A second-order system for polytime reasoning based on Grädel's theorem

Stephen Cook, Antonina Kolokolova
2003 Annals of Pure and Applied Logic  
We introduce a second-order system V1-Horn of bounded arithmetic formalizing polynomialtime reasoning, based on Gr adel's (Theoret. Comput. Sci. 101 (1992) 35) second-order Horn characterization of P. Our system has comprehension over P predicates (deÿned by Gr adel's second-order Horn formulas), and only ÿnitely many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates (such as Buss's S 1 2 or the second-order V 1 1 ), and hence are more powerful
more » ... than our system (assuming the polynomial hierarchy does not collapse), or use Cobham's theorem to introduce function symbols for all polynomial-time functions (such as Cook's PV and Zambella's P-def). We prove that our system is equivalent to QPV and Zambella's P-def. Using our techniques, we also show that V1-Horn is ÿnitely axiomatizable, and, as a corollary, that the class of ∀ b 1 consequences of S 1 2 is ÿnitely axiomatizable as well, thus answering an open question.
doi:10.1016/s0168-0072(03)00056-3 fatcat:xx73pukpxfg4fp7hjyxrpqopoa