A second-order theory for NL

We introduce a second-order theory V-Krom of bounded arithmetic for nondeterministic log space. This system is based on Gradel´s characterization of NL by second-order Krom formulae with only universal first-order quantifiers, which in turn is motivated by the result that the decision problem for 2-CNF satisfiability is complete for coNL (and hence for NL). This theory has the style of the authors´ theory Vi-Horn [APAL 124 (2003)] for polynomial time. Both theories use Zambella´s elegant second-order syntax, and are axiomatized by a set 2-BASIC of simple formulae, together with a comprehension scheme for either second-order Horn formulae (in the case of V₁-Horn), or second-order Krom (2CNF) formulae (in the case of V-Krom). Our main result for V-Krom is a formalization of the Immerman-Szelepcsenyi theorem that NL is closed under complementation. This formalization is necessary to show that the NL functions are Σ₁^B-definable in V-Krom. The only other theory for NL in the literature relies on the Immerman-Szelepcsenyi´s result rather than proving it.