Rūsiņš Freivalds

University of Latvia | LU · Faculty of Computing

Ehrenfeucht, Parikh and Rozenberg gave an interesting characterisation of the regular languages called the block pumping property. When requiring this property only with respect to members of the language but not with respect to nonmembers, one gets the notion of block pumpable languages. It is shown that these block pumpable are a more general con...

K. Iwama and R. Freivalds considered query algorithms where the black box contains a permutation. Since then several authors have compared quantum and deterministic query algorithms for permutations. It turns out that the case of n-permutations where n is an odd number is di�cult. There was no example of a permutation problem where quan- tization c...

B.A.Trakhtenbrot proved that in frequency computability (introduced by G. Rose) it is crucially important whether the frequency exceeds 1 2 . If it does then only recursive sets are frequency-computable. If the frequency does not exceed 1 2 then a continuum of sets is frequencycomputable. Similar results for �nite automata were proved by E.B. Kinbe...

We try to compare the complexity of deterministic, nonde- terministic, probabilistic and ultrametric finite automata for the same language. We do not claim to have final upper and lower bounds. Rather these results can be considered as experiments to find advantages of one type of automata versus another type.

New superimposed codes based on finite projective planes are proposed. These codes allow to construct efficient randomized query algorithms for some functions.

Frequency computation was introduced in [16]. Trakhtenbrot [17] proved the existence of a continuum of functions computable by frequency Turing machines with frequency 1/2 . In contrast, every function computable by a frequency Turing machine with frequency exceeding 1/2 is recursive. Essentially similar results for finite automata and other types...

Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. This paper introduces the notion of ultrametric query algorithms and shows an example of advantages of ultrametric query algorithms over deterministic, probabilistic and quantum query algorithms.

Probabilistic computations and frequency computations were invented for the same purpose, namely, to study possible advantages of technology involving random choices. Recently several authors have discovered close relationships of these generalizations of deterministic computations to computations taking advice. Various forms of computation taking...

We study active learning of classes of recursive functions by asking value queries about the target function f, where f is from the target class. That is, the query is a natural number x, and the answer to the query is f(x). The complexity measure in this paper is the worstcase number of queries asked. We prove that for some classes of recursive fu...

Elimination of potential hypotheses is a fundamental component of many learning processes. In order to understand the nature of elimination, herein we study the following model of learning recursive functions from examples. On any target function, the learning machine has to eliminate all, save one, possible hypotheses such that the missing one cor...

Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. This paper introduces the notion of ultrametric query algorithms and shows an example of advantages of ultrametric query algorithms over deterministic, probabilistic and quantum query algorithms.

We explore an alternative definition of query algorithms which proposes the use of p-adic numbers or their ordered tuples as amplitudes. The reader is introduced to the definition of p-adic numbers and their main properties and operations. Afterwards the reader is reminded of the notions of deterministic and randomized query algorithms, which are t...

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but complexity of probabilistic automata and complexity of ultrametric automata can differ very much.

Austinat, Diekert, Hertrampf, and Petersen [2] proved that every language L that is (m,n)-recognizable by a deterministic frequency automaton such that m > n/2 can be recognized by a deterministic finite automaton as well. First, the size of deterministic frequency automata and of deterministic finite automata recognizing the same language is compa...

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but complexity of probabilistic automata and complexity of ultrametric automata can differ very much.

We present examples where theorems on complexity of computation are proved using methods in algorithmic information theory. The first example is a non-effective construction of a language for which the size of any deterministic finite automaton exceeds the size of a probabilistic finite automaton with a bounded error exponentially. The second examp...

We present examples where theorems on complexity of computation are proved using methods in algorithmic information theory. The first example is a non-effective construction of a language for which the size of any deterministic finite automaton exceeds the size of a probabilistic finite automaton with a bounded error exponentially. The second examp...

Finite automata with random bits written on a separate 2-way readable tape can recognize languages not recognizable by probabilis-tic finite automata. This shows that repeated reading of random bits by finite automata can have big advantages over one-time reading of random bits.

In classical computation, a “write-only memory” (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can...

In this paper, we address the question whether a probabilistic finitestate automaton (pfa) could recognize a language not recognizable by a deterministic finite-state automaton in polynomial time. We show that there is such a language in a hyperbolic plane and prove that a 5-way error-bounded pfa can recognize it in polynomial time of the number of...

We introduce a notion of ultrametric �nite automata using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but the descriptional power of probabilistic automata and ultrametric automata can di�er very much.

K.Iwama and R.Freivalds [11] considered query algorithms where the black box contains a permutation. Since then several authors have compared quantum and deterministic query algorithms for permu- tations. It turns out that the case of n-permutations where n is an odd number is di�cult. There was no example of permutation problem where quantization...

String theory in physics and molecular biology use p-adic numbers as a tool to describe properties of microworld in a more adequate way. We consider a notion of automata and Turing machines using p-adic numbers to describe random branching of the process of computation.We prove that Turing machines of this type can have advantages in reversal compl...

Frequency computation was introduced in 1960-ies in an attempt to show that many properties of probabilistic algorithms can be simulated by purely deterministic computational devices. The aim of this paper is to show that for automata working on in�nite input tapes the situation is very much di�erent. Technically our main result shows that there is...

Prediction of functions is one of processes considered in in-ductive inference. There is a "black box" with a given total function f in it. The result of the inductive inference machine F (< f (0), f (1), · · · , f (n) >) is expected to be f (n + 1). Deterministic and probabilistic prediction of functions has been widely studied. Frequency computat...

A transducer is a finite-state automaton with an input and an output. We compare possibilities of nondeterministic and probabilis-tic transducers, and prove several theorems which establish an infinite hierarchy of relations computed by these transducers. We consider only left-total relations (where for each input value there is exactly one al-lowe...

Learning to write is a process where preverbal ideas (thoughts) are transformed into a written form. Such written forms produced vary from words to sentences to higher forms of discourse such as essays, reports, researches, reviews, poems, stories, dialogues, etc. These written forms that are generated, formed, and activated from thoughts are then...