Counting Proportions of Sets: Expressive Power with Almost Order

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.