A conjunctive logical characterization of nondeterministic linear time

Abstract

We denote by $\mathcal{G}$(f,0,s) the class of formulas of the form ∃g∀xφ, where g is a tuple of unary function symbols, χ is a first-order variable, and ϕ quantifier-free formula of signature {g, f, 0, s}.

We prove the reduction of $\mathcal{G}$(f, 0, s) to the subclass $\mathcal{G}_ \wedge ^ +$(f, 0, s) of formulas written without negation nor disjunction.

As a corollary we obtain a new logical characterization of the class NTIME_RAM(n): the problems of this class, and notably most of the natural NP-complete problems, can be considered as models of some sentence of the type ∃g ₁...∃g _p...∀x∧σ(x)=τ(x) (g _i : unary function symbols; σ, τ: compositions of such functions), that is, in some sense, of some system of functional equations.