Equational axioms of test algebra

Abstract

We have axiomatized test algebra equations and achieved a finite equational axiomatization relative to Kleene algebra. The previous section showed that the Kleene algebra component cannot be replaced by a finite set of test algebra equations.

There is thus a tradeoff: on the one hand we have a sound and complete axiomatization using an infinite number of equations (this paper), on the other we have a finite axiomatization that uses a П ²₀ -axiom ([17]). However, if we replace the Kleene algebra component of the test calculus with any theory that is sound in test algebra and implies all the valid Kleene algebra equations, we get another sound and complete theory. One such theory is presented in [7]. This theory consists of a finite number of equations and two quasi-equations: ab ≤ b → a*b ≤ b ba ≤ b → ba* ≤ b where x ≤ y abbreviates x + y = y (this theory is also used in [9] to define Kleene algebas). Thus, a finite axiomatization is possible with a purely П ⁰₁ -theory, in fact only quasi-equations are necessary. We do not need to reprove the completeness theorem: it is a simple matter of combining the theorem of this paper and that of [7].