H. Jung. On probabilistic tape complexity and fast circuits for matrix inversion problems. 11th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, 172:281--291, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of M (x) by START . With these definitions, it is clear that the probability that M (x) accepts is just the entry [START,ACC] of the hitting probability matrix. As explained earlier, this computation boils down to the matrix computation Q # = lim ##1 (I #Q) 1 for some substochastic Q. In [21], it is shown that if we choose # 2 Cs for some C 0, then Q # [i, j] T if and only if (I #Q) 1 [i, j] T . Thus we can reduce the problem Q # [i, j] T to the problem (I 1 [i, j] T for some strictly substochastic matrix R. For the halting classes, we can reduce them to the ....

H. Jung. On probabilistic tape complexity and fast circuits for matrix inversion problems. 11th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, 172:281--291, 1984.

....root, as well as exact computation of the following: 1. integer and polynomial quotient and remainder, 2. interpolation of rational polynomials, 3. banded matrix inverse, and 4. triangular Toeplitz matrix inverse. These problems can all be efficiently reduced to integer products; also see [3, 4, 12, 18]. Theorems 3.1 and 3.2 yield the characterization: Corollary 3.4. For S(n) P (n) n O(1) c1 Z P (n) S(n) c ; D(n) c1 Th(S(n) c ; D(n) For example, for S(n) n O(1) D(n) O(1) P (n) n O(1) we get Z P (n) n O(1) 1) Th(n O(1) 1) In other words, the ....

H. Jung, On probabilistic tape complexity and fast circuits for matrix inversion problems, in Proceedings of the 11th Colloquium on Automata, Languages, and Programming, 1984, pp. 281--291. Springer-Verlag LNCS vol. 172.

....of heads of a multihead probabilistic finite automaton and the bandwidth of its configuration transition matrix for an input string. Partially inspired by this relation and using the competition method of Freivalds [Fr81] in the setting of unbounded error computation, we improve a result of Jung [Ju84] and find an apparently easier log space complete problem for PL. In the final, we discuss possibilities for deterministic simulation of probabilistic automata in small space. 2 Preliminaries In this section, we define the concepts and notations used in the paper. A number b 2 [0; 1] is ....

....by the element from position (1; m) of the inverse of a m by m matrix that has bandwidth O(B) and can be obtained from M in log space. Comparing this element with 1 2 (i.e. deciding whether A accepts x or not) can be done in O(log n(log B log log n) space by a deterministic Turing machine [Ju84], Ma94] 2 This lemma gives us a motivation to find tight upper bounds for the bandwidth of configuration transition matrices associated with computations of probabilistic automata. Finding a solution to this problem is the topic of Lemmas 7 8 and Theorem 7. 5 Hartmanis [Ha72] used a similar ....

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Jung, H. On probabilistic tape complexity and fast circuits for matrix inversion problems. Proceedings, ICALP 1984, LNCS 172, pp. 281-291.

....if and only if the number of accepting paths of N on x is greater than its number of rejecting paths. In apparent contrast to P complete sets, sets in PL are decidable using very fast parallel computations [Jung 1985] The set of integer matrices with determinant greater than 0 is complete for PL [Jung 1984]. See also [Allender and Ogihara 1996] Probabilistic polynomial time, PP, is defined analogously. A classic PP complete problem is Majsat: given a Boolean formula, do the majority of assignments satisfy it For polynomially space bounded computations, PSPACE equals probabilistic PSPACE, and ....

Jung, H. 1984. On probabilistic tape complexity and fast circuits for matrix inversion problems. In Proceedings 11th ICALP (1984), pp. 281--291. Lecture Notes in Computer Science #172.

....are context free languages not included in S rat . Our main results are as follows: ffl The (proper) inclusion of S rat in Dspace(log n) which is optimal (i.e. S rat 6ae Dspace(o(log n) The previous upper bounds were Dspace(n) Dieu 1972] Wang 1992] and Dspace(log n log log n) [Jung 1984]. ffl Probabilistic Turing machines with space bound f(n) 2 O(log n) can be deterministically simulated in space O(min(c f(n) log n; log n(f(n) log log n) where c is a constant depending on the simulated probabilistic Turing machine. The best previously known simulation required space ....

....machines with space bound f(n) 2 O(log n) can be deterministically simulated in space O(min(c f(n) log n; log n(f(n) log log n) where c is a constant depending on the simulated probabilistic Turing machine. The best previously known simulation required space O(log n(f(n) log log n) [Jung 1984]. Of independent interest is our technique to compare numbers given in terms of their values modulo a sequence of primes, p 1 p 2 Delta Delta Delta p n = O(n a ) where a is some constant) in O(log n) deterministic space. Key words: stochastic language, probabilistic Turing machine, ....

[Article contains additional citation context not shown here]

Jung, H. On probabilistic tape complexity and fast circuits for matrix inversion problems. Proc. ICALP 1984, LNCS 172, pp. 281-291.

....if and only if the number of accepting paths of N on x is greater than its number of rejecting paths. In apparent contrast to P complete sets, sets in PL are decidable using very fast parallel computations [Jung 1985] The set of integer matrices with determinant greater than 0 is complete for PL [Jung 1984]. See also [Allender and Ogihara 1996] Probabilistic polynomial time, PP, is de ned analogously. A classic PP complete problem is Majsat: given a Boolean formula, do the majority of assignments satisfy it For polynomially space bounded computations, PSPACE equals probabilistic PSPACE, and ....

Jung, H. 1984. On probabilistic tape complexity and fast circuits for matrix inversion problems. In Proceedings 11th ICALP (1984), pp. 281-291. Lecture Notes in Computer Science #172.

....f , such that 8n 2 N , f(n) 2 (0; n] Band Mat Inv(f) denotes the following problem: for each positive integer n and a diagonally dominant n by n f(n) banded matrix A whose elements are n bit rational numbers, compare with 1 2 the element from the position (1; n) of the inverse of A. 1 In [Ju84], a particular form of this problem, Band Mat Inv(n) is denoted by MATIN. It is known that Band Mat Inv(n) is log space complete for PL [Ju84] The question we ask is how far can we reduce f and still have Band Mat Inv(f) log space complete for PL The answer to this question could have nice ....

....n by n f(n) banded matrix A whose elements are n bit rational numbers, compare with 1 2 the element from the position (1; n) of the inverse of A. 1 In [Ju84] a particular form of this problem, Band Mat Inv(n) is denoted by MATIN. It is known that Band Mat Inv(n) is log space complete for PL [Ju84]. The question we ask is how far can we reduce f and still have Band Mat Inv(f) log space complete for PL The answer to this question could have nice implications for deterministic simulations of probabilistic automata, when combined with the observations below. The currently most space efficient ....

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Jung, H. On probabilistic tape complexity and fast circuits for matrix inversion problems. Proceedings, ICALP 1984, Lecture Notes on Computer Science 172, pp. 281-291.

.... for C= L, and a variety of other problems regarding computation of the rank and determining if a system of linear equations is feasible are complete for L C=L [ABO96] Some other problems in linear algebra and problems involving Markov decision processes were shown to be complete for PL in [J84, MGA97]. One important problem whose complexity remains unresolved is the perfect matching problem. No deterministic NC algorithm is known for matching at all, but the probabilistic NC algorithm of [MVV87] can be used to show that, in the nonuniform setting, perfect matching is in coC=L and in PhiL. ....

H. Jung. On probabilistic tape complexity and fast circuits for matrix inversion problems. In Proc. ICALP '84, Lecture Notes in Computer Science 172, pp. 281--291, 1984.

....DSpace(log ) is the best upper bound for the class BPSpace(CON ) An interesting question is if the same holds for any sublogarithmic space bound. The best upper bound known for languages recognized by probabilistic machines in space S seems to be DSpace(min(2 S log ; log n(S llog ) see [Ju84] and [Ma94] However, the counting abilities of probabilistic machines are limited as has been shown recently by Freivalds and Karpinski using the language PALINDROME of all strings that are palindromes. Theorem 18 [FrKa94] PALINDROME 62 BPSpace(o(log ) What happens in the sublogarithmic ....

H. Jung, On probabilistic tape complexity and fast circuits for matrix inversion problems, Proc. 11th ICALP, 1984, 281-291.

....made to the proof of Theorem 6 is that the matrix A needs to record the transition matrix of machine M on input x with oracle S. This can clearly be computed in logspace, relative to oracle S. 3 The argument presented thus far (along with the Appendix) is closely patterned after an argument in [Ju84]. The paper [Ju84] presents a complete set for PL; the PL=PL(poly) theorem was not presented until [Ju85] 7 4 Closure Properties of GapL and PL One way of interpreting the results of the previous section is that PL is the class of languages that are reducible to the high order bit of ....

....of Theorem 6 is that the matrix A needs to record the transition matrix of machine M on input x with oracle S. This can clearly be computed in logspace, relative to oracle S. 3 The argument presented thus far (along with the Appendix) is closely patterned after an argument in [Ju84] The paper [Ju84] presents a complete set for PL; the PL=PL(poly) theorem was not presented until [Ju85] 7 4 Closure Properties of GapL and PL One way of interpreting the results of the previous section is that PL is the class of languages that are reducible to the high order bit of a #L function. In ....

H. Jung, On probabilistic tape complexity and fast circuits for matrix inversion problems, Proc. 11th ICALP, Lecture Notes in Computer Science 172, 1984, pp. 281--291. -- 18 --

....from Theorem 1 and Proposition 3. It is clear that PL(poly) is closed under complement; the usual proof that PP is closed under complement carries over to this case. Thus Corollary 4 and Theorem 1 show immediately that the following sets are complete for PL(poly) under log m reductions [26]: ffl The set of integer matrices with determinant 0. ffl The set of integer matrices with determinant 0. ffl f(A; m) DET(A) mg ffl f(A; m) DET(A) mg ffl f(A; B) DET(A) DET(B)g ffl f(A; B) DET(A) DET(B)g 3 A simple proof of Jung s Theorem In this section we define ....

H. Jung, On probabilistic tape complexity and fast circuits for matrix inversion problems, Proc. 11th ICALP, Lecture Notes in Computer Science 172, 1984, pp. 281--291.

....M x . For this operation we use the technique from [Mo79] where the graph accessibility problem restricted to graphs with bandwidth b(m) is solved deterministically in O(log m log b(m) space. We need this step to make sure that the matrix C, which will be defined later, is well defined. Jung [Ju84] used a logspace reduction to obtain a matrix close to C and well defined. However for our goals, the technique mentioned above is strong enough. Let P be the matrix obtained from M x by the previous operation. It is clear that we can compute every element of P (when needed) in O(log 2 m) ....

Jung, H. On probabilistic tape complexity and fast circuits for matrix inversion problems. Proceedings, ICALP 1984, LNCS 172, pp 281-291.

....[Ma94] If the transition probabilities are rational, it is possible to reduce such simulation to the comparison of appropriately computed large integers. We save space by computing and working with these integers modulo relatively small primes. Without our Theorem 1 below, Jung s related approach [Ju84] requires asymptotically more space, by a factor of Theta(log S k ) Results For perspective, let us first note that space efficient conversion in the other direction, from the binary system to the residue system, is straightforward. Proposition 1. From the binary representation of a number ....

H. Jung, On probabilistic tape complexity and fast circuits for matrix inversion problems (extended abstract), Lecture Notes in Computer Science (Automata, Languages and Programming, 11th Colloquium, Antwerp, Belgium, July 16--20, 1984) (J. Paredaens, ed.), vol. 172, Springer-Verlag, New York, 1984, pp. 281--291.

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Jung, H. On probabilistic tape complexity and fast circuits for matrix inversion problems. Proceedings, ICALP 1984, LNCS 172, pp. 281-291.

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H. Jung. On probabilistic tape complexity and fast circuits for matrix inversion problems. In Proceedings of the 11th International Colloquium on Automata, Languages and Programming, volume 172 of Lecture Notes in Computer Science, pages 281--291, 1984.

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