# Adas and the equational theory of if-then-else

## Abstract

Programming languages admit the ternary operator ifp thenf elseg wherep is a test which may not halt. Here,p ranges over a suitable 3-valued logic. Guzmán and Squier recently introduced “C-algebras” and an equational 3-valued generalization of Boolean algebra based on “or”, “andrd and “not”. We incorporate their results and introduce as well the concept of “ada” (for Algebra of Disjoint Alternatives) which results whenC-algebras are equipped with an oracle for the halting problem. The 3-element ada is functionally complete. For if-then-else over a Boolean algebra or over an ada, eight equations generate all. The resulting variety may be represented as a variety of modules over a Boolean algebra. Over aC-algebra, a ninth equation is required.

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Manes, E.G. Adas and the equational theory of if-then-else. Algebra Universalis 30, 373–394 (1993). https://doi.org/10.1007/BF01190447