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10
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 (7 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 ...
Computing With FirstOrder Logic
, 1995
"... We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtaine ..."
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Cited by 54 (13 self)
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We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...
Fixedpoint logics on planar graphs
, 1998
"... We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2 ..."
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Cited by 40 (12 self)
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We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2) Planar graphs can be characterized up to isomorphism in a logic with finitely many variables and counting. This answers a question of Immerman [7].(3) The class of planar graphs is definable in IFP. This answers a question of Dawar and Grädel [16].
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...
When do Fixed Point Logics Capture Complexity Classes?
 In Proceedings 10th IEEE Symposium on Logic in Computer Science
, 1995
"... We give examples of classes of rigid structures which are of unbounded rigidity but Least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LFP although sufficient for it to capture Boolean PTIM ..."
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Cited by 4 (1 self)
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We give examples of classes of rigid structures which are of unbounded rigidity but Least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LFP although sufficient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very different for nonzeroary queries. Next, we turn to the study of fixed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of finite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some sufficient conditions for PFP to capture PSPACE queries on classes of finite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to c...
Optimizing Active Databases using the Split Technique
 Proceedings 4th Intl. Conference on Database Theory (ICDT '92), LNCS 646
, 1992
"... A method to perform nonmonotonic relational rule computations is presented, called the split technique, The goal is to avoid redundant computations with rules that can insert and delete sets of tuples specified by the rule body. The method is independent of the control strategy that governs rule fir ..."
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Cited by 3 (2 self)
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A method to perform nonmonotonic relational rule computations is presented, called the split technique, The goal is to avoid redundant computations with rules that can insert and delete sets of tuples specified by the rule body. The method is independent of the control strategy that governs rule firing. Updatable relations are partitioned, as the computation progresses, into blocks of tuples such that tuples within a block are indiscernible from each other based on the computation so far. Results of previous rule firings are remembered as "relational equations" so that a new rule firing does not recompute parts of the result that can be determined from the existing equations. Seminaive evaluation falls out as a special case when all rules specify inserts. The method is amenable to parallelization.
The Role of Decidability in First Order Separations over Classes of Finite Structures
"... We establish that the decidability of the first order theory of a class of finite structures is a simple and useful condition for guaranteeing that the expressive power of FO + LFP properly extends that of FO on , unifying separation results for various classes of structures that have been studi ..."
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We establish that the decidability of the first order theory of a class of finite structures is a simple and useful condition for guaranteeing that the expressive power of FO + LFP properly extends that of FO on , unifying separation results for various classes of structures that have been studied. We then apply this result to show that it encompasses certain constructive pebble game techniques which are widely used to establish separations between FO and FO + LFP, and demonstrate that these same techniques cannot succeed in performing separations from any complexity class that contains DLOGTIME. 1.
Sharper Results on the Expressive Power of Generalized Quantifiers
, 1997
"... . In this paper we improve on some results of [3] and extend them to the setting of implicit definability. We show a strong necessary condition on classes of structures on which PSPACE can be captured by extending PFP with a finite set of generalized quantifiers. For IFP and PTIME the limitation of ..."
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. In this paper we improve on some results of [3] and extend them to the setting of implicit definability. We show a strong necessary condition on classes of structures on which PSPACE can be captured by extending PFP with a finite set of generalized quantifiers. For IFP and PTIME the limitation of expressive power of generalized quantifiers is shown only on some specific nontrivial classes. These results easily extend to implicit closure of these logics. In fact, we obtain a nearly complete characterization of classes of structures on which IMP (PFP ) can capture PSPACE if finitely many generalized quantifiers are also allowed. We give a new proof of one of the main results of [3], characterizing the classes of structures on which L ! 1;! (Q) collapses to FO(Q), where Q is a set of finitely many generalized quantifiers. This proof easily generalizes to the case of implicit definability, unlike the quantifier elimination argument of [3] which does not easily get adapted to implicit ...
Polynomially Orderable Classes of Structures
"... It is well known in descriptive computational complexity theory that fixpoint logic captures polynomial time on the class of ordered finite structures. The same is true on any class of structures on which a polynomial number of orders are definable in fixpoint logic. We call a class having this ..."
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It is well known in descriptive computational complexity theory that fixpoint logic captures polynomial time on the class of ordered finite structures. The same is true on any class of structures on which a polynomial number of orders are definable in fixpoint logic. We call a class having this property polynomially orderable. We investigate this property, and give examples of polynomially orderable classes of graphs and groups. 1 Introduction and summary In the field of descriptive computational complexity theory [11, 6, 1, 5], the complexity of computational problems is investigated in terms of the logics that can express them. Instances of a problem are represented as finite logical structures of some fixed similarity type. A standard result of the theory [10, 17] is that a property of ordered structures is recognizable in polynomial time if and only if it is definable in fixpoint logic. The restriction to ordered structures is important; many very simple polynomialtime pro...