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38
From nondeterministic Büchi and Streett automata to deterministic parity automata
 In 21st Symposium on Logic in Computer Science (LICS’06
, 2006
"... Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic au ..."
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Cited by 45 (3 self)
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Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic automaton for the same language. Dualizing the set of accepting states gives an automaton for the complement language. In the theory of automata on infinite words, determinization and complementation are much more involved. Safra provides determinization constructions for Büchi and Streett automata that result in deterministic Rabin automata. For a Büchi automaton with n states, Safra constructs a deterministic Rabin automaton with n O(n) states and n pairs. For a Streett automaton with n states and k pairs, Safra constructs a deterministic Rabin automaton with (nk) O(nk) states and n(k + 1) pairs. Here, we reconsider Safra’s determinization constructions. We show how to construct automata with fewer states and, most importantly, parity acceptance condition. Specifically, starting from a nondeterministic Büchi automaton with n states our construction yields a deterministic parity automaton with n 2n+2 states and index 2n (instead of a Rabin automaton with (12) n n 2n states and n pairs). Starting from a nondeterministic Streett automaton with n states and k pairs our construction yields a deterministic parity automaton with n n(k+2)+2 (k+1) 2n(k+1) states and index 2n(k + 1) (instead of a Rabin automaton with (12) n(k+1) n n(k+2) (k+1) 2n(k+1) states and n(k+1) pairs). The parity condition is much simpler than the Rabin condition. In applications such as solving games and emptiness of tree automata handling the Rabin condition involves an additional multiplier of n 2 n! (or (n(k + 1)) 2 (n(k + 1))! in the case of Streett) which is saved using our construction.
Games with secure equilibria
 In Logic in Computer Science
, 2004
"... Abstract. In 2player nonzerosum games, Nash equilibria capture the options for rational behavior if each player attempts to maximize her payoff. In contrast to classical game theory, we consider lexicographic objectives: first, each player tries to maximize her own payoff, and then, the player tr ..."
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Cited by 19 (7 self)
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Abstract. In 2player nonzerosum games, Nash equilibria capture the options for rational behavior if each player attempts to maximize her payoff. In contrast to classical game theory, we consider lexicographic objectives: first, each player tries to maximize her own payoff, and then, the player tries to minimize the opponent’s payoff. Such objectives arise naturally in the verification of systems with multiple components. There, instead of proving that each component satisfies its specification no matter how the other components behave, it often suffices to prove that each component satisfies its specification provided that the other components satisfy their specifications. We say that a Nash equilibrium is secure if it is an equilibrium with respect to the lexicographic objectives of both players. We prove that in graph games with Borel winning conditions, which include the games that arise in verification, there may be several Nash equilibria, but there is always a unique maximal payoff profile of a secure equilibrium. We show how this equilibrium can be computed in the case of ωregular winning conditions, and we characterize the memory requirements of strategies that achieve the equilibrium.
Optimal Bounds for Transformations of ωAutomata
, 1999
"... In this paper we settle the complexity of some basic constructions of omegaautomata theory, concerning transformations of automata characterizing the set of omegaregular languages. In particular we consider Safra's construction (for the conversion of nondeterministic Büchi automata into determinis ..."
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Cited by 17 (0 self)
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In this paper we settle the complexity of some basic constructions of omegaautomata theory, concerning transformations of automata characterizing the set of omegaregular languages. In particular we consider Safra's construction (for the conversion of nondeterministic Büchi automata into deterministic Rabin automata) and the appearance record constructions (for the transformation between different models of deterministic automata with various acceptance conditions). Extending results of Michel (1988) and Dziembowski, Jurdzi'nski, and Walukiewicz (1997), we obtain sharp lower bounds on the size of the constructed automata.
Trading memory for randomness
 In QEST
, 2004
"... Strategies in repeated games can be classified as to whether or not they use memory and/or randomization. We consider Markov decision processes and 2player graph games, both of the deterministic and probabilistic varieties. We characterize when memory and/or randomization are required for winning w ..."
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Cited by 16 (10 self)
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Strategies in repeated games can be classified as to whether or not they use memory and/or randomization. We consider Markov decision processes and 2player graph games, both of the deterministic and probabilistic varieties. We characterize when memory and/or randomization are required for winning with respect to various classes ofregular objectives, noting particularly when the use of memory can be traded for the use of randomization. In particular, we show that Markov decision processes allow randomized memoryless optimal strategies for all Müller objectives. Furthermore, we show that 2player probabilistic graph games allow randomized memoryless strategies for winning with probability 1 those Müller objectives which are upwardclosed. Upwardclosure means that if a set of infinitely repeating vertices is winning, then all supersets of are also winning. 1
Faster solution of Rabin and Streett games
 In Proc. 21st Symposium on Logic in Computer Science
, 2006
"... In this paper we improve the complexity of solving Rabin and Streett games to approximately the square root of previous bounds. We introduce direct Rabin and Streett ranking that are a sound and complete way to characterize the winning sets in the respective games. By computing directly and explicit ..."
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Cited by 12 (4 self)
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In this paper we improve the complexity of solving Rabin and Streett games to approximately the square root of previous bounds. We introduce direct Rabin and Streett ranking that are a sound and complete way to characterize the winning sets in the respective games. By computing directly and explicitly the ranking we can solve such games in time O(mn k+1 kk!) and space O(nk) for Rabin and O(nkk!) for Streett where n is the number of states, m the number of transitions, and k the number of pairs in the winning condition. In order to prove completeness of the ranking method we give a recursive fixpoint characterization of the winning regions in these games. We then show that by keeping intermediate values during the fixpoint evaluation, we can solve such games symbolically in time O(n k+1 k!) and space O(n k+1 k!). These results improve on the current bounds of O(mn 2k k!) time in the case of direct (symbolic) solution or O(m(nk 2 k!) k) in the case of reduction to parity games. 1
Weak automata for the linear time µcalculus
 Proc. 6th Int. Conf. on Verification, Model Checking and Abstract Interpretation, VMCAI’05, volume 3385 of LNCS
, 2005
"... Abstract. This paper presents translations forth and back between formulas of the linear time µcalculus and finite automata with a weak parity acceptance condition. This yields a normal form for these formulas, in fact showing that the linear time alternation hierarchy collapses at level 0 and not ..."
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Cited by 8 (1 self)
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Abstract. This paper presents translations forth and back between formulas of the linear time µcalculus and finite automata with a weak parity acceptance condition. This yields a normal form for these formulas, in fact showing that the linear time alternation hierarchy collapses at level 0 and not just at level 1 as known so far. The translation from formulas to automata can be optimised yielding automata whose size is only exponential in the alternation depth of the formula. 1
Logical Refinements of Church’s Problem
 In CSL 2007, LNCS 4646
, 2007
"... Abstract. Church’s Problem (1962) asks for the construction of a procedure which, given a logical specification ϕ on sequence pairs, realizes for any input sequence X an output sequence Y such that (X,Y)satisfies ϕ. Büchi and Landweber (1969) gave a solution for MSO specifications in terms of finite ..."
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Cited by 8 (6 self)
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Abstract. Church’s Problem (1962) asks for the construction of a procedure which, given a logical specification ϕ on sequence pairs, realizes for any input sequence X an output sequence Y such that (X,Y)satisfies ϕ. Büchi and Landweber (1969) gave a solution for MSO specifications in terms of finitestate automata. We address the problem in a more general logical setting where not only the specification but also the solution is presented in a logical system. Extending the result of Büchi and Landweber, we present several logics L such that Church’s Problem with respect to L has also a solution in L, and we discuss some perspectives of this approach.
Church’s Problem and a Tour through Automata Theory
"... Dedicated to Boris A. Trakhtenbrot, pioneer and teacher of automata theory for generations of researchers, on the occasion of his 85th birthday Abstract. Church’s Problem, stated fifty years ago, asks for a finitestate machine that realizes the transformation of an infinite sequence α into an infini ..."
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Cited by 7 (2 self)
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Dedicated to Boris A. Trakhtenbrot, pioneer and teacher of automata theory for generations of researchers, on the occasion of his 85th birthday Abstract. Church’s Problem, stated fifty years ago, asks for a finitestate machine that realizes the transformation of an infinite sequence α into an infinite sequence β such that a requirement on (α, β), expressed in monadic secondorder logic, is satisfied. We explain how three fundamental techniques of automata theory play together in a solution of Church’s Problem: Determinization (starting from the subset construction), appearance records (for stratifying acceptance conditions), and reachability analysis (for the solution of games). 1
Generalized parity games
 In FoSSaCS’07, LNCS 4423
, 2007
"... Abstract. We consider games where the winning conditions are disjunctions (or dually, conjunctions) of parity conditions; we call them generalized parity games. These winning conditions, while ωregular, arise naturally when considering fair simulation between parity automata, secure equilibria for ..."
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Cited by 5 (1 self)
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Abstract. We consider games where the winning conditions are disjunctions (or dually, conjunctions) of parity conditions; we call them generalized parity games. These winning conditions, while ωregular, arise naturally when considering fair simulation between parity automata, secure equilibria for parity conditions, and determinization of Rabin automata. We show that these games retain the computational complexity of Rabin and Streett conditions; i.e., they are NPcomplete and coNPcomplete, respectively. The (co)NPhardness is proved for the special case of a conjunction/disjunction of two parity conditions, which is the case that arises in fair simulation and secure equilibria. However, considering these games as Rabin or Streett games is not optimal. We give an exposition of Zielonka’s algorithm when specialized to this kind of games. The complexity of solving these games for k parity objectives with d priorities, n states, and m edges is O(n 2kd ·m) · (k·d)! d! k, as compared to O(n 2kd ·m)·(k·d)! when these games are solved as Rabin/Streett games. We also extend the subexponential algorithm for solving parity games recently introduced by Jurdziński, Paterson, and Zwick to generalized parity games. The resulting complexity of solving generalized parity games is n O( √ n) · (k·d)! d! k. As a corollary we obtain an improved algorithm for Rabin and Streett games with d pairs, with time complexity n O( √ n) · d!. 1