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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 35 (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...
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 26 (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.
Fixed Point Logics, Generalized Quantifiers, and Oracles
, 1995
"... The monotone circuit problem QMC is shown to be complete for fixed point logic IFP under quantifier free reductions. Enhancing the circuits with an oracle Q leads to a problem complete for IFP(Q). By contrast, if L is any extension of FO with generalized quantifiers, one can always find a Q such ..."
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Cited by 1 (1 self)
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The monotone circuit problem QMC is shown to be complete for fixed point logic IFP under quantifier free reductions. Enhancing the circuits with an oracle Q leads to a problem complete for IFP(Q). By contrast, if L is any extension of FO with generalized quantifiers, one can always find a Q such that IFP (Q) is not contained in L(Q). For L = FO(QMC ), we have L j IFP but L(Q) ! IFP (Q). The adjunction of Q reveals the difference between these two representations of the class of IFPdefinable queries. Also for partial fixed point logic, PFP a complete problem based on circuits is given, and, concerning the adjunction of further quantifiers, similar results as for IFP are proved. On ordered structures, where our results still hold, this reads as follows: For any given oracle Q, the complexity class PTIME Q (or PSPACE Q in a bounded oracle model) can be characterized by an extension of FO with a uniform sequence of quantifiers. However, there is no such logic L that sat...
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"... Introduction Complexity Theory investigates the computational resources required to solve decision problems. A decision problem is usually given as a set of strings. Typical measures of complexity are time or space, that is the number of steps, or the amount of tape cells respectively, used by a Tur ..."
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Introduction Complexity Theory investigates the computational resources required to solve decision problems. A decision problem is usually given as a set of strings. Typical measures of complexity are time or space, that is the number of steps, or the amount of tape cells respectively, used by a Turing machine that accepts this set of strings. In many cases however, decision problems represent structural properties, say the property of a graph having a Hamiltonian cycle. Hence, it is natural to generalize the notion of a decision problem from sets of strings to classes of finite structures (it is a generalization, because a string can also be seen as a finite structure).
Logic and Complexity without Order
, 1997
"... Introduction Descriptive Complexity can be thought of as differing from the more common view of computational complexity (i.e. measuring resource bounds on a machine) in two important respects: 1. We measure the complexity of a collection of structures as opposed to a collection of strings. This ca ..."
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Introduction Descriptive Complexity can be thought of as differing from the more common view of computational complexity (i.e. measuring resource bounds on a machine) in two important respects: 1. We measure the complexity of a collection of structures as opposed to a collection of strings. This can be seen as a generalisation, in that strings are a special case. It is also a useful generalisation in that the problems we are concerned with (graph problems, for instance) are naturally thought of as classes of structures, and must be encoded into strings in order to fit the mould of machine computations. 2. We measure the complexity of describing the collection as opposed to the complexity of computing it. So, the resources that are measured are logical resources (number and kind of quantifier, number of variables, etc.) as opposed to space, time and so on. Of course, the interest in descriptive complexity stems in large part from the fact that there is a close co
The Nature and Power of FixedPoint Logic with Counting
"... of fixed point logic with counting (FPC). This natural class is deeply entwined with the challenge of capturing the class of polynomialtime graph properties and determining the complexity of graph isomorphism — problems that remain open. Despite that, so much more is now known about the remarkable ..."
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of fixed point logic with counting (FPC). This natural class is deeply entwined with the challenge of capturing the class of polynomialtime graph properties and determining the complexity of graph isomorphism — problems that remain open. Despite that, so much more is now known about the remarkable power of FPC. Anuj Dawar explains in the following lovely survey.