## Paper: A Second-Order Theory for NL (at LICS 2004)

**Stephen A. Cook Antonina Kolokolova**

### Abstract

We introduce a second-order theory V-Krom of bounded arithmetic for nondeterministic log space. This system is based on Grädel's characterization of NL by second-order Krom formulate with only universal first-order quantifiers, which is 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 V₁-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-Szelepcsényi theorem that NL is closed under complementation. This formalization is necessary to show that the NL functions are \sum _1^B-definable in V-Krom. The only other theory for NL in the literature relies on the Immerman-Szelepcsényi's result rather than proving it.

### BibTeX

@InProceedings{CookKolokolova-ASecondOrderTheoryf, author = {Stephen A. Cook and Antonina Kolokolova}, title = {A Second-Order Theory for NL}, booktitle = {Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2004)}, year = {2004}, month = {July}, pages = {398--407}, location = {Turku, Finland}, publisher = {IEEE Computer Society Press} }