Terse, Superterse, and Verbose Sets

On checking versus evaluation of multiple queries

“…We show that from a result of [Rog67] it follows that the characteristic vector cost of K is exactly two. This strengthens the result of ( [BGG087], see also [Bar68]) that computing the solutions to two queries to K requires two queries to K. Other results on the characteristic vector cost of nonrecursive languages are also proven.…”

“…Pioneering work in this area was done by Barzdin' [Bar68]-and, more recently, by Beigel, Gasarch, Gill, and Owings [BGG087] for a recursion-theoretic context, and by Amir, Beigel, and Gasarch [Bei87,AG88] for a polynomial-time framework. Their approach to certain structural properties of hard sets studies the complexity of evaluating the values of queries-that is, of computing the answers to the queries.…”

“…We now propose and study a "checking answers" analog of the [AG88, Bar68,BGG087] theory of terseness.…”

Enumerations of the Kolmogorov function

“…Traditional approximations are a special case of this, in which the set of candidates is the set of numbers in a given range. This kind of approximation has been called enumeration complexity, see [1,2,4,5,7,8,12,18]. Bill Gasarch suggested the natural question whether we can approximate the Kolmogorov function in this sense, or more precisely, how many values does a Turing machine need to output before it is guaranteed that one of the values is C(x).…”

The communication complexity of enumeration, elimination, and selection

“…The complexity of doing k instances of a problem has been looked at in a variety of fields including decision trees [6,25], computability [5,13], complexity [2,7,19], straightline programs [10], and circuits [28]. …”