Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

A note on a canonical theory with undecidable unification and matching problem

Abstract: A natural example of a canonical theory with undecidable unification and matching problem is presented.

Composition Problems for Braids

Abstract: In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with 3 strands, B_3, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-hard for braids with only 3 strands. The membership problem is decidable for B_3, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least 5 strands, but decidability of these problems for B_4 remains open. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.

Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

Abstract: Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.

Two Undecidable Problems of Analysis

Abstract: In this article two undecidable problems belonging to the domain of analysis will be constructed. The basic idea is sketched as follows: Let us imagine an area B of functions (rational functions, trigonometric and exponential functions) and certain operations (addition, multiplication, integration over finite or infinite domains, etc.) and consider the smallest quantity M of functions which contains B and is closed with regard to the selected operations. The question will then be examined whether there is in M a function f( x)for which the predicate P( n)≡ � f( x)cos nxdx > 0 is not recursive. It will be shown that by suitably choosing the area B and the operations, the answer comes out positively. We will deal in general with complex functions of real variables, although one could with somewhat more effort carry out all considerations in the real domain. In the first example, new functions will be generated by means of the following operations: addition, multiplication, integration over finite intervals and the solution of Fredholm integral equations of the second kind. Following this, it will be shown that certain logically characterised functions can be represented as limits of functions of the area M. In these constructions care will be taken that the number of integral equations to be solved remains as small as possible (namely two). In the second example, instead of the solution of Fredholm integral equations, we permit integration over infinite intervals, and then prove for this instance the same theorems as in the first example, in the context of which considerable use will be made of the result of M. Davis, H. Putnam and J. Robinson (cf. (1)) on the unsolvability of exponential diophantine equations.