Weighted constraint satisfaction with set variables.

Siu Fai Keung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 79-83).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- (Weighted) Constraint Satisfaction --- p.1Chapter 1.2 --- Set Variables --- p.2Chapter 1.3 --- Motivations and Goals --- p.3Chapter 1.4 --- Overview of the Thesis --- p.4Chapter 2 --- Background --- p.6Chapter 2.1 --- Constraint Satisfaction Problems --- p.6Chapter 2.1.1 --- Backtracking Tree Search --- p.8Chapter 2.1.2 --- Consistency Notions --- p.10Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.14Chapter 2.2.1 --- Branch and Bound Search --- p.17Chapter 2.2.2 --- Consistency Notions --- p.19Chapter 2.3 --- Classical CSPs with Set Variables --- p.23Chapter 2.3.1 --- Set Variables and Set Domains --- p.24Chapter 2.3.2 --- Set Constraints --- p.24Chapter 2.3.3 --- Searching with Set Variables --- p.26Chapter 2.3.4 --- Set Bounds Consistency --- p.27Chapter 3 --- Weighted Constraint Satisfaction with Set Variables --- p.30Chapter 3.1 --- Set Variables --- p.30Chapter 3.2 --- Set Domains --- p.31Chapter 3.3 --- Set Constraints --- p.31Chapter 3.3.1 --- Zero-arity Constraint --- p.33Chapter 3.3.2 --- Unary Constraints --- p.33Chapter 3.3.3 --- Binary Constraints --- p.36Chapter 3.3.4 --- Ternary Constraints --- p.36Chapter 3.3.5 --- Cardinality Constraints --- p.37Chapter 3.4 --- Characteristics --- p.37Chapter 3.4.1 --- Space Complexity --- p.37Chapter 3.4.2 --- Generalization --- p.38Chapter 4 --- Consistency Notions and Algorithms for Set Variables --- p.41Chapter 4.1 --- Consistency Notions --- p.41Chapter 4.1.1 --- Element Node Consistency --- p.41Chapter 4.1.2 --- Element Arc Consistency --- p.43Chapter 4.1.3 --- Element Hyper-arc Consistency --- p.43Chapter 4.1.4 --- Weighted Cardinality Consistency --- p.45Chapter 4.1.5 --- Weighted Set Bounds Consistency --- p.46Chapter 4.2 --- Consistency Enforcing Algorithms --- p.47Chapter 4.2.1 --- "Enforcing Element, Node Consistency" --- p.48Chapter 4.2.2 --- Enforcing Element Arc Consistency --- p.51Chapter 4.2.3 --- Enforcing Element Hyper-arc Consistency --- p.52Chapter 4.2.4 --- Enforcing Weighted Cardinality Consistency --- p.54Chapter 4.2.5 --- Enforcing Weighted Set Bounds Consistency --- p.56Chapter 5 --- Experiments --- p.59Chapter 5.1 --- Modeling Set Variables Using 0-1 Variables --- p.60Chapter 5.2 --- Softening the Problems --- p.61Chapter 5.3 --- Steiner Triple System --- p.62Chapter 5.4 --- Social Golfer Problem --- p.63Chapter 5.5 --- Discussions --- p.66Chapter 6 --- Related Work --- p.68Chapter 6.1 --- Other Consistency Notions in WCSPs --- p.68Chapter 6.1.1 --- Full Directional Arc Consistency --- p.68Chapter 6.1.2 --- Existential Directional Arc Consistency --- p.69Chapter 6.2 --- Classical CSPs with Set Variables --- p.70Chapter 6.2.1 --- Bounds Reasoning --- p.70Chapter 6.2.2 --- Cardinality Reasoning --- p.70Chapter 7 --- Concluding Remarks --- p.72Chapter 7.1 --- Contributions --- p.72Chapter 7.2 --- Future Work --- p.74List of Symbols --- p.76Bibliography --- p.7

Constraint solving over multi-valued logics - application to digital circuits

Due to usage conditions, hazardous environments or intentional causes, physical and virtual systems are subject to faults in their components, which may affect their overall behaviour. In a ‘black-box’ agent modelled by a set of propositional logic rules, in which just a subset of components is externally visible, such faults may only be recognised by examining some output function of the agent. A (fault-free) model of the agent’s system provides the expected output given some input. If the real output differs from that predicted output, then the system is faulty. However, some faults may only become apparent in the system output when appropriate inputs are given. A number of problems regarding both testing and diagnosis thus arise, such as testing a fault, testing the whole system, finding possible faults and differentiating them to locate the correct one. The corresponding optimisation problems of finding solutions that require minimum resources are also very relevant in industry, as is minimal diagnosis. In this dissertation we use a well established set of benchmark circuits to address such diagnostic related problems and propose and develop models with different logics that we formalise and generalise as much as possible. We also prove that all techniques generalise to agents and to multiple faults. The developed multi-valued logics extend the usual Boolean logic (suitable for faultfree models) by encoding values with some dependency (usually on faults). Such logics thus allow modelling an arbitrary number of diagnostic theories. Each problem is subsequently solved with CLP solvers that we implement and discuss, together with a new efficient search technique that we present. We compare our results with other approaches such as SAT (that require substantial duplication of circuits), showing the effectiveness of constraints over multi-valued logics, and also the adequacy of a general set constraint solver (with special inferences over set functions such as cardinality) on other problems. In addition, for an optimisation problem, we integrate local search with a constructive approach (branch-and-bound) using a variety of logics to improve an existing efficient tool based on SAT and ILP

Constraint propagation in Mozart

This thesis presents constraint propagation in Mozart which is based on computational agents called propagators. The thesis designs, implements, and evaluates propagator-based propagation engines. A propagation engine is split up in generic propagation services and domain specific domain solvers which are connected by a constraint programming interface. Propagators use filters to perform constraint propagation. The interface isolates filters from propagators such that they can be shared among various systems. This thesis presents the design and implementation of a finite integer set domainsolver for Mozart which reasons over bound and cardinality approximations of sets.The solver cooperates with a finite domain solver to improve its propagation and expressiveness. This thesis promotes constraints to first-class citizens and thus, provides extra control over constraints. Novel programming techniques taking advantage of the first-class status of constraints are developed and illustrated.Diese Dissertation beschreibt Constraint-Propagierung in Mozart, die auf Berechnungsagenten, Propagierer genannt, basiert. Die Dissertation entwirft, implementiert und evaluiert Propagierer-basierte Propagierungsmaschinen. Eine Propagierungsmaschine ist aufgeteilt in generische Propagierungsdienste und domänenspezifische Domänenlöser, die durch eine Schnittstelle zur Constraint-Programmierung miteinander verbunden sind. Propagierer benutzen Filter, um Constraints zu propagieren. Die Schnittstelle isoliert Filter von Propagierern, so dass Programmkodes von Filtern von verschiedenen Systemen genutzt werden können. Diese Dissertation präsentiert den Entwurf und die Implementierung eines Domänenlösers über endliche Mengen von ganzen Zahlen für Mozart, die über Mengen- und Kardinalitätsschranken approximiert werden. Dieser kooperiert mit einem Löser über endlichen Bereichen, um die Propagierung und die Ausdrucksfähigkeit zu verbessern. Diese Dissertation erhebt Constraints zu emanzipierten Datenstrukturen und stellt auf dieseWeise zusätzliche Steuerungsmöglichkeiten über Constraints zur Verfügung. Des Weiteren werden neuartige Programmiertechniken für emanzipierte Constraints entwickelt und demonstriert

Constraint propagation in Mozart

This thesis presents constraint propagation in Mozart which is based on computational agents called propagators. The thesis designs, implements, and evaluates propagator-based propagation engines. A propagation engine is split up in generic propagation services and domain specific domain solvers which are connected by a constraint programming interface. Propagators use filters to perform constraint propagation. The interface isolates filters from propagators such that they can be shared among various systems. This thesis presents the design and implementation of a finite integer set domainsolver for Mozart which reasons over bound and cardinality approximations of sets.The solver cooperates with a finite domain solver to improve its propagation and expressiveness. This thesis promotes constraints to first-class citizens and thus, provides extra control over constraints. Novel programming techniques taking advantage of the first-class status of constraints are developed and illustrated.Diese Dissertation beschreibt Constraint-Propagierung in Mozart, die auf Berechnungsagenten, Propagierer genannt, basiert. Die Dissertation entwirft, implementiert und evaluiert Propagierer-basierte Propagierungsmaschinen. Eine Propagierungsmaschine ist aufgeteilt in generische Propagierungsdienste und domänenspezifische Domänenlöser, die durch eine Schnittstelle zur Constraint-Programmierung miteinander verbunden sind. Propagierer benutzen Filter, um Constraints zu propagieren. Die Schnittstelle isoliert Filter von Propagierern, so dass Programmkodes von Filtern von verschiedenen Systemen genutzt werden können. Diese Dissertation präsentiert den Entwurf und die Implementierung eines Domänenlösers über endliche Mengen von ganzen Zahlen für Mozart, die über Mengen- und Kardinalitätsschranken approximiert werden. Dieser kooperiert mit einem Löser über endlichen Bereichen, um die Propagierung und die Ausdrucksfähigkeit zu verbessern. Diese Dissertation erhebt Constraints zu emanzipierten Datenstrukturen und stellt auf dieseWeise zusätzliche Steuerungsmöglichkeiten über Constraints zur Verfügung. Des Weiteren werden neuartige Programmiertechniken für emanzipierte Constraints entwickelt und demonstriert

Modelling digital circuits problems with set constraints

Abstract. A number of diagnostic and optimisation problems in Electronics Computer Aided Design have usually been handled either by specific tools or by mapping them into a general problem solver (e.g. a propositional Boolean SAT tool). This approach, however, requires models with substantial duplication of digital circuits. In Constraint Logic Programming, the use of extra values in the digital signals (other than the usual 0/1) was proposed to reflect their dependency on some faulty gate. In this paper we present an extension of this modelling approach, using set variables to denote dependency of the signals on sets of faults, to model different circuits problems. We then show the importance of propagating constraints on sets cardinality, by comparing Cardinal, a set constraint solver that we implemented, with a simpler version that propagates these constraints similarly to Conjunto, a widely available set constraint solver. Results show speed ups of Cardinal of about two orders of magnitude, on a set of diagnostic problems. 1