Culprit Detection on Incremental Tractable Qualitative Temporal Reasoning Problems

......................................................................................... iii List of Figures ........................................................................................ vii Acknowledgement ....................................................................................ix Dedication ............................................................................................x Chapter 1 – Introduction..........................................................................1 1.1 – Problem overview...................................................................1 1.2 – Thesis organization.................................................................1 1.3 – Contribution ...........................................................................2 1.4 – Publications ............................................................................3 Chapter 2 – Background...........................................................................4 2.1 – Temporal reasoning ................................................................4 2.1.1 Point Reasoning ..........................................................4 2.1.2 Interval Reasoning ......................................................5 2.2 – Incremental Temporal Reasoning..........................................10 2.3 – Incremental Complexity issues .............................................11 2.4 – Problem Complexity .............................................................15 v 2.4.1 Point reasoning complexity .......................................15 2.4.2 Interval reasoning complexity ...................................15 Chapter 3 – Problem Establishment ......................................................16 3.1 – Disjunctive Normal Form .....................................................16 3.2 – Conjunctive Normal Form ....................................................16 3.2 – Culprit detection in the point algebra ....................................17 3.2.1 – The FindMinSet Algorithm ....................................20 3.2.2 – A Tractable problem...............................................23 3.2.3 – Implementation ......................................................34 3.3 – Culprit detection in the Interval Problem...............................36 3.3.1 – Reasoning on the whole algebra is NP-Hard ...........36 Chapter 4 – ORD-Horn algebra .............................................................38 4.1 – Definition .............................................................................38 4.2 – Literature..............................................................................39 4.3 – Decomposition into a point algebra problem .........................43 4.3.1 – Positive model........................................................43 4.3.2 – Negative model ......................................................46 4.4 – Algorithms............................................................................48 4.4.1 – Normalization Algorithm .......................................48 4.4.2 – Sorter Algorithm ....................................................55