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Most of the published material concerning "state minimization of incompletely specified machines" has been concerned with finding an algorithm which yields a minimum state machine M' which satisfies [ER, p. 132] a given machine M. A fundamental theorem, in this connection, is given in [PU]. We are concerned here primarily with properties which some or all machines M' inherit from M.

A transduction, in the sense of this paper, is a n-ary word relation (which may be a function) describable by a finite directed labeled graph. The class of all n-ary transductions is co-extensive with the Kleenean closure of finite n-ary relations. The 1-ary transductions are exactly the sets recognizable by finite automata. However, for n ≫ 1 the relations recognizable by finite automata constitute a proper subclass of the n-ary transductions. The closure properties of the class of transductions are studied. The decomposition of transductions into simpler ones is also studied.

The card-programmed calculator is briefly described. Detailed instructions are given for programming the machine for use with the NOL General Purpose Panel. The system permits 2 instructions per cycle to be completed and a wide range of transcendental functions to be computed in a fraction of a second. The results make possible the automatic solution of a wide variety of problems in connection with ordnance research and development.