We answer a question asked by J. F. Lynch by proving that existential monadic second-order logic with addition captures not only the class NTIME(n) but also the class NLIN (i.e., linear time on nondeterministic RAMs), so enlarging considerably the set of natural problems expressible in this logic, since most combinatorial NP-complete problems belong to NLIN. Moreover, our result still holds if the first-order part of the formulas is required to be ∀ * ∃ * , so improving the recent similar result by J. F. Lynch about NTIME(n).In addition, we explicitly state that a graph problem is recognizable in nondeterministic linear time O(n+ e) (where n and e are the numbers of vertices and edges, respectively) if and only if it can be defined in existential second-order logic with unary functions and only one variable on the vertices-edges domain.
This paper originates from the observation that many classical NP graph problems, including some NPcomplete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertexNLIN, which collects the graph problems computable on a nondeterministic RAM in time OðnÞ; where n is the number of vertices of the input graph G ¼ ðV ; EÞ; rather than its usual size jV j þ jEj: It appears that this class is robust (it is defined by a natural restrictive computational device; it is logically characterized by several simple fragments of existential second-order logic; it is closed under various combinatorial operators, including some restrictions of transitive closure) and meaningful (it contains many natural NP problems: connectivity, hamiltonicity, non-planarity, etc.). Furthermore, the very restrictive definition of vertexNLIN seems to have beneficial effects on our ability to answer difficult questions about complexity lower bounds or separation between determinism and nondeterminism. For instance, we prove that vertexNLIN strictly contains its deterministic counterpart, vertexDLIN, and even that it does not coincide with its complementary class, co-vertexNLIN. Also, we prove that several famous graph problems (e.g. planarity, 2-colourability) do not belong to vertexNLIN, although they are computable in deterministic time OðjV j þ jEjÞ: r
The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing a direct logical characterization of rudimentary languages and a representation result of second-order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity.Mathematics Subject Classification: 03C13, 03D20.')The authors wish to express their thanks to E. GRANDJEAN for suggesting the problem and for 2)e-mail: Malika.MoreQmath.unicaen.fr 3)e-mai1: oiiveQlogique.jussieu.fr many stimulating conversations.
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