Results 1  10
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13
Relativized Logspace and Generalized Quantifiers over Finite Structures
 Journal of Symbolic Logic
, 1997
"... The expressive power of first order logic with generalized quantifiers over finite ordered structures is studied. The following problem is addressed: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the q ..."
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Cited by 25 (4 self)
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The expressive power of first order logic with generalized quantifiers over finite ordered structures is studied. The following problem is addressed: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized by an oracle in C. We show that this is not always true. However, we derive sufficient conditions on complexity class C such that FO(Q) captures L C . These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures L NP . This answers a question raised by Blass and Gurevich. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q...
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.
Descriptive Complexity Theory over the Real Numbers
 LECTURES IN APPLIED MATHEMATICS
, 1996
"... We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field ..."
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Cited by 24 (8 self)
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We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field of reals and a finite set of functions from the finite structure into R. They are a special case of the metafinite structures introduced recently by Grädel and Gurevich. We argue that Rstructures provide the right class of structures to develop a descriptive complexity theory over R. We substantiate this claim by a number of results that relate logical definability on Rstructures with complexity of computations of BSSmachines.
Oracles and Quantifiers
, 1994
"... . We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence ..."
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Cited by 11 (4 self)
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. We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence between enhancing a logic with a Lindstr om quantifier and enhancing a complexity class with an oracle such that, if K is a set of structures, QK the associated Lindstrom quantifier and L a logic that captures a complexity class D, then the enhanced logic L[K] captures D K  the complexity class of machines in D using oracles for K. Our results are sensitive to the oracle computation model and hold in a natural modification of the unbounded model introduced by Buss [Bus88]. They do not hold in the, so called, space bounded oracle models or those that violate the `relativization thesis' of Buss. Our results generalize and extend previous results of Stewart [Ste93a, Ste93b] and Makowsky and...
Hierarchies in Classes of Program Schemes
, 1999
"... We begin by proving that the class of problems accepted by the program schemes of NPS is exactly the class of problems defined by the sentences of transitive closure logic (program schemes of NPS are obtained by generalizing basic nondeterministic whileprograms whose tests within while instruction ..."
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Cited by 9 (9 self)
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We begin by proving that the class of problems accepted by the program schemes of NPS is exactly the class of problems defined by the sentences of transitive closure logic (program schemes of NPS are obtained by generalizing basic nondeterministic whileprograms whose tests within while instructions are quantifierfree firstorder formulae). We then show that our program schemes form a proper infinite hierarchy within NPS whose analogy in transitive closure logic is a proper infinite hierarchy, the union of which is full transitive closure logic but for which every level of the hierarchy has associated with it a firstorder definable problem not in that level. We then proceed to add a stack to our program schemes, so obtaining the class of program schemes NPSS, and characterize the class of problems accepted by the program schemes of NPSS as the class of problems defined by the sentences of path system logic. We show that there is a proper infinite hierarchy within NPSS, with an analo...
Program schemes, arrays, Lindström quantifiers and zeroone laws
 PROCEEDINGS OF COMPUTER SCIENCE LOGIC 1999 , LECTURE NOTES IN COMPUTER SCIENCE VOLUME 1683, SPRINGERVERLAG
, 1999
"... We characterize the class of problems accepted by a class of program schemes with arrays, NPSA, as the class of problems defined by the sentences of a logic formed by extending firstorder logic with a particular uniform (or vectorized) sequence of Lindstrom quantifiers. A simple extension of a known ..."
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Cited by 7 (6 self)
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We characterize the class of problems accepted by a class of program schemes with arrays, NPSA, as the class of problems defined by the sentences of a logic formed by extending firstorder logic with a particular uniform (or vectorized) sequence of Lindstrom quantifiers. A simple extension of a known result thus enables us to prove that our logic, and consequently our class of program schemes, has a zeroone law. However, we use another existing result to show that there are problems definable in a basic fragment of our logic, and so also accepted by basic program schemes, which are not definable in boundedvariable infinitary logic. As a consequence, the class of problems NPSA is not contained in the class of problems defined by the sentences of partial fixedpoint logic even though in the presence of a builtin successor relation, both NPSA and partial fixedpoint logic capture the complexity class PSPACE.
The Expressive Power of Transitive Closure and 2way Multihead Automata
 Lecture Notes in Computer Science 626
, 1992
"... It is known, that over 1dimensional strings, the expressive power of 2way multihead (non) deterministic automata and (non) deterministic Transitive Closure formulas is (non) deterministic log space [Ib73, Im88]. However, the subset of formulas needed to simulate exactly k heads is unknown. It is a ..."
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Cited by 6 (2 self)
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It is known, that over 1dimensional strings, the expressive power of 2way multihead (non) deterministic automata and (non) deterministic Transitive Closure formulas is (non) deterministic log space [Ib73, Im88]. However, the subset of formulas needed to simulate exactly k heads is unknown. It is also unknown if the automata and formulas have the same expressive power over more general structures such as multidimensional grids. We define a reduction from khead automata to formulas of arity k, which works also for grids. The method used is a generalization of [Kl56], and the formulas obtained are a generalization of regular expressions to multihead automata and to grid languages. As simple applications, we use the reduction to show that the power of formulas of arity 1 over strings define (classical) regular languages, to give a simpler equivalent of the L=NL open problem, and to establish the equivalence of the automata and formulas over grids. Faculty of Computer Science, Technio...
Translation Schemes and the Fundamental Problem of Database Design
, 1996
"... . We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to reexamine the notion of dependency preserving decomposition and its generalization refinemen ..."
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Cited by 2 (2 self)
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. We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to reexamine the notion of dependency preserving decomposition and its generalization refinement. The most important aspect of this approach lies in laying the groundwork for a formulation of the Fundamental Problem of Database Design, namely to exhibit desirable differences between translationequivalent presentations of data and to examine refinements of data presentations in a systematic way. The emphasis in this paper is not on results. The main line of thought is an exploration of the use of an old logical tool in addressing the Fundamental Problem. Translation schemes allow us to have a new look at normal forms of database schemes and to suggest a new line of search for other normal forms. Furthermore we give a characterization of the embedded implicational dependencies (EID'...
Dependency Preserving Refinements and the Fundamental Problem of Database Design
, 1998
"... . We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to reexamine the notion of dependency preserving decomposition and its generalization refinemen ..."
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Cited by 1 (0 self)
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. We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to reexamine the notion of dependency preserving decomposition and its generalization refinement. We demonstrate the usefulness of this approach by recasting the theory of vertical and horizontal decompositions in our terminology. The most important aspect of this approach, however, lies in laying the groundwork for a formulation of the Fundamental Problem of Database Design, namely to exhibit desirable differences between translation equivalent presentations of data and to examine refinements of data presentations in a systematic way. The emphasis in this paper is not on results. The main line of thought is an exploration of the use of an old logical tool in addressing the Fundamental Problem. Translation schemes allow us also to have a new look at normal forms of database schemes an...
Program Schemes, Queues, the Recursive Spectrum and ZeroOne Laws
"... We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of r ..."
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Cited by 1 (1 self)
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We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of recursively solvable problems. The class of problems accepted by the program schemes of the class NPSQ(1) where only access to a queue, and not the additional numeric universe, is allowed is exactly the class of recursively solvable problems that are closed under extensions. We dene an innite hierarchy of classes of program schemes for which NPSQ(1) is the rst class and the union of the classes of which is the class NPSQ. We show that the class of problems accepted by the program schemes of NPSQ has a zeroone law and is the union of the classes of problems dened by the sentences of all vectorized Lindstrom logics formed using operators whose corresponding problems are recursively solvab...