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The deterministic algorithms on inputs of a given length are automatically finite (i.e. finiteness is a consequence of the input length, rather than something that needs to be assumed explicitly). But this bound holds on all lengths simultanouesly, and therefore on deterministic algorithms whose input is not a fixed length. The number of such algorithms is not finite. —David Eppstein (talk) 16:27, 19 March 2018 (UTC)[reply]
Isn't this only half of Yao's principle?
Namely,this page establishes the "easy direction" of Yao's principle, which says that considering arbitrary deterministic algorithms on a chosen distribution of instances is a valid proof technique. (This does not require the minimax theorem to be shown, and is pretty easy). But Yao's principle goes further, also showing that this is also the "right" thing to do (there is no loss in doing so), i.e. this lower bound technique is optimal (the proof of this relies on the minimax theorem, i.e. to show the inequality is an equality). See e.g. Goldreich's comment in http://drops.dagstuhl.de/opus/volltexte/2014/4733/ (Appendix A.1). Ceacy (talk) 19:33, 8 April 2018 (UTC)[reply]
