Ackermannian Integer Compression and the Word Problem for Hydra Groups

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions

Hydra groups

We give examples of CAT(0), biautomatic, free-by-cyclic, one-relator groups which have finite-rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules-versus-the-hydra battles in string-rewriting.Comment: 26 pages, 3 figure

Asymptotic invariants, complexity of groups and related problems

We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.Comment: 86 pages. Preliminary version, comments are welcome. v2: some references added, misprints fixed, some changes suggested by the readers are made. 88 pages. v3: more readers' suggestions implemented, index added, the list of references improved. This version is submitted to a journal. v4: The paper is accepted in Bulletin of Mathematical Science

Meaning versus Grammar

This volume investigates the complicated relationship between grammar, computation, and meaning in natural languages. It details conditions under which meaning-driven processing of natural language is feasible, discusses an operational and accessible implementation of the grammatical cycle for Dutch, and offers analyses of a number of further conjectures about constituency and entailment in natural language

Mathematical linguistics

but in fact this is still an early draft, version 0.56, August 1 2001. Please d