Program schemes with binary write-once arrays and the complexity classes they capture

We study a class of program schemes, NPSB, in which, aside from basic assignments, nondeterministic 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 Lindstr om logic; there are problems accepted by program schemes of NPSB that are not de nable in the bounded-variable in nitary logic L1!; all problems accepted by the program schemes of NPSB have a zero-one law; and on ordered structures, NPSB captures the complexity class LNP. The class of program schemes NPSB is actually the union of an in nite hierarchy of classes of program schemes. When we amend the semantics of our program schemes slightly, we nd that the classes of the resulting hierarchy capture the complexity classes i (where i 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 de ne only problems with zero-one laws (and so do not de ne some computationally trivial problems).