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MR2589423 (2010i:20085)20M20 (08A70)

Jackson, Marcel(5-LTRB); Stokes, Tim(NZ-WAIK)Semigroups with if-then-else and halting programs. (English summary)Internat. J. Algebra Comput.19 (2009),no. 7,937–961.

Summary: “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 obviousextension of this is to include the operation of composition of programs, which gives a semigroupof 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 andinstead focus on the case where composition of functions with predicates is also allowed (andwe argue there is good reason to take this approach). In the case of total functions—modelinghalting programs—we give a complete axiomatization for the theory in terms of a finite system ofequations. We obtain a similar result when an operation of equality test and/or fixed point test isincluded.”

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