  We study the algebraic theory of computable functions, which can be viewed as
arising from possibly non-halting computer programs or algorithms, acting on
some state space, equipped with operations of composition, {\em if-then-else}
and {\em while-do} defined in terms of a Boolean algebra of conditions. It has
previously been shown that there is no finite axiomatisation of algebras of
partial functions under these operations alone, and this holds even if one
restricts attention to transformations (representing halting programs) rather
than partial functions, and omits {\em while-do} from the signature. In the
halting case, there is a natural "fix", which is to allow composition of
halting programs with conditions, and then the resulting algebras admit a
finite axiomatisation. In the current setting such compositions are not
possible, but by extending the notion of {\em if-then-else}, we are able to
give finite axiomatisations of the resulting algebras of (partial) functions,
with {\em while-do} in the signature if the state space is assumed finite. The
axiomatisations are extended to consider the partial predicate of equality. All
algebras considered turn out to be enrichments of the notion of a (one-sided)
restriction semigroup.
