Mika Hirvensalo

Quantum Automata with Open Time Evolution

In this paper, a model for finite automaton with an open quantum evolution is introduced, and its basic properties are studied. It is shown that the (fuzzy) languages accepted by open evolution quantum automata obey various closure properties. More importantly, it is shown that major other models of finite automata, including probabilistic, measure once quantum, measure many quantum, and Latvian quantum automata can be simulated by the open quantum evolution automata without increasing the number of the states.

Undecidability Bounds for Integer Matrices using Claus Instances

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Binary (generalized) post correspondence problem

We give a new proof for the decidability of the binary Post Correspondence Problem (PCP) originally proved in 1982 by Ehrenfeucht, Karhumki and Rozenberg. Our proof is complete and somewhat shorter than the original proof although we use the same basic. Copyright 2002 Elsevier Science B.V. All rights reserved.

Positivity of second order linear recurrent sequences

We give a decision method for the Positivity Problem for second order recurrent sequences: it is decidable whether or not a recurrent sequence defined by un = aun-1 + bun-2 has only nonnegative terms.

Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages

We give constructions of small probabilistic and MO-type quantum automata that have undecidable emptiness problem for the cut-point languages.

Marked PCP is decidable

We show that the marked version of the Post Correspondence Problem, where the words on a list are required to differ in the first letter, is decidable. On the other hand, we prove that the PCP remains undecidable if we only require the words to differ in the first two letters. Thus we locate the decidability/undecidability-boundary between marked and 2-marked PCP.

GENERALIZED POST CORRESPONDENCE PROBLEM FOR MARKED MORPHISMS

We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphisms. This result gives as a corollary a shorter proof for the decidability of the binary PCP, proved in 1982 by Ehrenfeucht, Karhumaki and Rozenberg.

Improved matrix pair undecidability results

We improve the undecidability bounds for problems involving two integer matrices by showing that Scalar Reachability, Zero in the Right Upper Corner, Vector Reachability, and Zero in the Left Upper Corner are undecidable for dimensions of 9, 10, 11, and 13, respectively. Problems Scalar Reachability, Zero in the Right Upper Corner, and Vector Reachability were previously known undecidable for dimensions 18, 18, and 16, respectively.

Mortality for 2 ×2 matrices is NP-Hard

We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability of the mortality problem in two dimensions remains a long standing open problem although in dimension three is known to be undecidable as was shown by Paterson in 1970. We also show a lower bound on the minimum length solution to the Mortality Problem, which is exponential in the number of matrices of the generator set and the maximal element of the matrices.

On probabilistic and quantum reaction systems

The notion of reaction systems was introduced by Ehrenfeucht and Rozenberg (2004, 2007) [5,6]. In this article, we discuss the general properties of probabilistic reaction systems and introduce their quantum extension.