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@Techreport{BagnaraMHZ05TR,
  Author = "R. Bagnara and P. M. Hill and E. Mazzi and E. Zaffanella",
  Title = "Widening Operators for Weakly-Relational Numeric Abstractions",
  Number = 399,
  Type = "Quaderno",
  Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
  Year = 2005,
  Note = "Available at \url{http://www.cs.unipr.it/Publications/}",
  Abstract = "We discuss the construction of proper widening operators
              on several weakly-relational numeric abstractions. Our
              proposal differs from previous ones in that we actually
              consider the semantic abstract domains, whose elements
              are \emph{geometric shapes}, instead of the (more
              concrete) syntactic abstract domains of constraint
              networks and matrices.  Since the closure by entailment
              operator preserves geometric shapes, but not their
              syntactic expressions, our widenings are immune from the
              divergence issues that could be faced by the previous
              approaches when interleaving the applications of
              widening and closure.  The new widenings, which are
              variations of the \emph{standard widening} for convex
              polyhedra defined by Cousot and Halbwachs, can be made
              as precise as the previous proposals working on the
              syntactic domains.  The implementation of each new
              widening relies on the availability of an effective
              reduction procedure for the considered constraint
              description: we provide such an algorithm for the domain
              of \emph{octagonal shapes}."
}