It is proven that contrary to the deterministic and nondeterministic cases there is no recursive lower bound for Monte Carlo space constructible functions. The existence of small constructible bounds enables the separation of Monte Carlo space f(n) from probabilistic space g(n)(f(n)=o(g(n))) and—together with a new halting lemma for probabilistic machines with small space bounds—a hierarchy of “provable” Monte Carlo space classes with small bounds. We are also able to separate O(log log n) terminating Monte Carlo space in the sense of (Aleliunas et al., in “Proceedings, 20th IEEE Found. of Comput. Sci. 1979”, pp. 218–223; Welsh, Discrete Appl. Math. 5 (1983), 133–145) from NSPACE(log logn).

On the Monte Carlo space constructible functions and separation results for probabilistic complexity classes

Rather often difficult (and sometimes even undecidable) problems become easily decidable for tally languages, i.e. for languages in a single-letter alphabet. For instance, the class of languages recognizable by 1-way nondeterministic pushdown automata equals the class of the context-free languages, but the class of the tally languages recognizable by 1-way nondeterministic pushdown automata, contains only regular languages [LP81]. We prove that languages over one-letter alphabet accepted by randomized one-way 1-tape Monte Carlo pushdown automata are regular. However Monte Carlo pushdown automata can be much more concise than deterministic 1-way finite state automata.

Tally Languages Accepted by Monte Carlo Pushdown Automata

We study the most important probabilistic computation modes for pushdown automata. First we show that deterministic pushdown automata (pda) are weaker than Las Vegas pda, which in turn are weaker than one-sided-error pda. Next one-sided-error pda are shown to be weaker than (nondeterministic) pda. Finally bounded-error two-sided error pda and nondeterministic pda are incomparable. To show the limited power of bounded-error two-sided pda we apply communication arguments; in particular we introduce a non-standard model of communication which we analyze with the help of the discrepancy method. The power of randomization for pda is considerable, since we construct languages which are not deterministic context-free (resp. not context-free) but are recognizable with even arbitrarily small error by one-sided-error (resp. bounded-error) pda. On the other hand we show that, in contrast to many other fundamental models of computing, error probabilities can in general not be decreased arbitrarily: we construct languages which are recognizable by one-sided-error pda with error probability 12, but not by one-sided-error pushdown automata with error probability p<12. A similar result, with error probability 13, holds for bounded-error two-sided error pda.

http://thi.informatik.uni-frankfurt.de/~georg/journal3.pdf

On probabilistic pushdown automata

Postselection for quantum computing devices was introduced by S. Aaronson[2] as an excitingly efficient tool to solve long standing problems of computational complexity related to classical computing devices only. This was a surprising usage of notions of quantum computation. We introduce Aaronson's type postselection in quantum finite automata.

There are several nonequivalent definitions of quantumfinite automata. Nearly all of them recognize only regular languages but not all regular languages. We prove that PALINDROMES can be recognized by MM-quantum finite automata with postselection. At first we prove by a direct construction that the complement of this language can be recognized this way. This result distinguishes quantum automata from probabilistic automata because probabilistic finite automata with non-isolated cut-point 0 can recognize only regular languages but PALINDROMES is not a regular language.

https://www.lu.lv/fileadmin/user_upload/lu_portal/projekti/datorzinatnes_pielietojumi/seminari/3_seminars/publikacijas/publ/RMF_3

Postselection finite quantum automata

Although randomization is now a standard tool for the design of efficient algorithms or for building simpler systems, we are far from fully understanding the power of randomized computing. Hence it is advisable to study randomization for restricted models of computation.We followthis approach by investigating the power of randomization for pushdown automata.Our main results are as follows. First we show that deterministic pushdown automata are weaker than Las Vegas pushdown automata, which in turn are weaker than one-sided-error pushdown automata. Finally one-sided-error pushdown automata are weaker than (nondeterministic) pushdown automata.In contrast to many other fundamental models of computing there are no known methods of decreasing error probabilities.We show that such methods do not exist by constructing languages which are recognizable by one-sided-error pushdown automata with error probability 1/2, but not by one-sided-error pushdown automata with error probability p < 1/2. On the other hand we construct languages which are not deterministic context-free (resp. not context-free) but are recognizable with arbitrarily small error by one-sided-error (resp. bounded-error) pushdown automata.

Rūsinś Freivalds

Papers

Projections of languages recognizable by probabilistic and alternating finite multitape automata