  In this paper, we define and study variants of several complexity classes of
decision problems that are defined via some criteria on the number of accepting
paths of an NPTM. In these variants, we modify the acceptance criteria so that
they concern the total number of computation paths instead of the number of
accepting ones. This direction reflects the relationship between the counting
classes #P and TotP, which are the classes of functions that count the number
of accepting paths and the total number of paths of NPTMs, respectively. The
former is the well-studied class of counting versions of NP problems introduced
by Valiant (1979). The latter contains all self-reducible counting problems in
#P whose decision version is in P, among them prominent #P-complete problems
such as Non-negative Permanent, #PerfMatch, and #DNF-Sat, thus playing a
significant role in the study of approximable counting problems. We show that
almost all classes introduced in this work coincide with their `#accepting
paths'-definable counterparts, thus providing an alternative model of
computation for them. Moreover, for each of these classes, we present a novel
family of complete problems, which are defined via TotP-complete problems. This
way, we show that all the aforementioned classes have complete problems that
are defined via counting problems whose existence version is in P, in contrast
to the standard way of obtaining completeness results via counting versions of
NP-complete problems. To the best of our knowledge, prior to this work, such
results were known only for parity-P and C=P.
