[Paper Review] The Thompson-Higman monoids M_{k,i}: the J-order, the D-relation, and their complexity

Jean-Camille Birget|ArXiv.org|Apr 16, 2009

Commutative Algebra and Its Applications20 references3 citations

TL;DR

This paper investigates the Thompson-Higman monoids $M_{k,i}$, focusing on their Green's relations $ancyscript{J}$ and $ancyscript{D}$, and establishes the computational complexity of deciding these relations. It shows that $ancyscript{J}$-ordering is in P for standard generators, but coDP-complete for circuit-like generators, while $ancyscript{D}$-equivalence is $igoplus_{k-1}ullet{ m NP}$-complete, highlighting the role of advanced complexity classes in monoid structure.

ABSTRACT

The Thompson-Higman groups G_{k,i} have a natural generalization to monoids M_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features of M_{k,i} and Inv_{k,i} and investigate the computational complexity of decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of M_{k,1} are isomorphic to the groups G_{k,j} (1 \leq j \leq k-1); so we rediscover all the Thompson-Higman groups within M_{k,1}. The Green relations \leq_J and \equiv_D of M_{k,1} can be decided in deterministic polynomial time when the inputs are words over a finite generating set of M_{k,1}. When a circuit-like generating set is used for M_{k,1} then deciding \leq_J is coDP-complete. The multiplier search problem for \leq_J is xNPsearch-complete, whereas the multiplier search problems of \leq_R and \leq_L are not in xNPsearch unless NP = coNP. Deciding \equiv_D for M_{k,1} when the inputs are words over a circuit-like generating set, is \oplus_{k-1}.NP-complete. For Inv_{k,1} over a circuit-like generating set, deciding \equiv_D is \oplus_{k-1} P-complete.

Motivation & Objective

• To analyze the structural properties of the Thompson-Higman monoids $M_{k,i}$ and their inverse counterparts ${ m Inv}_{k,i}$, particularly focusing on Green's relations.
• To determine the computational complexity of deciding the $ancyscript{J}$-order and $ancyscript{D}$-relation in $M_{k,1}$ under different generating sets.
• To introduce and characterize the complexity class xNPsearch as a generalization of NPsearch to handle non-polynomially bounded solution spaces.
• To establish connections between monoid relations and circuit complexity, especially via circuit-like generating sets.
• To demonstrate that the image size and image size modulo $h$ problems for partial circuits are $igcircullet{ m NP}$-complete and $igoplus_hullet{ m NP}$-complete, respectively.

Proposed method

• Defining $M_{k,i}$ as the monoid of maximally extended right ideal homomorphisms between right ideals of $BA^*$, where $|A|=k$, $|B|=i$, using essential equality and maximal extension to ensure uniqueness.
• Using polynomial-time many-to-one search reductions to relate decision problems in $M_{k,1}$ to known complexity classes, particularly coDP and $igoplus_{k-1}ullet{ m NP}$.
• Introducing the class xNPsearch to capture search problems where solutions may not be polynomially bounded but verification is still feasible in polynomial time.
• Establishing reductions between the $ancyscript{J}$-order and $ancyscript{D}$-relation problems and known NP-hard problems, proving completeness results via verification and input-output reductions.
• Applying modular counting complexity classes to analyze the image size modulo $h$ problem, showing $igoplus_hullet{ m NP}$-completeness for $h eq 1$.
• Proving that the multiplier search problem for $ancyscript{J}$-order is xNPsearch-complete, while $ancyscript{R}$ and $ancyscript{L}$-order problems are not in xNPsearch unless NP = coNP.

Experimental results

Research questions

• RQ1 What is the computational complexity of deciding the $ancyscript{J}$-order in $M_{k,1}$ when inputs are words over a finite generating set?
• RQ2 How does the complexity of deciding the $ancyscript{D}$-relation in $M_{k,1}$ change when using a circuit-like generating set instead of a standard one?
• RQ3 Can the multiplier search problem for the $ancyscript{J}$-order be classified within known complexity classes, and what does this imply about the structure of solutions?
• RQ4 What is the role of the newly introduced xNPsearch class in capturing search problems with potentially unbounded solution sizes?
• RQ5 How do the image size and image size modulo $h$ problems for partial circuits relate to counting complexity classes like $igcircullet{ m NP}$ and $igoplus_hullet{ m NP}$?

Key findings

• Deciding the $ancyscript{J}$-order in $M_{k,1}$ is in P when inputs are words over a finite generating set.
• When a circuit-like generating set is used, deciding the $ancyscript{J}$-order becomes coDP-complete.
• The multiplier search problem for $ancyscript{J}$-order is xNPsearch-complete, indicating it lies outside standard NPsearch unless NP = coNP.
• The multiplier search problems for $ancyscript{R}$ and $ancyscript{L}$-orders are not in xNPsearch unless NP = coNP, suggesting they are fundamentally harder.
• Deciding the $ancyscript{D}$-relation in $M_{k,1}$ with a circuit-like generating set is $igoplus_{k-1}ullet{ m NP}$-complete.
• For ${ m Inv}_{k,1}$ over a circuit-like generating set, deciding $ancyscript{D}$-equivalence is $igoplus_{k-1}{ m P}$-complete.