## Paper: A Second-Order System for Polytime Reasoning Using Grädel's Theorem (at LICS 2001)

**Stephen A. Cook Antonina Kolokolova**

### Abstract

We introduce a second-order system V1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Grädel's [15] second-order Horn characterization of P. Our system has comprehension over P predicates (defined by Grädel'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 S12 or the second-order V11), 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 V1-Horn is finitely axiomatizable, and, as a corollary, that the class of \forall\Sigma_1^b consequences of S12 is finitely axiomatizable as well, thus answering an open question.

### BibTeX

@InProceedings{CookKolokolova-ASecondOrderSystemf, author = {Stephen A. Cook and Antonina Kolokolova}, title = {A Second-Order System for Polytime Reasoning Using Grädel's Theorem}, booktitle = {Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2001)}, year = {2001}, month = {June}, pages = {177--186}, location = {Boston, MA, USA}, publisher = {IEEE Computer Society Press} }