Abstract
We study the complexity of the decision problem of a variant of arithmetic with bounded quantifiers. By contrast with the usual asymptotic complexity results, where one only deals with suitably long sentences, our limitative results have a concrete character, so as to find application in physics. We develop the mathematical framework for our lower bounds up to the point where their physical meaning is evident. Imagining that Turing machineM materializes as a real computerM′, we give a rigorous formulation of the simplest space-time features ofM′, by essentially requiring that (i) each state ofM occupies some space, (ii) no signal between states and scanning head can travel at infinite speed. WhenM provides a decision procedure for arithmetic with bounded quantifiers, there exist stringent lower bounds for the “time”t(j) needed byM′ to solve the hardest problems of length smaller thanj, no matter the number of states ofM. We discuss several applications of our results.
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Mundici, D. Natural limitations of decision procedures for arithmetic with bounded quantifiers. Arch math Logik 23, 37–54 (1983). https://doi.org/10.1007/BF02023011
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DOI: https://doi.org/10.1007/BF02023011