We study the class coNPMV of complements of NPMV functions. Though defined symmetrically to NPMV, this class exhibits very different properties. We clarify the complexity of coNPMV by showing that it is essentially the same as that of NPMVNP. Complete functions for coNPMV are exhibited and central complexity-theoretic properties of this class are studied. We show that computing maximum satisfyi...

introductionpopulation data is easily accessible and shows the relationship between city size and other socio-economic factors in the city indirectly, so many geographers using this type of data to gain an understanding of the structure of the cities network and distribution of the people in the different levels of urban hierarchy. in fact, urban hierarchy is the best way of space organization ...

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

The bounded arithmetic theory S 2 is nitely axiomatized if and only if the polynomial hierarchy provably collapses. If T i 2 equals S i+1 2 then T i 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to p i+3 , and, in fact, to the Boolean hierarchy over p i+2 and to p i+1 =poly .

The ample hierarchy of geometries of stables theories is strict. We generalise the construction of the free pseudospace to higher dimensions and show that the n-dimensional free pseudospace is ω-stable n-ample yet not (n+ 1)-ample. In particular, the free pseudospace is not 3-ample. A thorough study of forking is conducted and an explicit description of canonical bases is exhibited.