An algorithm for recognition of n-collapsing words

A word w over a finite alphabet Σ is n-collapsing if for an arbitrary deterministic finite automaton A=〈Q,Σ,δ〉, the inequality |δ(Q,w)|≤|Q|−n holds provided that |δ(Q,u)|≤|Q|−n for some word u∈Σ+ (depending on A). We prove that the property of n-collapsing is algorithmically recognizable for any given positive integer n. We also prove that the language of all n-collapsing words is context-sensitive.