[Paper Review] Compressed word problems for inverse monoids

Markus Lohrey|arXiv (Cornell University)|Jun 6, 2011

semigroups and automata theory33 references4 citations

TL;DR

This paper investigates the compressed word problem for inverse monoids using straight-line programs (SLPs) to compress input words. It proves that the compressed word problem for free inverse monoids of rank at least two is complete for Π²ᵖ (the second level of the polynomial hierarchy), while for certain finite idempotent presentations, it is PSPACE-complete—despite the ordinary word problem being solvable in logspace. A PSPACE algorithm is also given for rational subset membership in free inverse monoids, which is shown to be PSPACE-complete.

ABSTRACT

The compressed word problem for a finitely generated monoid M asks whether two given compressed words over the generators of M represent the same element of M. For string compression, straight-line programs, i.e., context-free grammars that generate a single string, are used in this paper. It is shown that the compressed word problem for a free inverse monoid of finite rank at least two is complete for Pi^p_2 (second universal level of the polynomial time hierarchy). Moreover, it is shown that there exists a fixed finite idempotent presentation (i.e., a finite set of relations involving idempotents of a free inverse monoid), for which the corresponding quotient monoid has a PSPACE-complete compressed word problem. It was shown previously that the ordinary uncompressed word problem for such a quotient can be solved in logspace. Finally, a PSPACE-algorithm that checks whether a given element of a free inverse monoid belongs to a given rational subset is presented. This problem is also shown to be PSPACE-complete (even for a fixed finitely generated submonoid instead of a variable rational subset).

Motivation & Objective

• To analyze the computational complexity of the compressed word problem in inverse monoids, where input words are represented by straight-line programs (SLPs).
• To determine the precise complexity class of the compressed word problem for free inverse monoids of rank ≥ 2.
• To investigate whether the compressed word problem remains tractable for quotient monoids defined by finite idempotent presentations.
• To establish the complexity of rational subset membership in free inverse monoids under compressed input.
• To compare the algorithmic behavior of inverse monoids with that of free monoids and free groups, particularly regarding decidability and complexity.

Proposed method

• The paper uses Munn trees to represent elements of free inverse monoids, enabling efficient word problem solutions via tree gluing operations.
• It applies straight-line programs (SLPs) as a compressed representation of words, allowing exponential compression of input size.
• For the upper bound on the compressed word problem in free inverse monoids, it combines Munn’s word problem algorithm with Lipton and Zalcstein’s logspace result for free groups.
• For the lower bound, it reduces a variant of the SUBSETSUM problem to the compressed word problem using SLP-based encoding techniques.
• It constructs a tree automaton B that recognizes the tree language of Munn trees, using multisets of runs and labelings to simulate SLP-generated words.
• It proves PSPACE-completeness of rational subset membership by reducing the problem to tree automaton membership and using PSPACE-computable constructions.

Experimental results

Research questions

• RQ1 What is the computational complexity of the compressed word problem for a free inverse monoid of finite rank ≥ 2?
• RQ2 Can the compressed word problem for quotient monoids defined by finite idempotent presentations be more complex than the ordinary word problem?
• RQ3 Is the rational subset membership problem for free inverse monoids tractable under compressed input?
• RQ4 Does the use of SLPs as input representation significantly increase the complexity of algorithmic problems in inverse monoids compared to free monoids or groups?
• RQ5 Can the compressed word problem for inverse monoids be solved in polynomial time, or does it require higher complexity classes?

Key findings

• The compressed word problem for a free inverse monoid of rank at least two is complete for Π²ᵖ, the second universal level of the polynomial hierarchy.
• There exists a fixed finite idempotent presentation for which the compressed word problem of the quotient monoid is PSPACE-complete, despite the ordinary word problem being solvable in logspace.
• A PSPACE algorithm is presented for deciding whether a given element of a free inverse monoid belongs to a given rational subset.
• The rational subset membership problem for free inverse monoids is shown to be PSPACE-complete, even when the rational subset is fixed.
• The compressed word problem for free inverse monoids is strictly more complex than for free monoids (P-time) and free groups (P-time), as it reaches Π²ᵖ-completeness.
• The construction of a tree automaton B that recognizes Munn tree languages is shown to be computable in PSPACE, enabling the reduction to tree automaton membership.