Universal Automata with Uniform Bounds on Simulation Time

the re is no uniform bound on the loss of time incurred when some universal automaton globally simulates another automaton. More precisely, define an automaton to be a partial recursive function with decidable domain. Say that an automaton U locally simulates an automaton f via recursive, injective encoding cr with recursive time bound b iff for all z, z ' C IN the following holds: f ( z ) = z' ~ 9t[1 <~ t <~ b(z) A Uta(z) = a(z')]; i.e., one f-step can be simulated via cr by t U-steps, for some number t with 1 <~ t <~ b(z). We prove that, for any recursively enumerable family B of total recursive functions, there is no automaton which can locally simulate any automaton f via some injective encoding cr with some bound b from the given family B. In particular, there is neither a step-by-step simulator (taking B to be the family consisting only of the function 2z • 1) nor a real-time simulator (taking B to be the family of all constant functions). For an automaton f , define partial recursive functions