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Vesa Halava

Publications and source records attributed to Vesa Halava.

9 recordsLinked to original sources
arXiv2026

On the Complexity of the Bi-infinite Post Correspondence Problem

In the bi-infinite Post Correspondence Problem ($\Z$PCP), it is asked whether the same bi-infinite word can be constructed correspondingly from a given finite set of pairs of words. In this article, we study its complexity with respect to the arithmetical hierarchy and prove that it is in $\Si^0_2 \setminus (\Pi^0_1 \cup \Si^0_1)$ and, therefore, at the level 2 of the arithmetical hierarchy. For the proof, we present a sequence of reductions starting from the nonhalting of the Turing machine all the way to $\Z$PCP via infinite PCP, an $s$-shift infinite PCP and $s$-shift $\Z$PCP for all natural numbers $s$. In the process, we prove that the infinite PCP is undecidable for injective morphisms, and that the infinite injective PCP, $s$-shift infinite PCP, $s$-shift $\Z$PCP and the non-termination problem for (deterministic and reversible) semi-Thue systems are all $\Pi^0_1$-complete.

cs.FL↗
arXiv2023

Decision Problems on Copying and Shuffling

We study decision problems of the form: given a regular or linear context-free language $L$, is there a word of a given fixed form in $L$, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations.

cs.FL↗
arXiv2022

A recursive function coding number theoretic functions

We show that there exists a fixed recursive function $e$ such that for all functions $h\colon \mathbb{N}\to \mathbb{N}$, there exists an injective function $c_h\colon \mathbb{N}\to \mathbb{N}$ such that $c_h(h(n))=e(c_h(n))$, i.e., $h=c_h^{-1}ec_h$.

cs.DM↗
arXiv2021

On Bi-infinite and Conjugate Post Correspondence Problems

We study two modifications of the Post Correspondence Problem (PCP), namely 1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and 2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the bi-infinite PCP we show that it is in the class $\Sigma_2^0$ of the arithmetical hierarchy and for the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems.

cs.DM↗
arXiv2015

Weighted automata on infinite words in the context of Attacker-Defender games

We consider infinite-state Attacker-Defender games with reachability objectives. The results of the paper are twofold. Firstly we prove a new language-theoretic result for weighted automata on infinite words and show its encoding into the framework of Attacker-Defender games. Secondly we use this novel concept to prove undecidability for checking existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games.

cs.FL↗
arXiv2014

Another proof of undecidability for the correspondence decision problem - Had I been Emil Post

In 1946 Emil Leon Post (Bulletin of Amer. Math. Soc. 52 (1946), 264 - 268) defined a famous correspondence decision problem which is nowadays called the Post Correspondence Problem, and he proved that the problem is undecidable. In this article we follow the steps of Post, and give another, simpler and more straightforward proof of the undecidability of the problem using the same source of reduction as Post original did, namely, the Post Normal Systems.

cs.LO↗
arXiv2014

Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More

We study the decidability of three well-known problems related to integer matrix multiplication: Mortality (M), Zero in the Left-Upper Corner (Z), and Zero in the Right-Upper Corner (R). Let d and k be positive integers. Define M(k, d x d) as the following special case of the Mortality problem: given a set X of d -by-d integer matrices such that the cardinality of X is not greater than k, decide whether the d-by-d zero matrix belongs to X^+, where X^+ denotes the closure of X under the usual matrix multiplication. In the same way, define the Z(k, d x d) problem as: given an instance X of M(k, d x d) (the instances of Z(k, d x d) are the same as those of M(k, d x d)), decide whether at least one matrix in X^+ has a zero in the left-upper corner. Define R(k, d x d) as the variant of Z(k, d x d) where "left-upper corner" is replaced with "right-upper corner". In the paper, we prove that M(6, 3 x 3), M(4, 5 x 5), M(3, 9 x 9), M(2, 15 x 15), Z(5, 3 x 3), Z(3, 5 x 5), Z(2, 9 x 9), R(6, 3 x 3), R(5, 4 x 4), and R(3, 6 x 6) are undecidable. The previous best comparable results were the undecidabilities of M(7, 3 x 3), M(3, 13 x 13), M(2, 21 x 21), Z(7, 3 x 3), Z(2, 13 x 13), R(7, 3 x 3), and R(2, 10 x 10).

cs.DM↗