Symbolic Data Structure for sets of k-uples of integers Pierre Ganty 1 , C´ edric Meuter 1 , Laurent Van Begin 1 , Gabriel Kalyon 1 , Jean-Fran¸ cois Raskin 1 , and Giorgio Delzanno 2 1 D´ epartement d’Informatique, Universit´ e Libre de Bruxelles 2 Dipartimento di Informatica e Scienze dell’Informazione, Universit`a degli Studi di Genova Abstract. In this document we present a new symbolic data structure dedicated to the manipulation of (possibly infinite) sets of k-uples over integers, initially introduced in [Gan02]. This new data structure called Interval Sharing Tree (IST), is based on sharing trees [ZL95] where each node is labelled with an interval of integers. We present symbolic algo- rithm on IST for standard set operations and also introduce some specific operation that can be useful in the context of model-checking. 1 Introduction Many systems like multi-threaded programs and communication protocols are very complex to devise, hence error prone. For almost three decades a lot of re- search has been done in order to formalize and prove or refute properties on such systems. Given a formal description M of the system, we apply model checking methods [CGP99] to verify automatically that some properties hold on M. Those automatic methods manipulate (possibly) infinite sets of configurations of M. In model checking one problem we face off is the state explosion problem. This problem corresponds to a blow up in the size of the sets of configurations being manipulated. One way to overcome that problem is to use a compact symbolic representation of those sets. In our context assuming that those configurations are k-uples over integers, we define Interval Sharing Tree (IST) [Gan02] which is a symbolic representation of sets of k-uples. ISTs are directed acyclic graph- based data structures, where nodes are labelled with intervals of integers. This symbolic data structure benefits from several advantages. First, algorithms can be defined to manipulate ISTs symbolically (i.e. without exploring all path of the graph). Second, the size of an IST I may be exponentially smaller than the size of the set of k-uples it represent. Finally, an IST I can be (semantically) non- deterministic whereas other approaches facing to the state explosion problem traditionally use deterministic data structures (e.g. sharing trees [ZL95], BDD [CGP99], ...), thus allowing I to be more compact. The remainder of this document is structured as follows. First, in sec. 2, we introduce interval sharing trees and their semantics. We follow up, in sec. 3,