  We study a class of program schemes, NPSB, in which, aside from basic
assignments, non-deterministic guessing and while loops, we have access to
arrays; but where these arrays are binary write-once in that they are
initialized to `zero' and can only ever be set to `one'. We show, amongst other
results, that: NPSB can be realized as a vectorized Lindstrom logic; there are
problems accepted by program schemes of NPSB that are not definable in the
bounded-variable infinitary logic ${\cal L}^\omega_{\infty\omega}$; all
problems accepted by the program schemes of NPSB have a zero-one law; and on
ordered structures, NPSB captures the complexity class $[ L]^[{\scriptsize
NP\normalsize}]$. The class of program schemes NPSB is actually the union of an
infinite hierarchy of classes of program schemes. When we amend the semantics
of our program schemes slightly, we find that the classes of the resulting
hierarchy capture the complexity classes $\Sigma^p_i$ (where $i\geq 1$) of the
Polynomial Hierarchy PH. Finally, we give logical equivalences of the
complexity-theoretic question `Does NP equal PSPACE?' where the logics (and
classes of program schemes) involved define only problems with zero-one laws
(and so do not define some computationally trivial problems).
