Post’s Correspondence Problem: From Computer Science to Algebra

Abstract

AbstractIn this short survey we describe recent advances on the Post Correspondence Problem in group theory that were inspired by results in computer science. These algebraic advances can, in return, provide a source of interesting problems in more applied, computational settings.Post’s Correspondence Problem (PCP) is a classical decision problem in theoretical computer science that asks whether for a pair of free monoid morphisms \(g, h:\varSigma ^*\rightarrow \varDelta ^*\) there is any non-trivial \(x\in \varSigma ^*\) such that \(g(x)=h(x)\). One can similarly phrase a PCP for general groups, rather than free monoids, by asking whether pairs g, h of group homomorphisms agree on any inputs. This leads to interesting and unexpected (un)decidability results for PCP in groups.KeywordsPost Correspondence ProblemFree and hyperbolic groupsFree monoidsNilpotent groupsDecidability