SEMIGROUPS WITH if–then–else AND HALTING PROGRAMS

Abstract

The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included.

Axiomatization of if-then-else over possibly non-halting programs and tests

“…First we recall the concept of B-sets. The notion of a B-set was introduced by Bergman in [1] and elucidated by Jackson and Stokes in [10] to study the theory of halting programs equipped with the operation of if-then-else.…”

“…In [10] Jackson and Stokes also considered the case of modelling if-then-else over a collection of programs with composition by including an operation to capture the composition of programs with tests. Definition 1.2.…”

“…Further, they demonstrated how an algebra of non-halting tests could be constructed from Boolean tests in their setting. Jackson and Stokes proposed an alternative approach by considering an abstract collection of non-halting tests and posed the following problem:Characterize the algebras of computable functions associated with an abstract C-algebra of non-halting tests.The authors in [19] have approached the problem by adopting the approach of Jackson and Stokes in [10]. The notion of a C-set was introduced through which a complete axiomatization for if-then-else over a class of possibly non-halting programs and tests, where tests are drawn from an ada, was provided.In this paper, following the approach of Jackson and Stokes in [10], we extend the notion of C-sets to include composition of possibly non-halting programs and of these programs with possibly non-halting tests.…”

“…The authors in [19] have approached the problem by adopting the approach of Jackson and Stokes in [10]. The notion of a C-set was introduced through which a complete axiomatization for if-then-else over a class of possibly non-halting programs and tests, where tests are drawn from an ada, was provided.…”

Axioms for function semigroups with agreement quasi-order

“…Transformation semigroups and monoids equipped with abstract if-then-else operations indexed by the elements of a Boolean algebra were axiomatized in [10]. Amongst the many relations considered in [16], the kernel inclusion quasiorder relation , given by f g if and only if ker(f ) ⊆ ker(g), makes sense for transformations as well as for general (partial) functions.…”