A Study of NP-Completeness and Undecidable Word Problems in Semigroups

TL;DR

This paper investigates the limits of computational complexity and decidability in semigroups, demonstrating an algebraic structure with an undecidable word problem linked to non-recursive functions, highlighting fundamental computational boundaries.

Contribution

It introduces an associative calculus with an undecidable word problem connected to non-recursive functions, bridging complexity theory and algebraic undecidability.

Findings

Existence of an associative calculus with an undecidable word problem

Connection between non-recursive functions and algebraic undecidability

Insights into the limits of algorithmic solutions in mathematics

Abstract

In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$ implies the existence of a deterministic Turing machine solving $L$ in polynomial time $O(n^k)$. Central to our investigation is polynomial reducibility. Also, we demonstrate the existence of an associative calculus $A(\mathfrak{T})$ with an algorithmically undecidable word problem, where for a Turing machine $\mathfrak{T}$ computing a non-recursive function $E(x)$, we establish that $q_1 01^x v \equiv q_0 01^i v \Leftrightarrow x \in M_i$ for $i \in \{0,1\}$, where $M_i = \{x \mid E(x) = i\}$. This connection between computational complexity and algebraic undecidability illuminates the fundamental limits of algorithmic solutions in mathematics.