344 research outputs found

    Semigroups with if-then-else and halting programs

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    The "ifā€“thenā€“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 ifā€“thenā€“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 ā€” modeling halting programs ā€” 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

    Domain and range for angelic and demonic compositions

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    We give finite axiomatizations for the varieties generated by representable domain--range algebras when the semigroup operation is interpreted as angelic or demonic composition, respectively
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