Monadic logical definability of NP-complete problems
It is well known that monadic second-order logic with linear order captures exactly regular languages. On the other hand, if addition is allowed, then J.F.Lynch has proved that existential monadic secondorder logic captures at least all the languages in NTIME(n), and then expresses some NP-complete languages (e.g. knapsack problem).
It seems that most combinatorial NP-complete problems (e.g. traveling salesman, colorability of a graph) do not belong to NTIME(n). But it has been proved that they do belong to NLIN (the similar class for RAM's). In the present paper, we prove that existential monadic second-order logic with addition captures the class NLIN, so enlarging considerably the set of natural problems expressible in this logic. Moreover, we also prove that this logic still captures NLIN even if first-order part of the second-order formulas is required to be ∀*∃*, so improving the recent similar result of J.F.Lynch about NTIME(n).
Key wordsComputational complexity monadic second-order logic finite model theory nondeterminism NP-complete problem linear time random access machine
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