Constraint reasoning with incomplete knowledge: the tolerance propagation approach: Dissertation

Eero Hyvönen

Research output: ThesisDissertationMonograph

2 Citations (Scopus)


In the constraint satisfaction problem (CSP) formulation used in artificial intelligence, the variables related by the constraints may be initially unknown within their domains but are assumed to have some exact value in each solution. This approach leads to representational and computational problems when dealing with incompletely specified, i.e., underconstrained CSPs as the number of solutions explodes combinatorially or becomes infinite with infinite domains. I argue that this problem can be solved by generalizing the CSP formulation into what is called tolerance constraint satisfaction problem. A new reasoning scheme for solving the tolerance CSP, tolerance propagation (TP), has been designed and implemented. The idea of the tolerance approach is to aim at and reason with solutions and value assignments in which variables refer to sets and intervals, i.e., to tolerances, instead of exact values. This makes it possible to represent and reason with incomplete and partly inconsistent knowledge in a well defined way. Additional representational and computational power is obtained without losing the possibility of dealing with exact values that can be represented as singleton tolerances. The tolerance CSP cannot be solved by current constraint satisfaction techniques that are based on searching or synthesizing exact value solutions by constraint propagation or can find only locally consistent solutions. It is shown that the tolerance scheme generalizes current numerical exact value propagation systems and provides a rigid foundation for interval labeling systems based on interval arithmetic. Application of the scheme to propositional logics leads into logical tolerance (constraint) systems that can be used to produce more intuitive results than ordinary two , multiple , or multivalued logics when dealing with incomplete knowledge. In order to amalgamate numerical and logical tolerance reasoning, techniques are developed for representing and solving tolerance CSPs represented in first order predicate logic. The resulting system generalizes current constraint logic programming systems by making it possible to express quantified underconstrained logical problems more freely and to derive generalized tolerance valued solutions. Applications of tolerance systems include numerical equation solving, consistency maintenance systems, support for mixed initiative consultation mode in expert systems, and support for stepwise refinement in problem solving.
Original languageEnglish
QualificationDoctor Degree
Awarding Institution
  • Helsinki University of Technology
Award date15 Feb 1991
Place of PublicationEspoo
Print ISBNs951-38-3928-1
Publication statusPublished - 1991
MoE publication typeG4 Doctoral dissertation (monograph)


  • artificial intelligence
  • reasoning
  • decision making
  • constraints


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