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See detailCounting the solutions of Presburger equations without enumerating them
Boigelot, Bernard ULiege; Latour, Louis

in Theoretical Computer Science (2004), 313(1), 17-29

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 ... [more ▼]

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 problems. [less ▲]

Detailed reference viewed: 22 (5 ULiège)
Full Text
Peer Reviewed
See detailCounting the Solutions of Presburger Equations without Enumerating Them
Boigelot, Bernard ULiege; Latour, Louis

in Lecture Notes in Computer Science (2001), 2494

The Number Decision Diagram (NDD) has recently been proposed as a powerful representation system for sets of integer vectors. In particular, NDDs can be used for representing the sets of solutions of ... [more ▼]

The Number Decision Diagram (NDD) has recently been proposed as a powerful representation system for sets of integer vectors. In particular, NDDs can be used for representing the sets of solutions of arbitrary Presburger formulas, or the set of reachable states of some systems using unbounded integer variables. In this paper, we address the problem of counting the number of distinct elements in a set of vectors represented as an NDD. We give an algorithm that is able to perform 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 problems. [less ▲]

Detailed reference viewed: 21 (3 ULiège)