We prove a uniform effective density theorem as well an counting result for generic system comprising polynomial with mild homogeneous condition and several linear forms using Rogers' second moment formula the Siegel transform on space of unimodular lattices.

Motivated by the recent result of [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicat...

In the spring of 1989, Seinosuke Toda of the University of Electro-Communications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP To-89]. In this Structural Complexity Column, we will brieey review Toda's result, and explore how it relates to other topics of interest in computer science. In particular, we will introduce the reader to The Counting Hierarchy: a hierarch...

Rice's Theorem says that every nontrivial semantic property of programs is undecidable. It this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions.

Probabilistic inference via model counting has emerged as a scalable technique with strong formal guarantees, thanks to recent advances in hashing-based approximate counting. State-of-theart hashing-based counting algorithms use an NP oracle (SAT solver in practice), such that the number of oracle invocations grows linearly in the number of variables n in the input constraint. We present a new ...

This paper studies distributions which can be \approximated" by sampling algorithms in time polynomial in the length of their outputs. First, it is known that if polynomial-time samplable distributions are polynomial-time computable, then NP collapses to P. This paper shows by a simple counting argument that every polynomial-time samplable distribution is computable in polynomial time if and on...

Generalised Satisfiability Problems (or Boolean Constraint Satisfaction Problems), introduced by Schaefer in 1978, are a general class of problem which allow the systematic study of the complexity of satisfiability problems with different types of constraints. In 1979, Valiant introduced the complexity class parity P, the problem of counting the number of solutions to NP problems modulo two. Ot...

First-order model counting emerged recently as a novel reasoning task, at the core of efficient algorithms for probabilistic logics. We present a Skolemization algorithm for model counting problems that eliminates existential quantifiers from a first-order logic theory without changing its weighted model count. For certain subsets of first-order logic, lifted model counters were shown to run in...

We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for dimensions d =1,2,3, and 4 only.