1. What contributions have the authors mentioned in the paper "An improved tight closure algorithm for integer octagonal constraints⋆" ?
In this paper the authors present and fully justify an O ( n ) algorithm to compute the tight closure of a set of UTVPI integer constraints.
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2. What have the authors stated for future works in "An improved tight closure algorithm for integer octagonal constraints⋆" ?
The algorithm —which is based on the extension to integer-weighted octagonal graphs of the one the authors proposed for rational-weighted octagonal graphs [ 2, 3 ] — and its proof of correctness means the issue about the possibility of computing the tight closure at a computational cost that is asymptotically not worse than the cost of computing all-pairs shortest paths is finally closed.. Future work includes the investigation into such a combination, exploiting the ideas presented in this paper.. In the field of hardware and software verification, the integrality constraint that distinguishes integer-weighted from rational-weighted octagonal graphs can be seen as an abstraction of the more general imposition of a set of congruence relations.. Such a set can be encoded by an element of a suitable abstract domain such as the non-relational congruence domain of [ 10 ] ( that is, of the form x = a ( mod b ) ), the weakly relational zone-congruence domain of [ 17 ] ( that is, also allowing the form x − y = a ( mod b ) ), the linear congruence domain of [ 11 ], and the more general fully relational rational grids domain developed in [ 1 ].
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