A structure of finite signature is constructed so that: for all existential formulas ∃y⃗ φ(x⃗,y⃗) and for all tuples of elements ⃗ of the same length as the tuple x⃗, one can decide in a quadratic time depending only on the length of the formula, if ∃y⃗ φ(u⃗,y⃗) holds in the structure. In other words, the structure satisfies the relativized model-theoretic version of P=NP in the sense of . This is a model-theoretical approach to results of Hemmerling and Gaßner.
"Structure with fast elimination of quantifiers." J. Symbolic Logic 71 (1) 321 - 328, March 2006. https://doi.org/10.2178/jsl/1140641177