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Wadge hierarchy and Veblen hierarchy. Part II: Borel sets of infinite rank
, 1998
"... We consider Borel sets of the form A ` ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a "normal form" of A, by finding a Borel set\Omega\Gamma ff) such that A and\Omega\Gamma ff) continuously reduce to each other. We do so by d ..."
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Cited by 39 (12 self)
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We consider Borel sets of the form A ` ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a "normal form" of A, by finding a Borel set\Omega\Gamma ff) such that A and\Omega\Gamma ff) continuously reduce to each other. We do so by defining Borel operations which are homomorphic to the first Veblen ordinal functions of base required to compute the Wadge degree of the set A: the ordinal ff.
The complexity of Nash equilibria in infinite multiplayer games
 In Proceedings of the 11th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2008
, 2008
"... Abstract. We study the complexity of Nash equilibria in infinite (turnbased, qualitative) multiplayer games. Chatterjee & al. showed the existence of a Nash equilibrium in any such game with ωregular winning conditions, and they devised an algorithm for computing one. We argue that in applicati ..."
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Cited by 18 (8 self)
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Abstract. We study the complexity of Nash equilibria in infinite (turnbased, qualitative) multiplayer games. Chatterjee & al. showed the existence of a Nash equilibrium in any such game with ωregular winning conditions, and they devised an algorithm for computing one. We argue that in applications it is often insufficient to compute just some Nash equilibrium. Instead, we enrich the problem by allowing to put (qualitative) constraints on the payoff of the desired equilibrium. Our main result is that the resulting decision problem is NPcomplete for games with coBüchi, parity or Streett winning conditions but fixedparameter tractable for many natural restricted classes of games with parity winning conditions. For games with Büchi winning conditions we show that the problem is, in fact, decidable in polynomial time. We also analyse the complexity of strategies realising a Nash equilibrium. In particular, we show that pure finitestate strategies as opposed to arbitrary mixed strategies suffice to realise any Nash equilibrium of a game with ωregular winning conditions with a qualitative constraint on the payoff. 1
Competitive Paging And DualGuided OnLine Weighted Caching And Matching Algorithms
, 1991
"... This thesis presents research done by the author on competitive analysis of online problems. An online problem is a problem that is given and solved one piece at a time. An online strategy for solving such a problem must give the solution to each piece knowing only the current piece and preceding ..."
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Cited by 12 (0 self)
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This thesis presents research done by the author on competitive analysis of online problems. An online problem is a problem that is given and solved one piece at a time. An online strategy for solving such a problem must give the solution to each piece knowing only the current piece and preceding pieces, in ignorance of the pieces to be given in the future. We consider online strategies that are competitive (guaranteeing solutions whose costs are within a constant factor of optimal) for several combinatorial optimization problems: paging, weighted caching, the kserver problem, and weighted matching. We introduce variations on the standard model of competitive analysis for paging: allowing randomization, allowing resourcebounded lookahead, and loose competitiveness, in which performance over a range of fast memory sizes is considered and noncompetitiveness is allowed provided the fault rate is insignificant. Each variation leads to substantially better competitive ratios. We prese...
Strategy construction in infinite games with Streett and Rabin chain winning conditions
 In TACAS 96, volume 1055 of Lect. Notes in Comp. Sci
, 1996
"... We consider finitestate games as a model of nonterminating reactive computations. A natural type of specification is given by games with Streett winning condition (corresponding to automata accepting by conjunctions of fairness conditions). We present an algorithm which solves the problem of progra ..."
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Cited by 10 (3 self)
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We consider finitestate games as a model of nonterminating reactive computations. A natural type of specification is given by games with Streett winning condition (corresponding to automata accepting by conjunctions of fairness conditions). We present an algorithm which solves the problem of program synthesis for these specifications. We proceed in two steps: First, we give a reduction of Streett automata to automata with the Rabin chain (or parity) acceptance condition. Secondly, we develop an inductive strategy construction over Rabin chain automata which yields finite automata that realize winning strategies. For the step from Rabin chain games to winning strategies examples are discussed, based on an implementation of the algorithm. 1 Introduction In recent years, methods of automatic verification for finitestate programs have been applied successfully, which have clearly reached the level of practical use. For the existing automata theoretic results on finitestate program sy...
Facets of Synthesis: Revisiting Church’s Problem
"... Abstract. In this essay we discuss the origin, central results, and some perspectives of algorithmic synthesis of nonterminating reactive programs. We recall the fundamental questions raised more than 50 years ago in “Church’s Synthesis Problem ” that led to the foundation of the algorithmic theory ..."
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Abstract. In this essay we discuss the origin, central results, and some perspectives of algorithmic synthesis of nonterminating reactive programs. We recall the fundamental questions raised more than 50 years ago in “Church’s Synthesis Problem ” that led to the foundation of the algorithmic theory of infinite games. We outline the methodology developed in more recent years for solving such games and address related automata theoretic problems that are still unresolved. 1
An InfiniteGame Semantics for WellFounded Negation in Logic Programming
"... We present an infinitegame characterization of the wellfounded semantics for functionfree logic programs with negation. Our game is a simple generalization of the standard game for negationless logic programs introduced by M.H. van Emden (1986, Journal of Logic Programming, 3(1), 3753) in which ..."
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Cited by 6 (2 self)
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We present an infinitegame characterization of the wellfounded semantics for functionfree logic programs with negation. Our game is a simple generalization of the standard game for negationless logic programs introduced by M.H. van Emden (1986, Journal of Logic Programming, 3(1), 3753) in which two players, the Believer and the Doubter, compete by trying to prove (respectively disprove) a query. The standard game is equivalent to the minimum Herbrand model semantics of logic programming in the sense that a query succeeds in the minimum model semantics iff the Believer has a winning strategy for the game which begins with the Doubter doubting this query. The game for programs with negation that we propose follows the same rules as the standard one, except that the players swap roles every time the play “passes through ” negation. We start our investigation by establishing the determinacy of the new game by using some classical tools from the theory of infinitegames. Our determinacy result immediately provides a novel and purely gametheoretic characterization of the semantics of negation in logic programming. We proceed to establish the connections of the game semantics to the existing semantic approaches for logic programming with negation. For this purpose, we first define a refined version of the game that uses degrees of winning and losing for the two players. We then demonstrate that this refined game corresponds exactly to the infinitevalued minimum model semantics of negation (Rondogiannis & Wadge, 2005, ACM TOCL, 6(2), 441467). This immediately implies that the unrefined game is equivalent to the wellfounded semantics (since the infinitevalued semantics is a refinement of the wellfounded semantics).
An Infinite Hierarchy of Temporal Logics over Branching Time
 INFORMATION AND COMPUTATION
, 2000
"... Many temporal logics were suggested as branching time specification formalisms during the last 20 years. These logics were compared against each other for their expressive power, model checking complexity and succinctness. Yet, unlike the case for linear time logics, no canonical temporal logic of b ..."
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Cited by 6 (3 self)
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Many temporal logics were suggested as branching time specification formalisms during the last 20 years. These logics were compared against each other for their expressive power, model checking complexity and succinctness. Yet, unlike the case for linear time logics, no canonical temporal logic of branching time was agreed upon. We offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified `yardstick' to measure these logics. We define
The Steel Hierarchy of Ordinal Valued Borel Mappings
, 2003
"... Given well ordered countable sets of the form , we consider Borel mappings from ! with countable image inside the ordinals. The ordinals and ! are respectively equipped with the discrete topology and the product of the discrete topology on . The Steel wellordering on such mappings is de ..."
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Given well ordered countable sets of the form , we consider Borel mappings from ! with countable image inside the ordinals. The ordinals and ! are respectively equipped with the discrete topology and the product of the discrete topology on . The Steel wellordering on such mappings is dened by i there exists a continuous function f such that (x) (x) holds for any x 2 ! . It induces a hierachy of mappings which we give a complete description of. We provide, for each ordinal , a mapping T() whose rank is precisely in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by . These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.