We prove that a very basic class of program schemes augmented with
access to a queue and an additional numeric universe within which counting is
permitted, so that the resulting class is denoted NPSQ
_+
(1), is such that the
class of problems accepted by these program schemes is exactly the class of
recursively enumerable problems. The class of problems accepted by the program
schemes of the class NPSQ(1) where only access to a queue, and not the
additional numeric universe, is allowed is exactly the class of recursively
enumerable problems that are closed under extensions. We define an infinite
hierarchy of classes of program schemes for which NPSQ(1) is the first class
and the union of the classes of which is the class NPSQ.We show that the class
of problems accepted by the program schemes of NPSQ is the union of the classes
of problems defined by the sentences of all vectorized Lindström
logics formed using operators whose corresponding problems are recursively
enumerable and closed under extensions, and, as a result, has a zero-one law.
Moreover, we also show that this class of problems can be realized as the class
of problems defined by the sentences of a particular vectorized
Lindström logic. Finally, we show how our results can be applied
to yield logical characterizations of complexity classes and provide logical
analogues to a number of inequalities and hypotheses from computational
complexity theory involving (non-deterministic) complexity classes ranging from
NP through to ELEMENTARY.
