Recursive Properties of Abstract Complexity Classes
Journal of the ACM (JACM)
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
WEAK MONADIC SECOND ORDER THEORY OF SUCCESSOR IS NOT ELEMENTARY-RECURSIVE
WEAK MONADIC SECOND ORDER THEORY OF SUCCESSOR IS NOT ELEMENTARY-RECURSIVE
The complexity of propositional linear temporal logics
Journal of the ACM (JACM)
Cosmological lower bound on the circuit complexity of a small problem in logic
Journal of the ACM (JACM)
Expressive Languages for Path Queries over Graph-Structured Data
ACM Transactions on Database Systems (TODS)
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The complexity of decision procedures for the Weak Monadic Second-Order Theories of the Natural Numbers are considered. If only successor is allowed as a primitive, then every alternation of second-order quantifiers causes an exponential increase in the complexity of deciding the validity of a formula. Thus a heirarchy similar in form to Kleene's arithmetic heirarchy may be shown to correspond to the Ritchie functions. On the other hand, if first-order less-than is allowed as a primitive, one existential quantifier suffices for arbitrarily complex (in the Ritchie heirarchy) decision problems. This leads to a normal form, in which every sentence in the theory is equivalent in polynomial time to a sentence with less-than but only one existential second-order quantifier.