The R- and L-orders of the Thompson-Higman monoid M_{k,1} and their complexity

TL;DR

This paper analyzes the structure and computational complexity of the R- and L-preorders in the Thompson-Higman monoid M_{k,1}, revealing dense orderings and complexity classifications under different generating sets.

Contribution

It characterizes the R- and L-preorders of M_{k,1} and establishes their complexity classifications, including Pi_2^P-completeness and coNP-completeness results.

Findings

R-preorder decision problem is Pi_2^P-complete with circuit-like generators.

L-preorder decision problem is coNP-complete with circuit-like generators.

Orderings are dense within classes despite the monoid's simple class structure.

Abstract

We study the monoid generalization M_{k,1} of the Thompson-Higman groups, and we characterize the R- and the L-preorder of M_{k,1}. Although M_{k,1} has only one non-zero J-class and k-1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <_R is dense (even within an L-class), and <_L is dense (even within an R-class). We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of M_{k,1}, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a "circuit-like" generating set, the R-preorder decision problem of M_{k,1} is Pi_2^P-complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Pi_2^P-complete, whereas the injectiveness problem is coNP-complete.