Syntactic approximations to computational complexity classes
Argimiro Arratia
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Abstract
We present a formal syntax of approximate formulas suited for the logic with counting quantifiers SOLP. This logic was studied by us in [1] where, among other properties, we showed: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and fragments of it capture classes like P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments, using a combinatorial tool adapted to these languages. The purpose of the approximate formulas presented here, is to provide a syntactic approximation to logics contained in SOLP with built-in order, that should be complementary of the semantic approximation based on almost orders, by producing approximating logics where problems are described within a small counting error. We state and prove a Bridge Theorem that links expressibility in fragments of SOLP over almostordered structures to expressibility with respect to approximate formulas for the corresponding fragments over...
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This paper presents a syntax of approximate formulae suited for the logic with counting quantifiers SOLP. This logic was formalised by us in where, among other properties, we showed the following facts: (i) In the presence of a built-in (linear) order, SOLP can describe NP-complete problems and some of its fragments capture the classes P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments, using a combinatorial tool adapted to these languages.
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Lecture Notes in Computer Science, 2006
We present a second order logic of proportional quantifiers, SOLP, which is essentially a first order language extended with quantifiers that act upon second order variables of a given arity r, and count the fraction of elements in a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of different problems of complexity up to NP-hard, and fragments within our logic capture complexity classes as NL and P, with auxiliary ordering relation. When restricted to monadic second order variables our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers, for it does not has the bounded number of degrees property. Moreover, we show in this almost ordered setting the existence of an infinite hierarchy inside our monadic language. We extend our inexpressibility result to an almost ordered (not necessarily monadic) fragment of SOLP, which in the presence of full order captures P. To obtain all our inexpressibility results we developed combinatorial games appropriate for these logics.
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