Yuri V. Matiyasevich

Window subsequence problems for compressed texts

Given two strings (a text t of length n and a pattern p) and a natural number w, window subsequence problems consist in deciding whether p occurs as a subsequence of t and/or finding the number of size (at most) w windows of text t which contain pattern p as a subsequence, i.e. the letters of pattern p occur in the text window, in the same order as in p, but not necessarily consecutively (they may be interleaved with other letters). We are searching for subsequences in a text which is compressed using Lempel-Ziv-like compression algorithms, without decompressing the text, and we would like our algorithms to be almost optimal, in the sense that they run in time O(m) where m is the size of the compressed text. The pattern is uncompressed (because the compression algorithms are evolutive: various occurrences of a same pattern look different in the text).

Some Decision Problems for Traces

The notion of a word, considered as an element of a free monoid, has been long ago generalized to the notion of a trace, an element of a partially commutative monoId. Traces turned out to be useful tool for studying concurrency.

Simultaneous rigid E-unification and related algorithmic problems

The notion of simultaneous rigid E-unification was introduced in 1987 in the area of automated theorem proving with equality in sequent-based methods, for example the connection method or the tableau method. Recently, simultaneous rigid E-unification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid E-unification and show the connections between this notion and some algorithmic problems of logic and computer science.

Solving word equations modulo partial commutations

It is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanins theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz.

Decision problems for semi-Thue systems with a few rules

For several decision problems about semi-Thue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the U-Termination Problem, the Accessibility Problem and the Common-Descendant Problem are undecidable for 3 rules semi-Thue systems. As a corollary we obtain the undecidability of the Post-Correspondence Problem for 7 pairs of words.

Solving Trace Equations Using Lexicographical Normal Forms

Very recently, the second author showed that the question whether an equation over a trace monoid has a solution or not is decidable [11,12]. In the original proof this question is reduced to the solvability of word equations with constraints, by induction on the size of the commutation relation. In the present paper we give another proof of this result using lexicographical normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints, using a new result on lexicographical normal forms.

Word Problem for Thue Systems with a Few Relations

The history of investigations on the word problem for Thue systems is presented with the emphasis on undecidable systems with a few relations. The best known result, a Thue system with only three relations and undecidable word problem, is presented with details. Bibl. 43 items.

Approximation of Riemann’s Zeta Function by Finite Dirichlet Series: A Multiprecision Numerical Approach

The finite Dirichlet series of the title are defined by the condition that they vanish at as many initial zeros of the zeta function as possible. It turns out that such series can produce extremely good approximations to the values of Riemann’s zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large-scale high-precision numerical experiments. So far, we have found no theoretical explanation for the observed phenomena.

Some arithmetical restatements of the Four Color Conjecture

The Four Colour Conjecture is reformulated as a statement about non-divisibility of certain binomial coefficients. This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in finding closed forms for binomial summations.

Hilbert's tenth problem and paradigms of computation

This is a survey of a century long history of interplay between Hilberts tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.