 
Summary: Hierarchies in classes of program schemes
Argimiro A. ArratiaQuesada,
Departamento de Matematicas, Universidad Simon Bol var,
Apartado 89000, Caracas 1080A, Venezuela
Savita R. Chauhany,
Department of Computer Science,
University of Wales Swansea, Swansea SA2 8PP, U.K.
Iain A. Stewartz,
Department of Mathematics and Computer Science,
University of Leicester, Leicester LE1 7RH, U.K.
Abstract
We begin by proving that the class of problems accepted by the program schemes of NPS
is exactly the class of problems de ned by the sentences of transitive closure logic (program
schemes of NPS are obtained by generalizing basic nondeterministic whileprograms whose
tests within while instructions are quanti erfree rstorder formulae). We then show that
our program schemes form a proper in nite hierarchy within NPS whose analogy in transitive
closure logic is a proper in nite hierarchy, the union of which is full transitive closure logic but
for which every level of the hierarchy has associated with it a rstorder de nable problem not
in that level. We then proceed to add a stack to our program schemes, so obtaining the class
of program schemes NPSS, and characterize the class of problems accepted by the program
