# Deciding verbose languages with linear advice

## Abstract

A language *A* is verbose if for some *k* there is a Turing machine *M* that for every input of *k* words *w*, ... , *w*_{k} computes a bitstring of length *k* that is not the characteristic string *X*_{ A }(*w*_{1}, ... , *w*_{k}). A language *A* is p-verbose (or *A* ∈ P-verb) if *M* is polynomially time bounded. Linear advice is sufficient to decide p-verbose languages in linear exponential time. Even languages that are linear-exponential time Turing reducible with linearly many queries to a p-verbose language are in E/lin. In [BL97] the special case of Turing reductions with a bounded number of queries to a p-selective language was investigated. Their results are extended to the general case of bounded Turing reductions to p-verbose languages.

In particular, it is shown that E_{lin-T}(P-verb) ⊂ E/lin; and EXP/poly is characterized as EXP_{ poly-T }(P-verb). On the other hand for fixed *c* and *k* it holds that E\(\nsubseteq\)*P*_{ cn-T }(P-*k*-verb) and \(EXP \nsubseteq E_{n^c } \left( { \nsubseteq - k - verb} \right)\).

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