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26
Towards Tractable Algebras for Bags
, 1993
"... Bags, i.e. sets with duplicates, are often used to implement relations in database systems. In this paper, we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in ..."
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Cited by 59 (4 self)
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Bags, i.e. sets with duplicates, are often used to implement relations in database systems. In this paper, we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in the language extends its expressive power, and increases its complexity. We consider two main issues, namely (i) the impact of the depth of bag nesting on the expressive power, and (ii) the complexity and the expressive power induced by the algebraic operations. We show that the bag algebra is more expressive than the nested relation algebra (at all levels of nesting), and that the difference may be subtle. We establish a hierarchy based on the structure of algebra expressions. This hierarchy is shown to be highly related to the properties of the powerset operator. Invited to a special issue of the Journal of Computer and System Sciences selected from ACM Princ. of Database Systems,...
Infinitary Logic and Inductive Definability over Finite Structures
 Information and Computation
, 1995
"... The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abi ..."
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Cited by 56 (6 self)
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The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...
Feasible Computation through Model Theory
, 1993
"... The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as ..."
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Cited by 36 (7 self)
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The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of firstorder logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...
The Expressive Power of Finitely Many Generalized Quantifiers
 Information and Computation
, 1995
"... We consider extensions of first order logic (FO) and fixed point logic (FP) by means of generalized quantifiers in the sense of Lindstrom. We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature. This strengthens results in [10] and [15]. We als ..."
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Cited by 24 (5 self)
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We consider extensions of first order logic (FO) and fixed point logic (FP) by means of generalized quantifiers in the sense of Lindstrom. We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature. This strengthens results in [10] and [15]. We also prove a stronger version of this result for PSPACE, which enables us to establish a weak version of a conjecture formulated in [16]. These results are obtained by defining a notion of element type for bounded variable logics with finitely many generalized quantifiers. Using these, we characterize the classes of finite structures over which the infinitary logic L ! 1! extended by a finite set of generalized quantifiers Q is no more expressive than first order logic extended by the quantifiers in Q . 1 Introduction Computational complexity measures the complexity of a problem in terms of the resources, such as time, space, or hardware, required to solve the problem relative to a given ma...
Semantics and Expressiveness Issues in Active Databases
 J. OF COMPUTER AND SYSTEM SCIENCES
, 1995
"... A formal framework is introduced for studying the semantics and expressiveness of active databases. The power of various abstract trigger languages is characterized, and related to several major active database prototypes such as ARDL, HiPAC, Postgres, Starburst, and Sybase. This allows to forma ..."
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Cited by 23 (1 self)
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A formal framework is introduced for studying the semantics and expressiveness of active databases. The power of various abstract trigger languages is characterized, and related to several major active database prototypes such as ARDL, HiPAC, Postgres, Starburst, and Sybase. This allows to formally compare the expressiveness of the prototypes. The results provide insight into the programming paradigm of active databases, the interplay of various features, and their impact on expressiveness and complexity.
Definable Relations and FirstOrder Query Languages over Strings
"... We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relati ..."
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Cited by 22 (8 self)
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We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their modeltheory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,
First Order Logic, Fixed Point Logic and Linear Order
, 1995
"... The Ordered conjecture of Kolaitis and Vardi asks whether fixedpoint logic differs from firstorder logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures o ..."
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Cited by 16 (0 self)
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The Ordered conjecture of Kolaitis and Vardi asks whether fixedpoint logic differs from firstorder logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a modeltheoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexitytheoretic implications of this line of research.
How to Define a Linear Order on Finite Models
, 1997
"... We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, pa ..."
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Cited by 14 (1 self)
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We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, partial fixpoint logic PFP, infinitary logic L ! 1! with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the upper and lower bounds established here can not be made substantially tighter, unless outstanding conjectures in complexity theory are resolved at the same time. Research of L. Hella was partially supported by a grant from the University of Helsinki. y Research of Ph. Kolaitis was partially supported by a 1993 John Simon Guggenheim Fellowship and by NSF Grants No. CCR9108631, CCR9307758, and INT9024681 z Research of K. Luosto was partially supported by a grant from the Emil Aaltonen Foundation. 1 Intro...
Asymptotic Probabilities of Languages with Generalized Quantifiers
 In Proc. IEEE Symp. of Logic in Computer Science
, 1994
"... We study the impact of adding certain families of generalized quantifiers to firstorder logic (FO) on the asymptotic behavior of sentences. All our results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers. For a class K of finite structures clos ..."
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Cited by 8 (2 self)
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We study the impact of adding certain families of generalized quantifiers to firstorder logic (FO) on the asymptotic behavior of sentences. All our results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers. For a class K of finite structures closed under isomorphism, the quantifier QK is said to be strongly monotonic, sm, if membership in the class is preserved under a loose form of extensions. Our first theorem (0/1 law for FO with any set of sm quantifiers) subsumes a previous criterion for proving that almost no graphs satisfy a given property [BH79]. We also establish a 0/1 law for FO with Hartig quantifiers (equicardinality quantifiers) and a limit law for a fragment of FO with Rescher quantifiers (expressing inequalities of cardinalities) . The proofs of these last two results combine standard combinatorial enumerations with more sophisticated techniques from complex analysis. We also prove that the 0/1 law fails for the exten...