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Implicit Real Vector Automata Degbomont, Jean-François Doctoral thesis (2013) This thesis introduces a new data structure, the Implicit Real Vector Automaton (IRVA), suited for representing symbolically polyhedra, i.e., regions of n-dimensional space defined by finite Boolean ... [more ▼] This thesis introduces a new data structure, the Implicit Real Vector Automaton (IRVA), suited for representing symbolically polyhedra, i.e., regions of n-dimensional space defined by finite Boolean combinations of linear inequalities. IRVA can represent exactly arbitrary convex and non-convex polyhedra, including features such as open and closed boundaries, unconnected parts, and non-manifold components. In addition, they provide efficient procedures for deciding whether a point belongs to a given polyhedron, and determining the polyhedron component (vertex, edge, facet, ...) that contains a point. An advantage of IRVA is that they can easily be minimized into a canonical form, which leads to a simple and efficient test for equality between represented polyhedra. Elementary IRVA representing primitive polyhedra, such as linear (in)equations and vector spaces are easily constructed and algorithms have been developed for computing Boolean combinations as well as projections of polyhedra represented by IRVA. These algorithms are illustrated by complete examples of executions as a support for the comprehension of their mechanisms. Another contribution is a first prototype implementation of an IRVA library, containing functions for building and manipulating arbitrary n-dimensional polyhedra. We reinforce the presentation of the implementation by discussing some design choices. Such choices include the use of exact arithmetic. Finally, experimental results are presented and discussed. These experiments pave the way to future adaptations and improvements. [less ▲] Detailed reference viewed: 81 (36 ULg)Automata-Based Symbolic Representations of Polyhedra Boigelot, Bernard ; Brusten, Julien ; Degbomont, Jean-François in Lecture Notes in Computer Science (2012, March), 7183 Detailed reference viewed: 67 (23 ULg)Implicit Real Vector Automata Boigelot, Bernard ; Brusten, Julien ; Degbomont, Jean-François in Electronic Proceedings in Theoretical Computer Science [=EPTCS] (2010), 39 This paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data ... [more ▼] This paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data structure for representing such sets, based on an implicit and concise encoding of a known structure, the Real Vector Automaton. The resulting formalism provides a canonical representation of polyhedra, is closed under Boolean operators, and admits an efficient decision procedure for testing the membership of a vector. [less ▲] Detailed reference viewed: 103 (41 ULg)Partial Projection of Sets Represented by Finite Automata, with Application to State-Space Visualization Boigelot, Bernard ; Degbomont, Jean-François in Lecture Notes in Computer Science (2009), 5457 Detailed reference viewed: 56 (29 ULg) |
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