Universality and semicomputability for nondeterministic programming languages over abstract algebras

The Universal Function Theorem (UFT) originated in 1930s with the work of Alan Turing, who proved the existence of a universal Turing machine for computations on strings over a finite alphabet. This stimulated the development of stored-program computers. Classical computability theory, including the UFT and the theory of semicomputable sets, has been extended by Tucker and Zucker to abstract manysorted algebras, with algorithms formalized as deterministic While programs. This paper investigates the extension of this work to the nondeterministic programming languages While consisting of While programs extended by random assignments, as well as sublanguages of While formed by restricting the random assignments to booleans or naturals only. It also investigates the nondeterministic language GC of guarded commands. There are two topics of investigation: (1) the extent to which the UFT holds over abstract algebras in these languages; (2) concepts of semicomputability for these languages, and the extent to which they coincide with semicomputability for the deterministic While language.