Polynomial-Time Fraisse Limits via Fixed Point Theorems: From Model Theory to Trustworthy AI

Andrey Nechesov, Vadim Puzarenko

Keywords: p-computability, Fraisse limits;polynomial computability; Gandy's fixed point theorem; model theory; trustworthy AI; digital twins; smart cities

TL;DR: We establish sufficient conditions under which the Fraisse limit of a countable chain of finite p-computable algebraic structures is itself a p-computable structure.

Abstract: We establish sufficient conditions under which the Fraisse limit of a countable chain of finite p-computable algebraic structures is itself a p-computable structure. Our approach synthesizes two pillars: the polynomial analogue of Gandy's fixed point theorem (PAG-theorem), which guarantees that inductively defined predicate extensions remain polynomially decidable, and its functional variant (FPAG-theorem), which ensures that recursively defined functions on hereditary-finite list superstructures preserve polynomial complexity bounds. The central result---the P-Fraisse Theorem}---provides a unified criterion involving a Delta_0^p-operator for generators, a Delta_0^p-operator for predicates, and a functional boundary condition for operations. We demonstrate that classical Fraisse limits---the random graph (Rado graph), the countable dense linear order without endpoints (\eta-order), the countable atomless Boolean algebra, and the countable Ershov algebra---all admit p-computable presentations under natural encodings. These results bridge classical model-theoretic constructions with the computational complexity requirements of trustworthy artificial intelligence, digital twin infrastructures for smart cities, and formally verified decision-support systems, where polynomial computational complexity serves as a fundamental reliability criterion.

Submission Number: 130