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Categorical Unification
Umeå University, Faculty of Science and Technology, Computing Science.
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with different aspects towards many-valued unification which have been studied in the scope of category theory. The main motivation of this investigation comes from the fact that in logic programming, classical unification has been identified as the provision of coequalizers in Kleisli categories of term monads. Continuing in that direction, we have used categorical instrumentations to generalise the classical concept of a term. It is expected that this approach will provide an appropriate formal framework for useful developments of generalised terms as a basis for many-valued logic programming involving an extended notion of terms.

As a first step a concept for generalised terms has been studied. A generalised term is given by a composition of monads that again yields a monad, i.e. compositions of powerset monads with the term monad provide definitions for generalised terms. A composition of monads does, however, not always produce a monad. In this sense, techniques for monads composition provide a helpful tool for our concerns and therefore the study of these techniques has been a focus of this research.

The composition of monads make use of a lot of equations. Proofs become complicated, not to mention the challenge of understanding different steps of the equations. In this respect, we have studied visual techniques and show how a graphical approach can provide the support we need.

For the purpose of many-valued unification, similarity relations, generalised substitutions and unifiers have been defined for generalised terms.

Place, publisher, year, edition, pages
2004. , 91 p.
Series
Report / UMINF, ISSN 0348-0542 ; 04.09
Keyword [en]
Datalogi, Monad compositions, generalised terms, many-valued logic
Keyword [sv]
Datalogi
National Category
Computer Science
Research subject
Computing Science
Identifiers
URN: urn:nbn:se:umu:diva-245ISBN: 91-7305-642-1 (print)OAI: oai:DiVA.org:umu-245DiVA: diva2:142802
Public defence
2004-05-24, MA121, MIT-huset, Umeå University, Umeå, 13:15
Opponent
Supervisors
Available from: 2004-04-22 Created: 2004-04-22Bibliographically approved
List of papers
1. Setfunctors, L-fuzzysetcategories, and generalizedterms
Open this publication in new window or tab >>Setfunctors, L-fuzzysetcategories, and generalizedterms
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2002 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, Vol. 43, no 6-7, 693-705 p.Article in journal (Refereed) Published
Abstract [en]

In this work, we generalize previous constructions of fuzzysetcategories, introduced in [1], by consideringL-fuzzysets in which the values of the characteristic functions run on a completely distributive lattice, rather than in the unit real interval. Later, these L-fuzzysets are used to define the L-fuzzycategories, which are proven to be rational. In the final part of the paper, the L-fuzzyfunctors given by the extension principles are provided with a structure of monad which is used, together with the functorial definition of theterm monad, to provide monad compositions as a basis for a notion of generalisedterms.

Place, publisher, year, edition, pages
Elsevier, 2002
Keyword
Extension principles, Generalised terms, Monad compositions
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-3898 (URN)10.1016/S0898-1221(01)00314-5 (DOI)
Available from: 2004-04-22 Created: 2004-04-22 Last updated: 2012-05-14Bibliographically approved
2. A graphical approach to monad compositions
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2001 In: Electronic Notes in Theoretical Computer Science, Vol. 40, 16 pages- p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-3899 (URN)
Available from: 2004-04-22 Created: 2004-04-22Bibliographically approved
3. Powersets of Terms and Composite Monads
Open this publication in new window or tab >>Powersets of Terms and Composite Monads
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2004 (English)In: Technical Report, UMINF, ISSN 0348-0542, no 7Article in journal (Refereed) Published
Abstract [en]

Composing various powerset functors with the term monad gives rise to the concept of generalised terms. This in turn provides a technique for handling many-valued sets of terms in a framework of variable substitutions, thus being the prerequisite for categorical unification in many-valued logic programming using an extended notion of terms. As constructions of monads involve complicated calculations with natural transformations, proofs are supported by a graphical approach that provides a useful tool for handling various conditions, such as those well-known for distributive laws.

Place, publisher, year, edition, pages
Umeå: Department of Computing Science, 2004
Keyword
Composite monads, generalised terms, many-valued logic
Identifiers
urn:nbn:se:umu:diva-3900 (URN)
Available from: 2004-04-22 Created: 2004-04-22 Last updated: 2011-04-08Bibliographically approved
4. A categorical approach to unification of generalised terms
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2002 In: Electronic Notes in Theoretical Computer Science, Vol. 66, no 5, 11 pages- p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-3901 (URN)
Available from: 2004-04-22 Created: 2004-04-22Bibliographically approved
5. Similarities between powersets of terms
Open this publication in new window or tab >>Similarities between powersets of terms
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2004 (English)In: Fuzzy sets and systems (Print), ISSN 0165-0114, Vol. 144, no 1, 213-225 p.Article in journal (Refereed) Published
Abstract [en]

Generalisation of the foundational basis for many-valued logic programming builds upon generalised terms in the form of powersets of terms. A categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions and compositions thereof. In this paper, substitutions and unifiers appear as constructs in Kleisli categories related to particular composed powerset term monads. Specifically, we show that a frequently used similarity-based approach to fuzzy unification is compatible with the categorical approach, and can be adequately extended in this setting; also some examples are included in order to illuminate the definitions.

Place, publisher, year, edition, pages
Elsevier, 2004
Keyword
Similarities, Fuzzy unification, Category theory and unification, Generalised terms
National Category
Computer and Information Science
Identifiers
urn:nbn:se:umu:diva-3902 (URN)10.1016/j.fss.2003.10.021 (DOI)
Available from: 2004-04-22 Created: 2004-04-22 Last updated: 2012-06-14Bibliographically approved

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