K. R. Reischuk. Einfuhrung in die Komplexitatstheorie. Teubner, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....p(Q) c and t(Q) O(log k ) where c is a constant and k a natural number. Furthermore, a is said to be efficient wrt A iff T (Q) t(Q) Theta p(Q) 1=log k and t(Q) O(log k ) where k is again a natural number. Throughout the paper we will assume the PRAM computational model (see e.g. [11, 12]) As shown in [5] unsatisfiability in AL is in P . In this paper we are interested in the question of whether unsatisfiability in AL is also in NC and, if this is the case, whether there is an efficient or optimal parallel algorithm checking unsatisfiability. These problems are tackled in the ....

K. R. Reischuk. Einfuhrung in die Komplexit atstheorie. B.G. Teubner, Stuttgart, 1990.

....in our formal model. In this article we are interested in relating the computational power of various kinds of SNN s to other computational models. We employ for that purpose the common notion of a real time simulation from computational complexity theory (see e.g. Leong 81] Paul 84] Reischuk 90] Maass 96] One says that M 0 simulates M in real time if M 0 can simulate each step in a computation of M with a fixed number of computation steps (i.e. the simulation of later computation steps of M does not require more steps of M 0 than the simulation of the first ones) It is ....

K. R. Reischuk. (1990) Einfuhrung in die Komplexitatstheorie. Teubner (Stuttgart).

....unary minimum, i.e. P 2:5 = PM 2:5 . P = PSPACE iff P 2:5 is closed under limited unary iteration, i.e. P 2:5 = PO 2:5 . PH = PSPACE iff PM 2:5 is closed under limited unary iteration, i.e. PM 2:5 = PO 2:5 . Proof: If P = NP then the polynomial hierachy collapses: P = PH (see [9]) and then P 2:5 = PM 2:5 . Vice versa, if the latter identity holds then by the theorems 24, 27, and 28 we have P = PH and hence P = NP . Similarly, P = PSPACE iff P 2:5 = PO 2:5 and PH = PSPACE iff PM 2:5 = PO 2:5 . Clearly, from F 1 = F 2 follows (F 1 ) F 2 ) So it ....

K.R. Reischuk. Einfuhrung in die Komplexitatstheorie. B.G. Teubner Stuttgart, 1990.

.... Delta s 2 (n) Delta e t(n) e 2 Deltat(n) Deltalog p 2 Delta Deltac Deltas(n) 7) In this equation s(n) is much smaller than t(n) Thus the bound of Eq. 7 is in the complexity class exponential polynomial ExP(f(n) with ExP(f(n) ff(n)jf(n) e p(n) g, and p(n) is a polynomial (Reischuk, 1990, Garey and Johnson, 1979) This means that the magnitude of the function on the right side of Eq. 7 is upper bounded by a constant times f(n) for all large n. Empirical Case jVT j jVS j. In this case we suppose that jVT j is much smaller than jVS j. Let fi = 1 2 Delta jP1 j Delta 1 jV T ....

Reischuk, K., 1990. Einf uhrung in die Komplexit atstheorie.

....one non single edge, and the variables represents the non single edges of the hypergraph. Since these variables are required to belong to a set with two elements, e.g. f reduce to single edge with white vertices , reduce to single edge with black vertices g, the problem is NP complete (see [10]) Surprisingly, for a class of hypergraphs with at most two non single edges in every cycle a sequence of operations can be found efficiently, and hence, an ordering of indices to solve the scheduling problem without backtracking if such an ordering exists. Theorem 3.7 Let G be the hypergraph of ....

....the value 1 is stated for black vertices. For two non simple edges e i and e j in a cycle, accordingly to white and black vertices, one of the following inequalities is derived: v i v j 1 or Gamma v i Gamma v j Gamma1: This variant of integer programming is solved in polynomial time (see [10]) Hypergraphs with this property are called 2 non simple edges hypergraphs. Consider the hypergraph in Figure 1, there are two cycles, i.e. v 1 C 5 v 2 C 6 v 3 C 4 and v 5 C 1 v 8 C 2 v 7 C 6 v 6 C 5 . The both cycles contains two non single edges C 5 and C 6 . Since there is a solution to the ....

K. R. Reischuk, Einfuhrung in die Komplexitatstheorie, B. G. Teubner, Stuttgart, Germany, 1990.

....first operates as Z 1 . Whenever Z 1 terminates, Z 2 is called where the input of Z 2 is the output of Z 1 . We also use the notation Z n 1 for Z 1 Z 1 : Z 1 z n times . 3 Probabilistic circuits 3 3 Probabilistic circuits A probabilistic circuit is a deterministic circuit (see [4], pp. 73) with a partition In = In D [ In P , In D In P = of the input nodes. The input nodes in In D (the deterministic input nodes) receive the input of the computation. The nodes in In P (the probabilistic input nodes) are assigned uniformly at random 0 or 1. The number of all nodes but ....

.... languages L f0; 1g the following holds: if L is decided by a deterministic Turing machine M = K; Sigma; ffi; s) in time T : IN IN, then there is a family fC n g n2IN of deterministic circuits which decides L and satisfies size(C n ) jKj j Sigmaj) c T (n) log T (n) Proof: See [4], pp. 84 91. Lemma 2 There are c; d 2 IN such that for all homogenous polynomial coin tossing machines M = K; Sigma; ffi; s) p) with output length : IN IN and running time bounded by T 2 IN[X ] there is a polynomial family fC n g n2IN of probabilistic circuits such that f Pi C jxj (x; ....

K. R. Reischuk. Einfuhrung in die Komplexitatstheorie. Teubner, 1990.

.... size circuits of depth O(d(n) In [12] Ruzzo proved (in a more general setting) the following result: Proposition 14 For a(n) log n holds : L uniform NC 1 DLOGT IME uniform Size Depth(n O(1) a(n) Delta log n) Ruzzo s simulation result can be related to the following result (see [11]) Proposition 15 Let AT ISP(t(n) s(n) denote the class of languages recognized by alternating Turing machines that are t(n) time bounded and s(n) space bounded. If s(n) Omega Gamma 52 n) is space constructible then for a(n) log n holds DSPA CE(n) AT ISP(s(n) Delta a(n) s(n) It is ....

K. R. Reischuk. Einfuhrung in die Komplexitatstheorie. B. G. Teubner, Stuttgart, 1990.

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K. R. Reischuk. Einfuhrung in die Komplexitatstheorie. Teubner, 1990.

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R. Reischuk. Einfuhrung in die Komplexitatstheorie. Teubner, Stuttgart, Germany, 1990.

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