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I t is of interest to design a universal Turing machine smaller than any ever previously published in the literature. According to Shannon's suggestion [1], the product of the number of states and the number of symbols would be an appropriate measure of the size of a Universal Turing machine (U.T.M.) . We present here one machine with a product of 40--five symbols and eight s ta tes and one with a product of 30--five symbols and six s ta tes--provided that symbols can be printed on the infinitely many squares of the input tape. The former is correctly an ordinary Universal Turing machine, but the latter is a slightly extended one. According to Davis ' definition [2] of the Turing machine, the input condition must be expressed as follows: the input tape is always finite but can be extended by a certain given rule (presented below). The machine must have the property that , whenever it is about to run off an end of the tape, a row of new squares in which appear certain given symbols is spliced onto the end of the tape. Some published results are listed in Table 1.

5-Symbol 8-State and 5-Symbol 6-State Universal Turing Machines