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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 powerfuldoi:10.1016/s0168-0072(03)00056-3 fatcat:xx73pukpxfg4fp7hjyxrpqopoa