The termination problem for semi-Thue systems asks whether all derivations for a given word in a given semi-Thue system are finite, i.e., all derivations terminate after finite number of steps. This problem is known to be undecidable, there is a standard reduction of the halting problem of the Turi ng machines into termination problem; moreover, one can fix a semi-Thue system and still have the undecidability. In 1996 Sénizergues and the second author gave a construction for a 3-rule semi-Thue system with undecidable termination problem. However, in their construction the words of one of the rules are very long. Using some ideas of Tseijtin we give a construction for a semi-Thue system with low number of short rules having undecidable termination problem. Namely, we construct a semi-Thue system with 24 rules over 8 letter alphabet with rule words of length at most 5, and the termination problem for this semi-Thue system is undecidable. Moreover, this system is universal, that is, it can simulate any semi-Thue system.

Institute of St. Petersburg.The second event took place in Turku on September 25-28

Given two strings, text t of length n, and pattern p = p1 : : : pk of length k, and given a natural number w, the subsequence matching problem consists in finding the number of size w windows of text t which contain pattern p as a subsequence, i.e. the letters p1 ; : : : ; pk occur in the window, in the same order as in p, but not necessarily consecutively (they may be interleaved with other letters). Subsequence matching is used for finding frequent patterns and association rules in databases. We generalize the Knuth-Morris-Pratt (KMP) pattern matching algorithm; we define a non-conventional kind of RAM, the MP--RAMs which model more closely the microprocessor operations; we design an O(n) on-line algorithm for solving the subsequence matching problem on MP--RAMs. Keywords: Subsequence matching, algorithms, frequent patterns, episode matching, datamining. 1 Introduction We address the following problem. Given a text t of length n and a pattern p = p 1 \\Delta \\Delta \\Delta p k of l...

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.

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.

Abstract Let be a first-order structure; we denote by DEF( ) the set of all first-order definable relations and functions within . Let π be any one-to-one function from ℕ into the set of prime integers. Let ∣ and • be respectively the divisibility relation and multiplication as function. We show that the sets DEF(ℕ, π, ∣) and DEF(ℕ, π, •) are equal. However there exists function π such that the set DEF(ℕ, +, ∣), or, equivalently, DEF(ℕ, π, •) is not equal to DEF(ℕ, +, •). Nevertheless, in all cases there is an {π, •}-definable and hence also {π, |}-definable structure over π which is isomorphic to 〈ℕ, +, •〉. Hence theories TH(ℕ, π, ∣) and TH(ℕ, π, •) are undecidable. The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice. The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the form where a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns. Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.