Counting Quantifiers, Successor Relations, and Logarithmic Space

Counting Quantifiers, Successor Relations, and Logarithmic Space

Partition Relations for Successor Cardinals

This paper investigates the relations K+ --t (a): and its variants for uncountable cardinals K. First of all, the extensive literature in this area is reviewed. Then, some possibilities afforded by large cardinal hypotheses are derived, for example, if K is measurable, then K + + (K + K + 1, a): for every a < K +. Finally, the limitations imposed on provability in ZFC by L and by relative consi...

Constraint Satisfaction with Counting Quantifiers

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃ which assert the existence of at least j elements such that the ensuing property holds. These are natural variants of CSPs in the mould of quantified CSPs (QCSPs). Namely, ∃ := ∃ and ∃ := ∀ (for the domain of size n) We observe that a single counting quantifier ∃ strictly between ∃ and ∀ a...

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs i...

Counting quantifiers, subset surjective functions, and counting CSPs

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation reductions between counting constraint satisfaction problems (CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined...