Comments on: Two Puzzles in Recursion Theory: Verbose Sets and Terse Sets http://xorshammer.com/2008/10/16/two-puzzles-in-recursion-theory-verbose-sets-and-terse-sets/ Some things in mathematical logic that I find interesting Wed, 18 Jan 2012 18:13:09 +0000 hourly 1 http://wordpress.com/ By: mkoconnor http://xorshammer.com/2008/10/16/two-puzzles-in-recursion-theory-verbose-sets-and-terse-sets/#comment-132 Fri, 17 Oct 2008 21:38:21 +0000 http://xorshammer.wordpress.com/?p=194#comment-132 Whoops, thanks. I fixed it so that the Turing machines in the posing of the problem also take no input.

You’re right, though, it’s inessential: Let K_0 be the set of Turing machines which halt with no input (say they start on a blank tape or something) and K_1 be the set of Turing machines which halt when given their own code as put. Then there are computable functions f and g such that for all n, n\in K_0 iff f(n)\in K_1 and n\in K_1 iff g(n) \in K_0.

There’s not even really any fancy trickery going on here. Just let the f(n)th Turing machine erase whatever input its given and then simulate the nth Turing machine, and let the g(n)th Turing machine first write n on the input tape and then simulate the nth Turing machine.

]]> By: Kenny Easwaran http://xorshammer.com/2008/10/16/two-puzzles-in-recursion-theory-verbose-sets-and-terse-sets/#comment-131 Fri, 17 Oct 2008 13:21:33 +0000 http://xorshammer.wordpress.com/?p=194#comment-131 You seem to switch from running Turing machines on their own codes as input in the posing of the puzzles to running Turing machines with no input in the solution – I assume that some sort of diagonalization or recursion theorem trickery means that this is an inessential point?

Also, it’s pretty funny that “Possible related posts” corrently includes “Strategies for Adjusting to Puberty?”

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