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58
Reasoning about The Past with TwoWay Automata
 In 25th International Colloqium on Automata, Languages and Programming, ICALP ’98
, 1998
"... Abstract. The pcalculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the pcalculus is EXPTIMEcomplete ..."
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Cited by 160 (14 self)
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Abstract. The pcalculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the pcalculus is EXPTIMEcomplete. This upper bound, however, is known for a version of the logic that has only forward modalities, which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability problem of the pcalculus with both forward and backward modalities. To get this result we develop a theory of twoway alternating automata on infinite trees. 1
On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 83 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
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 55 (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...
Fixpoint Logics, Relational Machines, and Computational Complexity
 In Structure and Complexity
, 1993
"... We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have t ..."
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Cited by 40 (5 self)
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We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic  while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...
Inductive Definability with Counting on Finite Structures
 IN PROC. OF COMPUTER SCIENCE LOGIC 92
, 1993
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BisimulationInvariant Ptime and HigherDimensional µCalculus
 THEORETICAL COMPUTER SCIENCE
, 1998
"... Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are ffl recognizable in polynomial time, and ffl closed under bisimulation equivalence. It is shown that the class of these bisimulationinvariant Ptime queries has a ..."
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Cited by 26 (4 self)
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Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are ffl recognizable in polynomial time, and ffl closed under bisimulation equivalence. It is shown that the class of these bisimulationinvariant Ptime queries has a natural logical characterization. It is captured by the straightforward extension of propositional µcalculus to arbitrary finite dimension. Bisimulationinvariant Ptime, or the modal fragment of Ptime, thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures. It is also shown that higherdimensional µcalculus is undecidable for satisfiability in finite structures, and even \Sigma 1 1 hard over general structures.
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 25 (6 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...
A Query Language for NC
 In Proceedings of 13th ACM Symposium on Principles of Database Systems
, 1994
"... We show that a form of divide and conquer recursion on sets together with the relational algebra expresses exactly the queries over ordered relational databases which are NC computable. At a finer level, we relate k nested uses of recursion exactly to AC k , k 1. We also give corresponding resul ..."
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Cited by 20 (11 self)
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We show that a form of divide and conquer recursion on sets together with the relational algebra expresses exactly the queries over ordered relational databases which are NC computable. At a finer level, we relate k nested uses of recursion exactly to AC k , k 1. We also give corresponding results for complex objects. 1 Introduction NC is the complexity class of functions that are computable in polylogarithmic time with polynomially many processors on a parallel random access machine (PRAM). The query language for NC discussed here is centered around a form of divide and conquer recursion (dcr ) on finite sets which has obvious potential for parallel evaluation and can easily express, for example, transitive closure and parity. Divide and conquer with parameters e; f; u defines the unique function ', notation dcr (e; f; u), taking finite sets as arguments, such that: '(;) def = e '(fyg) def = f(y) '(s 1 [ s 2 ) def = u('(s 1 ); '(s 2 )) when s 1 " s 2 = ; For parity, we t...
Querying Spatial Databases via Topological Invariants
 In PODS'98
, 1998
"... The paper investigates the use of topological annotations (called topological invariants) to answer topological queries in spatial databases. The focus is on the translation of topological queries against the spatial database into queries against the topological invariant. The languages considered ..."
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Cited by 19 (2 self)
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The paper investigates the use of topological annotations (called topological invariants) to answer topological queries in spatial databases. The focus is on the translation of topological queries against the spatial database into queries against the topological invariant. The languages considered are firstorder on the spatial database side, and fixpoint + counting, fixpoint, and firstorder on the topological invariant side. In particular, it is shown that fixpoint + counting expresses precisely all the ptime queries on topological invariants; if the regions are connected, fixpoint expresses all ptime queries on topological invariants. 1 Introduction Spatial data is an increasingly important part of database systems. It is present in a wide range of applications: geographic information systems, video databases, medical imaging, CADCAM, VLSI, robotics, etc. Different applications pose different requirements on query languages and therefore on the kind of spatial information th...
Almost Everywhere Equivalence Of Logics In Finite Model Theory
 BULLETIN OF SYMBOLIC LOGIC
, 1996
"... We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More pr ..."
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Cited by 19 (1 self)
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We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L 0 are two logics and # is an asymptotic measure on finite structures, then L j a.e. L 0 (#) means that there is a class C of finite structures with #(C ) = 1 and such that L and L 0 define the same queries on C. We carry out a systematic investigation of j a.e. with respect to the uniform measure and analyze the j a.e. equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework.