986
"... this paper, we consider circuits of bounded depth in the basis f; \Phig. ..."
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Cited by 77 (0 self)
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this paper, we consider circuits of bounded depth in the basis f; \Phig.
Polynomial size proofs of the propositional pigeonhole principle
 Journal of Symbolic Logic
, 1987
"... Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems ha ..."
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Cited by 72 (7 self)
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Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = coNP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which
The Polynomial Method in Circuit Complexity
 In Proceedings of the 8th IEEE Structure in Complexity Theory Conference
, 1993
"... The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polyno ..."
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Cited by 68 (4 self)
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The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of ANDOR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...
On the compilability and expressive power of propositional planning formalisms
, 1998
"... The recent approaches of extending the GRAPHPLAN algorithm to handle more expressive planning formalisms raise the question of what the formal meaning of “expressive power ” is. We formalize the intuition that expressive power is a measure of how concisely planning domains and plans can be expressed ..."
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Cited by 67 (10 self)
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The recent approaches of extending the GRAPHPLAN algorithm to handle more expressive planning formalisms raise the question of what the formal meaning of “expressive power ” is. We formalize the intuition that expressive power is a measure of how concisely planning domains and plans can be expressed in a particular formalism by introducing the notion of “compilation schemes ” between planning formalisms. Using this notion, we analyze the expressiveness of a large family of propositional planning formalisms, ranging from basic STRIPS to a formalism with conditional effects, partial state specifications, and propositional formulae in the preconditions. One of the results is that conditional effects cannot be compiled away if plan size should grow only linearly but can be compiled away if we allow for polynomial growth of the resulting plans. This result confirms that the recently proposed extensions to the GRAPHPLAN algorithm concerning conditional effects are optimal with respect to the “compilability ” framework. Another result is that general propositional formulae cannot be compiled into conditional effects if the plan size should be preserved linearly. This implies that allowing general propositional formulae in preconditions and effect conditions adds another level of difficulty in generating a plan.