Reference : Counting the solutions of Presburger equations without enumerating them
Scientific journals : Article
Engineering, computing & technology : Computer science
Counting the solutions of Presburger equations without enumerating them
Boigelot, Bernard mailto [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Informatique >]
Latour, Louis [ > > ]
Theoretical Computer Science
Elsevier Science
Yes (verified by ORBi)
The Netherlands
[en] Presburger arithmetic ; automata ; counting ; symbolic representation systems
[en] The Number Decision Diagram (NDD) has recently been introduced
as a powerful representation system for sets of integer vectors. NDDs
can notably be used for handling sets defined by arbitrary Presburger
formulas, which makes them well suited for representing the set of
reachable states of finite-state systems extended with unbounded
integer variables. In this paper, we address the problem of counting
the number of distinct elements in a set of numbers or, more
generally, of vectors, represented by an NDD. We give an algorithm
that is able to produce an exact count without enumerating explicitly
the vectors, which makes it capable of handling very large sets. As an
auxiliary result, we also develop an efficient projection method that
allows to construct efficiently NDDs from quantified formulas, and
thus makes it possible to apply our counting technique to sets
specified by formulas. Our algorithms have been implemented in the
verification tool LASH, and applied successfully to various counting
This work was partially funded by a grant of the "Communauté française de Belgique - Direction de la recherche scientifique - Actions de recherche concertées", and by the European Commission (FET project ADVANCE, contract No IST-1999-29082)
This is the author's version of the work. The published version is available online at

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