925 research outputs found
Random equations in nilpotent groups
In this paper we study satisfiability of random equations in an infinite
finitely generated nilpotent group G. We show that the set SAT(G,k) of all
equations in k > 1 variables over G which are satisfiable in G has an
intermediate asymptotic density in the space of all equations in k variables
over G. When G is a free abelian group of finite rank, we compute this density
precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1
the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting
sets in groups are either zero or one. The results of this paper provide new
examples of algebraically significant sets of intermediate asymptotic density.Comment: 25 page
Generalized Craig Interpolation for Stochastic Boolean Satisfiability Problems with Applications to Probabilistic State Reachability and Region Stability
The stochastic Boolean satisfiability (SSAT) problem has been introduced by
Papadimitriou in 1985 when adding a probabilistic model of uncertainty to
propositional satisfiability through randomized quantification. SSAT has many
applications, among them probabilistic bounded model checking (PBMC) of
symbolically represented Markov decision processes. This article identifies a
notion of Craig interpolant for the SSAT framework and develops an algorithm
for computing such interpolants based on a resolution calculus for SSAT. As a
potential application area of this novel concept of Craig interpolation, we
address the symbolic analysis of probabilistic systems. We first investigate
the use of interpolation in probabilistic state reachability analysis, turning
the falsification procedure employing PBMC into a verification technique for
probabilistic safety properties. We furthermore propose an interpolation-based
approach to probabilistic region stability, being able to verify that the
probability of stabilizing within some region is sufficiently large
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
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