On a Relation Between Algebraic Programs and Turing Machines

i? recent years the need for a rigorous formaliza!ion’hbf the mathematical properties of computer programs has become increasingly evident. Problems in program proving, for exampie, require that a precise meanir!.: be given to otherwise intuitive notions. One such formalization has been suggested by Milner [6] and has been successfully used in areas such as program and machine verification (see, for example, the S machine experiment [2]). In this pap& we shall investigate the implications of this formalization from a global perspective, with the objective of showing that (under suitable restrictions) the formalized program is equivalent to the model proposed by Turing, i.e., the Turing machine. While the program approach may often be more feasible from a practical point of view, the equivalence of the two models serves to make theoretical results proven in one COIItext applicable in the other. After the prequisite delinitions, we shall first establish (independent of Turing machine considerations) the existence of a universal upper bound on programs. We shall then establish a correspondence between Turing macl,jnes and programs, a correspondence which in faci is an orderisomorphism between the partially ordered sets (under simulation) of Turing mat:hin;:s and program.