A second-order system for polynomial-time reasoning based on Graedel's theorem

摘要

We introduce a second-order system V_1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Graedel's second-order Horn characterization of P. Our system has comprehension over P predicates (defined by Graedel's second-order Horn formulas), and only finitely many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates (such as Buss's S_2^1 or the second-order V_1^1), and hence are more powerful 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 V_1-Horn is finitely axiomatizable, and, as a corollary, that the class of orallSigma_1^b consequences of S^1_2 is finitely axiomatizable as well, thus answering an open question.