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Numbering Turing Programs
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[QUOTE="jgens, post: 3705137, member: 107320"] [h2]Homework Statement [/h2] Assign a code number n(P) to every Turing program P. [h2]Homework Equations[/h2] N/A [h2]The Attempt at a Solution[/h2] Let p[SUB]i[/SUB] denote the ith prime number. Let Q = {q[SUB]0[/SUB],q[SUB]1[/SUB],...} be internal states, let {B,1} denote tape symbols and let {L,R} denote direction symbols. Assign a code number to each of these symbols as follows: n(B) = p[SUB]1[/SUB], n(1) = p[SUB]2[/SUB], n(L) = p[SUB]3[/SUB], n(R) = p[SUB]4[/SUB], n(q[SUB]k[/SUB]) = p[SUB]k+5[/SUB]. Now suppose L is a line of a Turing program P. Then L = (q[SUB]i[/SUB],s,q[SUB]j[/SUB],s',X) where s,s' in {B,1} and X in {L,R}. Assign a code number to L as follows: n(L) = p[SUB]1[/SUB][SUP]n(q[SUB]i[/SUB])[/SUP]p[SUB]2[/SUB][SUP]n(s)[/SUP]p[SUB]3[/SUB][SUP]n(q[SUB]j[/SUB])[/SUP]p[SUB]4[/SUB][SUP]n(s')[/SUP]p[SUB]5[/SUB][SUP]n(X)[/SUP]. Lastly suppose that P = {L[SUB]1[/SUB],...,L[SUB]k[/SUB]} is a Turing program. Assign a code number to P as follows: n(P) = p[SUB]n(L[SUB]1[/SUB])[/SUB][SUP]n(L[SUB]1[/SUB])[/SUP]...p[SUB]n(L[SUB]k[/SUB])[/SUB][SUP]n(L[SUB]k[/SUB])[/SUP]. This assigns each syntactic object we consider to a unique natural number. Moreover, this assignment is effective. Does this work? [/QUOTE]
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