A.K. Chandra, D.C. Kozen and L.J. Stockmeyer, "Alternation", Journal of the ACM 28 (1981), 114-133.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....machines is known to havethe effect of turning time into space: It was shown in [Pa1] that the languages decidable bypolynomial time bounded stochastic Turing machines comprise all of PSPACE. The lemma essentially says that, as is the case with the alternating machines (another much studied variant[CHK]) stochastic machines also turn space into time one exponential higher. We first show that anyspace bounded stochastic Turing machine can be simulated in exponential time. The simulation algorithm is simple: Wevisit the exponentially many configurations of the machine one by one, in decreasing ....

A. Chandra, D. Harel, D. Kozen "Alternation," J.ACM 28,pp. 114--133, 1981.

....computational complexity has been focused on the study of how quantitatively measured resources a#ect computation what can and cannot be decided when algorithms are allowed bounded resources which grow with the size of the input. The intuitive notions of nondeterminism [13] versus alternation [2], and time [5] versus space [6] as resources have been cornerstones in the development of the theory. Complexity theorists have placed a strong emphasis on understanding computable sets via the time and space requirements of deterministic, nondeterministic, or alternating machines recognizing ....

....we can exploit our access the solution space to find more than just one solution, or two or three or exponentially many solutions. Rather, we can simulate alternating polynomial time with the solution space and our logspace machine. Recall that alternating polynomial time is equivalent to PSPACE [2]. Theorem 8.1 PSPACE = L P I. Proof. L P I PSPACE is true since the L P I model always uses no more than polynomial space. For the other direction, it su#ces to prove that for any S AP (alternating polynomial time) we can decide S using a L P I machine. AP , without loss of ....

A. Chandra, D. Kozen, L. Stockmeyer, "Alternation," Journal of the ACM, 28:114133, 1981.

....procedure we can develop a labeling corresponding to the non loss outcome, and show the theorem to be true for non loss outcome as well. Moreover, both labeling (for win and non loss outcome) can be computed in deterministic time 2 O(S(n) 2 The following theorem is due to Chandra, Stockmeyer [36]: Corollary 4.3.1 (CS76) For any S(n) log(n) ASPACE(S(n) ATIME(EXP(S(n) ASPACE(S(n) DTIME(EXP(S(n) Proof: The proof follows from Lemma 4.3.2 and Theorem 4.3.1 in conjunction with the upper bound results of Chandra et al. 36] 2 22 5 Eliminating Incomplete Information This ....

..... 2 The following theorem is due to Chandra, Stockmeyer [36] Corollary 4.3.1 (CS76) For any S(n) log(n) ASPACE(S(n) ATIME(EXP(S(n) ASPACE(S(n) DTIME(EXP(S(n) Proof: The proof follows from Lemma 4.3.2 and Theorem 4.3. 1 in conjunction with the upper bound results of Chandra et al. [36]. 2 22 5 Eliminating Incomplete Information This section provides methods for converting games of incomplete information to perfect information games by making the incomplete information content explicit. Subsequently, we can use algorithms for deciding games of perfect information for deciding ....

A. K. Chandra and L. J. Stockmeyer, "Alternation", in Proceedings of 17th Annual IEEE Symposium on Foundations of Computer Science, pages 98-108, October 1976. 37

....parallel, etc. were associated with corresponding models of computations (non deterministic Turing machine, parallel random access machine, etc. The need for a formal computational model to address the computational aspects of games was fulfilled by Chandra, Kozen, and Stockmeyer [11] with the Alternating Turing Machine (A TM) Subsequently, this model has been extended and enhanced to model more intricate 5 games. Reif [1, 2] extended A TM model to incorporate private and blindfold two player games by introducing private alternating Turing machine (PA TM) and blind ....

....because a player can deduce certain characteristic of the board when the player attempts to make a move that is termed illegal due to information that previously not known to them. In this paper, we present extensions of the alternating machines of Reif [1, 2] and Chandra, Kozen, and Stockmeyer [11]. Our machines model multiplayer games of incomplete information. MPA k TM is a machine model that corresponds to a (k 1) player multiplayer (team) game of incomplete information with k existential players and one universal player. The states of the machine are labeled with tuples: each ....

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A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer, "Alternation", J. of ACM, 28(1): pages 114-133, 1981.

....only existential (universal) states is not closed undercomp= 8W tation. We further show a few otherphe erties of these automata. 2. Preliminaries A 2 way alternating multi counter automaton (2amca) M is a generalization of a 2 way nondeterministic multicounter automaton in the same sense as [8]. The state set of M ispLOz7O]W78 into universal and existential # From now on, logarithms are base 2. 286 IEICE TRANS. INF. SYST. VOL.E83 D, NO.2 FEBRUARY 2000 states. Intuitively, in a universal state MspL7 into some submachines which act inp]W 7W8] and in an existential state M ....

A.K. Chandra, D.C. Kozen, and L.J. Stockmeyer, "Alternation, " J. ACM, vol.28, no.1, pp.114--133, 1981.

....proof of Lemma 2 can be extended in a straightforward manner to nite tree automata. The only di erence will be that the algorithm will do universal choices when the arity of function symbols (letters) in the component automata is 1. This leads to alternating PSPACE, and thus, by the result of Chandra, Kozen Stockmeyer [1981], to EXPTIME upper bound for the constrained product nonemptiness problem of TAs. Although we will not use this fact, it is worth noting that the constrained product nonemptiness problem is also PSPACE hard, and this so already for DFAs (or monadic DTAs) It is easy to see that T n i=1 L(A i ) ....

Chandra, A., Kozen, D. & Stockmeyer, L. (1981), `Alternation', Journal of the Association for Computing Machinery 28(1), 114-133.

....to the obtained results and pass on the resulting value to the state by which it was activated. A word w is accepted if the starting state computes the value of 1 and it is rejected otherwise. Below we briefly recall the definition of AFA and their equational representation. For more details see [5, 8, 14]. We denote by the symbol B the two element Boolean algebra B = f0; 1g; 0; 1) Let Q be a finite set. Then B Q is the set of all mappings of Q into B. Note that u 2 B Q can be considered as a vector of jQj entries, indexed by elements of Q, with each entry being from B. For u 2 B Q ....

A.K. Chandra, D.C. Kozen, L.J. Stockmeyer, "Alternation", Journal of the ACM 28 (1981) 114-133.

....Boolean valued functions on 2 S . Let 2 B S and S 0 S. Then (S 0 ) is the Boolean value of when the states in S 0 are assigned 1 and the states in S Gamma S 0 are assigned 0. Formulas of the form s or :s, where s 2 S, are called atomic formulas. An alternating finite automaton [BL80, CKS81] (abbr. AFA) A is a tuple A = Sigma; S; ae; 0 ; F ) where Sigma is the input alphabet, S is the set of states fs 1 ; s mg, ae : S Theta Sigma B S is the transition function that associates with each state and letter a Boolean formula in B S , 0 2 B S is the start formula, and F ....

....there is a child y of x labeled by . A node x labeled by x is accepting if x (F ) 1. The run forest is accepting if x is accepting whenever x is a leaf. A accepts w if it has an accepting run forest on w. Afa s define regular languages. Nevertheless, it follows from the results in [Le81] [CKS81] that they can be exponentially more succinct than NFA s. That is, given any n state AFA, one can construct an 2 n state NFA that accepts the same language. Furthermore, for each n there is an n states AFA A, such that the language defined by A is not definable by any NFA with less than 2 n ....

A.K. Chandra, D.C. Kozen, and L.J. Stockmeyer, "Alternation", J. ACM 28(1981), pp. 114--133.

....If the time unit is the average time for a state transition, then T (jxj) is an upper bound on the average time for the network to settle. If the network is synchronous, then T (jxj) is the actual completion time. 3 Alternating machines Alternation was defined by Chandra, Kozen and Stockmeyer [1] as a general model for parallel computation. Because of its close relationship to logic programming, alternation has been suggested as a good paradigm to write and reason about parallel algorithms [5, 3] It is closely related to many other models of parallel computation [2] The fundamental ....

....label Reject in Gamma . It is possible, but not necessary, to add functions other than and to the definition of an alternating machine. In particular, negation (interchanging Accept and Reject) may be simulated using only and by introducing positive and negative versions of each state [1]. Any other function can be obtained using , and : An alternating machine M computes a function g if M can recognize each output symbol; i.e. M accepts from configuration Out(k; a) with x as the input string if and only if the kth symbol of g(x) is a, where Out is a predetermined mapping from ....

A. K. Chandra, D. C. Kozen and L. J. Stockmeyer, "Alternation," J. Assoc. Comput. Mach. 28 (1981) 114--133. Page 6

.... it a good formalization of the notion of feasible computation [19] Arguments for this view include the robustness of P it is largely independent of machine model, and is closed under composition and Boolean operations and its many machine independent, logical characterizations (e.g. [7, 13, 15, 28, 29, 31, 24, 33, 34, 40, 41, 45, 51]) Unfortunately, general polynomials are too large to be considered feasible. A program that runs in time Theta(n 17 ) has little practical value. Many classes within P have been studied, including those defined by circuit complexity, or those defined on Turing machines or RAM models by ....

....definitions that do not depend on an arbitrary bound on nondeterminism. 4.1 An Alternating Machine Model Our first characterization comes from a modified alternating machine, which allows blocks of nondeterminism in a simple, natural way. The standard model of an alternating Turing machine [15] chooses (existentially or universally) one of two possible next states at each step. However, alternation can also be achieved by existentially or universally choosing a head position. This provides the desired (log n) bit blocks of nondeterminism. Definition 11 A length alternating Turing ....

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A. Chandra, D. Kozen, and L. Stockmeyer, "Alternation," J. Assoc. Comput. Mach. 28 (1981) 114-133.

....[10] and Papadimitriou [16] We also assume familiarity with elementary game theory, in particular with the notions of perfect information and perfect recall. The few game theoretic notions that we use are defined precisely in, e.g. 9] 2 Alternation and Randomized Players Chandra et al. [5] proved a fundamental result about the connection between games and complexity that serves as the starting point for most of the results surveyed in this paper. In the Alternating Polynomial Time computational model, there are two computationally unbounded players P 1 and P 0 and a polynomial time ....

.... P 0 ; V ) to be an Alternating Polynomial Time machine for the language L, it must have the property that, if x 2 L, V always outputs ACCEPT (i.e. P 1 has a winning strategy) and, if x 62 L, V always outputs REJECT (i.e. P 0 has a winning strategy) The fundamental result of Chandra et al. [5] is that Alternating Polynomial Time is equal to PSPACE: Languages that correspond to zero sum, perfect information, polynomial depth games are exactly those recognizable by Turing Machines that use polynomial space. The fundamental correspondence between PSPACE and perfect information games is ....

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A. Chandra, D. Kozen, and L. Stockmeyer, "Alternation," Journal of the Association for Computing Machinery, 28 (1981), pp. 114-133.

....is known to have the effect of turning time into space: It was shown in [Pa1] that the languages decidable by polynomial time bounded stochastic Turing machines comprise all of PSPACE. The lemma essentially says that, as is the case with the alternating machines (another much studied variant [CHK]) stochastic machines also turn space into time one exponential higher. We first show that any space bounded stochastic Turing machine can be simulated in exponential time. The simulation algorithm is simple: We visit the exponentially many configurations of the machine one by one, in decreasing ....

A. Chandra, D. Harel, D. Kozen "Alternation," J.ACM 28, pp. 114--133, 1981.

....definitions that do not depend on an arbitrary bound on nondeterminism. 4.1 An Alternating Machine Model Our first characterization comes from a modified alternating machine, which allows blocks of nondeterminism in a simple, natural way. The standard model of an alternating Turing machine [15] chooses (existentially or universally) one of two possible next states at each step. However, alternation can also be achieved by existentially or universally choosing a head position. This provides the desired (log n) bit blocks of nondeterminism. Using a head position to represent log(n) ....

....tape k s to an arbitrary nonblank square, and changes to the next state. This transition counts as a single step of the computation. Acceptance and rejection are determined for each possible configuration of a lengthalternating Turing machine in the same way as for standard alternating machines [15]. The indeterminism depth of a configuration in an LATM computation tree is the number of its ancestor configurations in the tree which are indeterministic. The indeterminism depth of an entire LATM computation tree is the maximum indeterminism depth of all its configurations. The time bound of an ....

A. Chandra, D. Kozen, and L. Stockmeyer, "Alternation," J. Assoc. Comput. Mach. 28 (1981) 114-133.

....is known to have the effect of turning time into space: It was shown in [Pa1] that the languages decidable by polynomial time bounded stochastic Turing machines comprise all of PSPACE. The lemma essentially says that, as is the case with the alternating machines (another much studied variant [CHK]) stochastic machines also turn space into time one exponential higher. We first show that any space bounded stochastic Turing machine can be simulated in exponential time. The simulation algorithm is simple: We visit the exponentially many configurations of the machine one by one, in decreasing ....

A. Chandra, D. Harel, D. Kozen "Alternation," J.ACM 28, pp. 114--133, 1981.

....artificial intelligence [Ro93] distributed computing [BL85] security and privacy [FGY93] and lower bounds [Ya77] Most importantly for structural complexity theorists, games have been used to characterize complexity classes in illuminating ways. For example, a classical result of Chandra et al. [CKS81] tells us that PSPACE is characterized by two person, perfect information games in which the length of a played game is polynomial in the length of the description of the initial position. In this paper, we present a general framework in which to study the connection between game theory and ....

....see all of the communication between the referee and the other player) or imperfect recall (does not remember all the information that he himself once knew) This framework allows us to 1. Unify and generalize the game theoretic aspects of ealier work on the complexity of interactive computation [BFL91, CKS81, FRS94, FST88, LFKN92, Sh91]. For example, game theory is a natural framework in which to state the equivalence of oracle proof systems and multiprover proof systems given in [FRS94] because oracles and teams of noncommunicating provers are both examples of players with imperfect recall. An oracle has imperfect recall ....

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A. Chandra, D. Kozen, and L. Stockmeyer, "Alternation," Journal of the ACM, 28 (1981), pp. 114--133.

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A.K. Chandra, D.C. Kozen and L.J. Stockmeyer, "Alternation", Journal of the ACM 28 (1981), 114-133.

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A. Chandra, D. Kozen, L. Stockmeyer. "Alternation ". Journal of the ACM, 28:114--133, 1981.

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A. Chandra, D. Kozen & L. Stockmeyer, "Alternation," J. ACM 28.1 (1981), 114 -- 133.

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A. Chandra, D. Kozen, and L. Stockmeyer, "Alternation," J. Assoc. Comput. Mach. 28 (1981) 114-133.

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A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer "Alternation", J. of ACM, 28(1): pages 114-133, 1981.

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A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer "Alternation", J. of ACM, 28(1): pages 114-133, 1981.

No context found.

A. K. Chandra and L. J. Stockmeyer, "Alternation", in Proceedings of 17th Annual IEEE Symposium on Foundations of Computer Science, pages 98-108, October 1976.

No context found.

A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer "Alternation", J. of ACM, 28(1): pages 114-133, 1981.

No context found.

A. K. Chandra and L. J. Stockmeyer, "Alternation", in Proceedings of 17th Annual IEEE Symposium on Foundations of Computer Science, pages 98-108, October 1976.

No context found.

A. K. Chandra, D. C. Kozen, L. J. Stockmeyer, "Alternation", JACM 28 (1981), 114-133.

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