Theory of Computing Systems

, Volume 56, Issue 4, pp 662–685 | Cite as

Two-Way Automata Characterizations of L/poly Versus NL



Let L/poly and NL be the standard complexity classes, of languages recognizable in logarithmic space by Turing machines which are deterministic with polynomially-long advice and nondeterministic without advice, respectively. We recast the question whether \(\mathsf {L}/\mathsf {poly}~{\supseteq }~{\mathsf {NL}}\) in terms of deterministic and nondeterministic two-way finite automata (2dfas and 2nfas). We prove it equivalent to the question whether every s-state unary2nfa has an equivalent poly(s)-state 2dfa, or whether a poly(h)-state 2dfa can check accessibility in h-vertex graphs (even under unary encoding) or check two-way liveness in h-tall, h-column graphs. This complements two recent improvements of an old theorem of Berman and Lingas. On the way, we introduce new types of reductions between regular languages (even unary ones), use them to prove the completeness of specific languages for two-way nondeterministic polynomial size, and propose a purely combinatorial conjecturethat implies \(\mathsf {L}/\mathsf {poly}~{\nsupseteq }~\mathsf {NL}\).


Two-way finite automata Logarithmic space Structural complexity Descriptional complexity 



The first author has been supported by a Marie Curie Intra-European Fellowship (pief-ga-2009-253368) within the European Union Seventh Framework Programme (fp7/2007-2013).


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Carnegie Mellon University in Qatar - Education CityDohaQatar
  2. 2.DI, Università degli Studi di MilanoMilanoItaly

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