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The complexity class NP can be logically characterized both through existential second order logic SO∃, as proven by Fagin, and through simulating a Turing machine via the satisfiability problem of propositional logic SAT, as proven by Cook. Both theorems involve encoding a Turing machine by a formula in the corresponding logic and stating that a model of this formula exists if and only if the Turing machine halts, i.e. the formula is satisfiable iff the Turing machine accepts its input.arXiv:1410.4925v1 fatcat:opzk733isfcntgssmcepys4e7q