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Kp, if dyadic(p) is a normal TM code and k p = the number of tapes used in TMp ; 0, otherwise is computable in O(lpl) time. FUrthermore, the work space bounds on the functions and routines above are as follows. Each of the functions and routines of parts (b), (c), (d), (f), (g), and (i) are computable in simultaneous linear time and constant work space. 2 x from part (k)) are computable in simultaneous linear time and logarithmic work space. The function AX. 1 Details of the Model 37 O(lxl) work space.

We define 101 = 1 to simplify notation within 0 expressions. lxl+ ·ly4) = O(Ax,Y·lxl·lyl). We briefly note some useful equalities and inequalities having to do with 1·1. By our definition of 1·1 and binary, we have that Ibinary(x)1 = 1 + logx. for all x > 0, (2) 1,1, it follows that, for each x> 0: Ixl = Ibinary(x) I = 1 + logx. By our definitions of dyadic, binary, and (3) x x + 1 is a power of 2 + 1 is not a power of 2 =} =} Ixl = Ibinary(x)I-1 = logx. (4) By the definition of (', , '), it follows by induction that, for all n > 0, and all Xl, ...

LEMMA 7To verify this in reasonable detail involves more details of the Hennie and Stearns construction than we want to discuss here. These details are straightforward, if grubby, and we leave them to the interested reader. 46 Deterministic Multi-tape Thring Machines (c) On a given input, TMnorm(p) halts if and only if it halts by going into state 1. Furthermore, whenever TMnorm(p) halts, all its work tapes are blank. PROOF. Suppose P is a normal code. Let k be the number of tapes of TMpo We sketch a TM M p that: (i) computes the same partial function as TMp; and, (ii) on a given input, M p halts if and only if it halts by going into state 1.

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