On the Structure of Log-Space Probabilistic Complexity Classes

We investigate hierarchical properties and log-space reductions of languages recognized by log-space probabilistic Turing machines, Arthur-Merlin games, and Games against Nature with log-space probabilistic verifiers. For each log-space complexity class, we decompose it into a hierarchy based on corresponding multihead two-way finite automata and we (eventually) prove the separation of the hierarchy levels (even over one letter alphabet); furthermore, we show log-space reductions of each log-space complexity class to low levels of its corresponding hierarchy. We find probabilistic (and ``probabilistic+nondeterministic'') variants of Savitch's maze threading problem which are log-space complete for PL (and respectively P) and can be recognized by two-head one-way and one-way one-counter finite automata with probabilistic (probabilistic and nondeterministic) states

On the Structure of Log-Space Probabilistic Complexity Classes

We investigate hierarchical properties and log-space reductions of languages recognized by log-space probabilistic Turing machines, Arthur-Merlin games and Games against Nature with log-space probabilistic verifiers. For each log-space complexity class, we decompose it into a hierarchy based on corresponding multihead two-way finite automata and we (eventually) prove the separation of the hierarchy levels (even over one letter alphabet); furthermore, we show log-space reductions of each log-space complexity class to low levels of its corresponding hierarchy. We find probabilistic (and "probabilistic+nondeterministic") variants of Savitch's maze threading problem which are log-space complete for PL (and respectively P) and can be recognized by two-head one-way and one-way one-counter finite automata with probabilistic (probabilistic and nondeterministic) states. This research was supported by the National Science Foundation under Grant No. CDA 8822724. 1 Results We focus on classes d..

Connections among space-bounded and multihead probabilistic automata

We show that the heads of multihead unbounded-error or bounded-error or one-sided-error probabilistic finite automata are equivalent alternatives to the storage tapes of the corresponding probabilistic Turing machines (Theorem 1). These results parallel the classic ones concerning deterministic and nondeterministic automata. Several important properties of logarithmic-space (nondeterministic and probabilistic) Turing machines follow trivially (Observations 1--3) from the more refined versions that we prove in the setting of multihead finite automata (Theorems 2--4)

Decreasing the Bandwidth of a Transition Matrix

Adapting the competitions method of Freivalds to the setting of unbounded-error probabilistic computation, we prove that, for any ffl 2 (0; 1], Band-Mat-Inv(n ffl ) is log-space complete for the class of languages recognized by log-space unbounded-error probabilistic Turing machines (PL). This extends the result of Jung that Band-Mat-Inv(n) is log-space complete for PL, and may open new possibilities for space-efficient deterministic simulation of space-bounded probabilistic Turing machines. Key words: computational complexity, log-space probabilistic Turing machines, log-space complete problem. This research was supported by the National Science Foundation under Grant No. CDA 8822724. 1 Introduction For an arbitrary function 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 ..

Space-Efficient Deterministic Simulation of Probabilistic Automata

Given a description of a probabilistic automaton (one-head probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The question is interesting even in the case of one-head one-way probabilistic finite automata (PFA). We call (rational) stochastic languages (S^>_{rat}) the class of languages recognized by PFAs whose transition probabilities and cutpoints (i.e., recognition thresholds) are rational numbers. The class S^>_{rat} contains context-sensitive languages that are not context free, but on the other hand there are context-free languages not included in S^>_{rat}. Our main results are as follows: (1) The (proper) inclusion of S^>_{rat} in Dspace(log n), which is optimal (i.e., S^>_{rat} \not \subset Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972; Wang 1992] and Dspace(log n log log n) [Jung 1984]. (2) Probabilistic Turing machines with space bound f(n) \in 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 < \cdots < p_n = O(n^a) (where a is some constant) in O(log n) deterministic space

Closure Properties of Stochastic Languages

The study of probabilistic one-way finite-state automata has been sparsely spread over at least thirty-three years, and across media that are not all readily accessible in the West at the present time. In a uniform setting, we present properties of the classes of languages specifiable by such automata. We present results found in a wide and nonhomogeneous literature together with original results. Along with probabilistic automata, we also study a clean generalized version. The classes of languages accepted by such automata are defined in terms of numerical cutpoints and the functions that these automata compute. Our main results are some closure properties for the classes and functions. We give a technique for proving stochasticity and we apply it in the case of some well-known languages

Connections Among Space-Bounded and Multihead Probabilistic Automata

We show that the heads of multihead unbounded-error or bounded-error or one-sidederror probabilistic finite automata are equivalent alternatives to the storage tapes of the corresponding probabilistic Turing machines (Theorem 1). These results parallel the classic ones concerning deterministic and nondeterministic automata. Several important properties of logarithmic-space (nondeterministic and probabilistic) Turing machines follow trivially (Observations 1--3) from the more refined versions that we prove in the setting of multihead finite automata (Theorems 2--4). This research was supported by the National Science Foundation under Grant No. CDA 8822724. 1 Background It is known that the heads of multihead (deterministic and nondeterministic) finite automata represent a storage-alternative to the work-tape of (deterministic and nondeterministic) logarithmic-space Turing machines. These results were noted in early 70&apos;s [Har72]. In this paper we show that similar relations hold in th..

Properties of multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata with log-space constructible transition probabilities and describe a technique to simulate these automata by standard log-space probabilistic Turing machines. Next we represent log-space probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and show their (deterministic log-space) reducibility to the second levels of these hierarchies. We relate the number of heads of a multihead probabilistic finite automaton to the bandwidth of its configuration transition matrix for an input string; partially based on this relation we find an apparently easier log-space complete problem for PL (the class of languages recognized by log-space unbounded-error probabilistic Turing machines), and explore possibilities for a space-efficient deterministic simulation of probabilistic automata