The word problem for free partially commutative groups

Computer Science – Theory and Applications

Schleimer

Abstract.A compressed variant of the word problem for finitely generated groups, where the input word is given by a context-free grammar that generates exactly one string (also called a straight-line program), is studied. It is shown that finite extensions and free products preserve the complexity of the compressed word problem and that the compressed word problem for a graph group can be solved in polynomial time. Using these results together with connections between the compressed word problem and the (classical) word problem allows to obtain new upper complexity bounds for certain automorphism groups and group extensions.

“…Graph groups were studied e.g. in [15]; they are also known as free partially commutative groups [12,42], right-angled Artin groups [6,9], and semifree groups [1].…”

Section: Trace Monoids and Graph Groupsmentioning

confidence: 99%

“…This leads to a linear time solution for the word problem of G(Σ, I) [12,42]. An example for a derivation using the trace rewriting system R is:…”

Section: Graph Groups and Graph Productsmentioning

confidence: 99%

Efficient Computation in Groups Via Compression

Computer Science – Theory and Applications

Schleimer

“…Since R is length-reducing, R is terminating. By [9,28], R is also confluent. For traces u, v ∈ M(Σ ±1 , I) we have u = v in G(Σ, I) if and only if NF R (u) = NF R (v).…”

Section: Lemma 4 ([6]) the Trace U ⊔ V Exists If And Only Ifmentioning

confidence: 99%

“…For traces u, v ∈ M(Σ ±1 , I) we have u = v in G(Σ, I) if and only if NF R (u) = NF R (v). Using these facts, it was shown in [9,28] that the word problem for G(Σ, I) can be solved in linear time (on the RAM model).…”

Section: Lemma 4 ([6]) the Trace U ⊔ V Exists If And Only Ifmentioning

confidence: 99%

See 1 more Smart Citation

Compressed Conjugacy and the Word Problem for Outer Automorphism Groups of Graph Groups

Haubold

Lecture Notes in Computer Science

Mathissen

2010

Partially Commutative Inverse Monoids

Diekert

Lecture Notes in Computer Science

Miller

2006

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.