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Relational Queries Computable in Polynomial Time
 Information and Control
, 1986
"... We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several ..."
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Cited by 269 (17 self)
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We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation. This proves that the fixed point query hierarchy suggested by Chandra and Harel collapses at the first fixed point level. It is also a general result showing that in finite model theory one application of fixed point suffices. Introduction and Summary Query languages for relational databases have received considerable attention. In 1972 Codd showed that two natural languages for queries  one algebraic and the other a version of first order predicate calculus  have identical powers of expressibility, [Cod72]. Query languages which are as expressive as Codd's Relational Calculus are sometimes called complete. This term is misleading however becau...
Alternating refinement relations
 In Proceedings of the Ninth International Conference on Concurrency Theory (CONCUR’98), volume 1466 of LNCS
, 1998
"... Abstract. Alternating transition systems are a general model for composite systems which allow the study of collaborative as well as adversarial relationships between individual system components. Unlike in labeled transition systems, where each transition corresponds to a possible step of the syste ..."
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Cited by 123 (16 self)
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Abstract. Alternating transition systems are a general model for composite systems which allow the study of collaborative as well as adversarial relationships between individual system components. Unlike in labeled transition systems, where each transition corresponds to a possible step of the system (which may involve some or all components), in alternating transition systems, each transition corresponds to a possible move in a game between the components. In this paper, we study refinement relations between alternating transition systems, such as “Does the implementation refine the set £ of specification components without constraining the components not in £? ” In particular, we generalize the definitions of the simulation and trace containment preorders from labeled transition systems to alternating transition systems. The generalizations are called alternating simulation and alternating trace containment. Unlike existing refinement relations, they allow the refinement of individual components within the context of a composite system description. We show that, like ordinary simulation, alternating simulation can be checked in polynomial time using a fixpoint computation algorithm. While ordinary trace containment is PSPACEcomplete, we establish alternating trace containment to be EXPTIMEcomplete. Finally, we present logical characterizations for the two preorders in terms of ATL, a temporal logic capable of referring to games between system components. 1
Describing Graphs: a FirstOrder Approach to Graph Canonization
, 1990
"... In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting ..."
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Cited by 57 (7 self)
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In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting quantifiers". We give efficient canonization algorithms for graphs characterized by Ck or Lk . It follows from known results that all trees and almost all graphs are characterized by C2 .
DiscreteTime Control for Rectangular Hybrid Automata
"... Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in realnumbered time, and the controller samples the plant state and changes the control state discretely, only at the integer poi ..."
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Cited by 56 (8 self)
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Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in realnumbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have nite bisimilarity quotients when all control transitions happen at integer times, even if the constraints on the derivatives of the variables vary between control states. This is in contrast with the conventional model where control transitions may happen at any real time, and already the reachability problem is undecidable. Based on the nite bisimilarity quotients, we give an exponential algorithm for the symbolic samplingcontroller synthesis of rectangular automata. We show our algorithm to be optimal by proving the problem to be EXPTIMEhard. We also show that rectangular automata form a maximal class of systems for which the samplingcontroller synthesis problem can be solved algorithmically.
DynFO: A Parallel, Dynamic Complexity Class
 Journal of Computer and System Sciences
, 1994
"... Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of compu ..."
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Cited by 50 (4 self)
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Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic FirstOrder Logic (DynFO). This is the set of properties that can be maintained and queried in firstorder logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in DynFO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...
Concurrent Reachability Games
, 2008
"... We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objecti ..."
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Cited by 49 (20 self)
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We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zerosum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type1 states, player 1 has a deterministic strategy to always reach the target. From type2 states, player 1 has a randomized strategy to reach the target with probability 1. From type3 states, player 1 has for every real ε> 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type1 states in linear time, and type2 and type3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.
Rectangular Hybrid Games
 In CONCUR 99, LNCS 1664
, 1999
"... In order to study control problems for hybrid systems, we generalize hybrid automata to hybrid games  say, controller vs. plant. If we specify the continuous dynamics by constant lower and upper bounds, we obtain rectangular games. We show that for rectangular games with objectives expressed in Lt ..."
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Cited by 30 (4 self)
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In order to study control problems for hybrid systems, we generalize hybrid automata to hybrid games  say, controller vs. plant. If we specify the continuous dynamics by constant lower and upper bounds, we obtain rectangular games. We show that for rectangular games with objectives expressed in Ltl (linear temporal logic), the winning states for each player can be computed, and winning strategies can be synthesized. Our result is sharp, as already reachability is undecidable for generalizations of rectangular systems, and optimal  singly exponential in the size of the game structure and doubly exponential in the size of the Ltl objective. Our proof systematically generalizes the theory of hybrid systems from automata (singleplayer structures) [9] to games (multiplayer structures): we show that the successively more general infinitestate classes of timed, 2d rectangular, and rectangular games induce successively weaker, but still finite, quotient structures called game bisimilarity, game similarity, and game trace equivalence. These quotients can be used, in particular, to solve the Ltl control problem.
Descriptive Complexity: a Logician's Approach to Computation
 Notices of the American Mathematical Society
, 1995
"... this article is complete for NSPACE[log n].) ..."