A Mathematical Framework for a General Purp ose Constraint Management System by Steven James Carden Submitted in accordance with the requirements for the degree of Do ctor of Philosophy The University of Leeds Scho ol of Computer Studies June The candidate conrms that the work submitted is his own and that appropriate credit has b een given where reference has b een made to the work of others Abstract The use of constraints in engineering for designing complex mo dels is very p op ular Current constraint solvers are divided into two broad classes general and domain specic Those that are general can handle very general constraint prob lems but are typically slow while those that are domain sp ecic can handle only a sp ecic typ e of problem but are typically fast For example numerical algorithms are slow but general whilst lo cal propagation techniques are fast but limited to simple problems It is generally acknowledged that there is a close coupling b etween engineering constraints and geometric constraints in the design pro cess and so the solution of constraint problems consisting of engineering and geometric constraints is an imp or tant research issue Some authors attempt to overcome the expressive limitations of domain sp ecic solvers by using hybrid systems which try to nd a balance b etween the sp eed of domain sp ecic solvers and the generality of general solvers Previous research at the University of Leeds has led to the development of a num b er of domain sp ecic solvers that are capable of solving geometric and engineering constraint problems separately In particular the Leeds solvers are incremental and can nd solutions when a new constraint is added very quickly This thesis investi gates the use of a hybrid of the various Leeds solvers with an aim to interactively solving constraint problems in engineering design This hybrid would have the sp eed advantages of the domain sp ecic solvers and the expressiveness of a more general solver In order for the hybrid to b e constructed commonalities of existing engineer ing constraint solvers must b e identied A characterisation of existing constraint solvers leads to the identication of a numb er of issues that need to b e addressed b efore the hybrid can b e built In order to examine these issues a framework for the constraint satisfaction process is presented that allows abstractions of constraint denition constraint rep resentation and constraint satisfaction Using the constraint satisfaction framework it is p ossible to study the quality of solution of constraint solvers This leads to the identication of imp ortant problems in current constraint solvers The constraint pro cess framework leads to a study of the use of various paradigms of col laboration within the hybrid such as sequential parallel and concurrent The study of the quality of solution allows concrete statements to b e made ab out the hybrid collab orations A new incremental constraint solver is presented that uses the hybrid collab oration paradigms and provides a rst step towards a p owerful engineering constraint solver i Acknowledgments First of all I would like to thank Pete Dew my sup ervisor Although I was jealous of more ordered pro jects I enjoyed the relatively free hand with just enough rop e to hang myself Pete was usually there with an encouraging word and just enough guidance to keep me on the right track Either that or I nally convinced him that I was right Terrence Fernando my second sup ervisor was also a great supp ort to me as was the whole Virtual Working Environment group esp ecially Martin Thompson Mingxian Fa YungTeng Tsai and Edgard Lamounier The p eople in BGT and the AI lab also kept me comparatively sane and cheerful esp ecially on trips to the Pennines Russ Bubley was a huge help throughout the three years I have known him and shared an oce with him We continually b ounced ideas o each other and whilst I tried not to b e to o ignorant of what he was talking ab out he usually managed to put me straight on my pro ject His capitulation at squash was also particularly gratifying My housemates Mark Fred and Stuart were great fun to b e with and weve had some great parties My parents and siblings Jim Carol Neil and Claire put up with many a demon stration at the dinner table ab out what constraints actually were I think theyre still none the wiser And nally I would like to thank Alice who b esides b eing wonderful and sym pathetic all the way through also pro ofread my thesis even though it must have sounded like gobbledigo ok to her ii Contents Intro duction Ob jectives of this thesis Incremental constraint solvers Thesis organisation Related Work The theory of constraints Dimensions Decomp osition of constraint problems Hybrid constraint solvers Solution spaces Constraints in engineering design Constraint solvers General constraint solvers Numerical solvers Symb olic solvers Finite domain constraint solvers Backtracking Forwardchecking Other nite domain research Geometric constraint solvers Underconstrained geometric constraint solvers Wellconstrained geometric constraint solvers Overconstrained geometric constraint solvers Functional constraint solvers Overconstrained functional constraint solvers Maintenance and physical constraint solvers Conclusions iii Solving Problems by Decomp osition Examples of current constraint solvers DCM INCES IGCS Connectivity Analysis Decomp osition strategies Examples of decomp osition strategies Decomp osition to domain sp ecic subproblems Advantages of decomp osition strategies Limitations of decomp osition strategies Incremental issues in decomp osition strategies Conclusions Ordering strategies Examples of ordering strategies Ordering strategies for a constraint solver Incremental issues in ordering strategies Conclusions Solution of subproblems Examples of solution of subproblems Solving using domain sp ecic knowledge Incremental issues in solving subproblems Conclusions Conclusions Constraint Denition Entities Constraints Constraint problems Constraint solvers Dimensions Denition of dimensions Constrainedness Conclusions iv Constraint Representation Representing entities and constraints Finitedomain entities and constraints Innitedomain entities and constraints Representing constraint problems Example constraint representation schemes Algebraic representation Relationship graph representation Undirected graph representation Hyp ergraph representation Bipartite representation Valid representation schemes Reductions Conclusions Constraint Satisfaction Constraint solution Solution spaces A framework for the solution pro cess Solution steps Prop erties of solution steps Solution pro cesses Solution pro cesses always head towards a solution Solution pro cess prop erties Using lo cal prop erties to draw conclusions ab out pro cesses Consequences of the Lo calGlobal Theorem Enrichment of the constraint satisfaction framework Constraint priorities Variabledriven satisfaction Backtracking Incremental satisfaction Conclusions Hybrid Collab oration Using domain sp ecic knowledge in constraint solvers Using domain sp ecic knowledge is fast Using domain sp ecic knowledge is not enough v Hybrid constraint solvers BALI Enhanced solution spaces A simple example hybrid constraint solver Paradigms of collab oration Sequential hybrids Limitations of serial hybrids Solver collab oration language An example of many solvers in serial Case study The solvers used Results Conclusions New Directions Decomp osition strategy Ordering strategy Solution and recombination Advantages of the ErepIGCS hybrid Limitations of the ErepIGCS hybrid Incremental implications of new solver .