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Helgesson, Robert
Publications (7 of 7) Show all publications
Eklund, P., Ángeles Galán, M., Helgesson, R. & Kortelainen, J. (2014). Fuzzy terms. Fuzzy sets and systems (Print), 256, 211-235
Open this publication in new window or tab >>Fuzzy terms
2014 (English)In: Fuzzy sets and systems (Print), ISSN 0165-0114, E-ISSN 1872-6801, Vol. 256, p. 211-235Article in journal (Refereed) Published
Abstract [en]

In this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty; more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures. There are basically two ways of invoking uncertainty for terms. On one hand, we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad. On the other hand, we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty. This is the distinction between 'computing with fuzzy' and 'fuzzy computing' and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic. This paper also shows how the notion of signature often needs to be expanded to levels of signatures, in particular when dealing with type constructors. Such levels allow us to strictly delineate, e.g., primitive operations, type terms, and value level terms. Levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
term monads, quantale, algebra, category theory
National Category
Computer Sciences Mathematics
Identifiers
urn:nbn:se:umu:diva-96490 (URN)10.1016/j.fss.2013.02.012 (DOI)000343783600009 ()
Available from: 2014-12-01 Created: 2014-11-21 Last updated: 2018-06-07Bibliographically approved
Eklund, P., Galán, M. Á., Helgesson, R. & Kortelainen, J. (2013). From Aristotle to Lotfi. In: Seising, Rudolf; Trillas, Enric; Moraga, Claudio; Termini, Settimo (Ed.), On Fuzziness: A Homage to Lotfi A. Zadeh – Volume 1 (pp. 147-152). Berlin Heidelberg: Springer Berlin/Heidelberg, 298
Open this publication in new window or tab >>From Aristotle to Lotfi
2013 (English)In: On Fuzziness: A Homage to Lotfi A. Zadeh – Volume 1 / [ed] Seising, Rudolf; Trillas, Enric; Moraga, Claudio; Termini, Settimo, Berlin Heidelberg: Springer Berlin/Heidelberg, 2013, Vol. 298, p. 147-152Chapter in book (Other academic)
Place, publisher, year, edition, pages
Berlin Heidelberg: Springer Berlin/Heidelberg, 2013
Series
Studies in Fuzziness and Soft Computing
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-66719 (URN)10.1007/978-3-642-35641-4_23 (DOI)978-3-642-35640-7 (ISBN)
Available from: 2013-03-01 Created: 2013-03-01 Last updated: 2018-06-08Bibliographically approved
Helgesson, R. (2013). Generalized General Logics. (Doctoral dissertation). Umeå: Umeå Universitet
Open this publication in new window or tab >>Generalized General Logics
2013 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

Logic as a vehicle for sound reason has a long and lustrous history and while most developments follow the traditional notions of binary truth and crisp sentences, great efforts have been placed into the problem of reasoning with uncertainties. To this end the field of "fuzzy logic" is now of great importance both theoretically and practically. The present monograph seeks to extend and clarify the treatment of non-classical notions of logic and, more broadly, information representation in general. This is done using two theoretical developments presented with additional discussions concerning possible applications.

The first theoretical development takes the form of a novel and strictly categorical term monad that readily allows for a multitude of non-classical situations and extensions of the classical term concept. For example, using monad composition we may represent and perform substitutions over many-valued sets of terms and thereby represent uncertainty of information. As a complementary example, we may extend this term monad to incorporate uncertainty on the level of variables and indeed the operators themselves. These two notions are embodied inthe catchphrases "computing with fuzzy" and "fuzzy computing".

The second theoretical development is a direct generalization of the notion of general logics, a successful categorical framework to describe and interrelate the various concepts included under theumbrella term of 'logic'. The initial leap towards general logics was the introduction of institutions by Goguen and Burstall. This construction cover the semantic aspects of logics and in particular, axiomatizes the crucial satisfaction relation. Adding structures for, e.g., syntactic entailment and proof calculi, Meseguer established general logics as a framework capable of describing a wide range of logics. We will in our generalization further extend general logics to more readily encompass non-classical notions of truth such as in fuzzy logics and logics operating over non-classical notions of sets of sentences.

Abstract [sv]

Logik är ett gammalt och kraftfullt verktyg för att föra korrekta resonemang och människor har under lång tid studerat och tillämpat olika logiska system. Dessa studier har till stor del fokuserat på logik som innefattar ett binärt sanningsbegrepp, det vill säga logik där en utsaga är antingen sann eller falsk. Givetvis är detta en väldigt begränsad vy då vi i vår vardag ofta måste värdera sanning på en betydligt mer flexibel skala. På samma sätt måste vi ofta resonera och fatta beslut utifrån information som på olika sätt kan vara osäker. En djupare förståelse för dessa otraditionella logiker är därför av stort intresse till exempel i expert-system som mekaniskt föreslår beslut utifrån osäker information. I denna monografi beskriver vi två matematiska ramverk som tillåter oss att beskriva dessa typer av otraditionella logiker och otraditionell representation av information i allmänhet. Dessa teoretiska konstruktioner är kompletterade med diskussioner relaterade till praktiska och vidare teoretiska tillämpningar.

Den första av de två teoretiska konstruktionerna är en strikt kategoriteoretiskt definierad termmonad som kan beskriva termer och variabelsubstitution för en mängd olika otraditionella termkonstruktioner. Till exempel, genom monadkomposition kan vi representera flervärda mängder av termer och detta ger oss möjlighet att representera osäker information. Alternativt kan vi konstruera en termmonad som uttrycker osäkerhet på ett mer fundamental sätt genom att termernas ingående variabler och operatörer i sig kan ha associerad osäkerhet.

Den andra teoretiska konstruktionen är en direkt generalisering av det matematiska ramverket "generell logik" som axiomatiserar de olika komponenter som ofta faller under logikbegreppet. I generell logik beskrivs dock dessa utifrån en traditionell vy och för att beskriva otraditionella logiker krävs därför ofta obekväma representationer. Den nämnda generaliseringen syftar därför till att förenkla konstruktionen av mer exotiska logiker inom ett ramverk som till stor del liknar generell logik.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2013
Series
Report / UMINF, ISSN 0348-0542 ; 13.07
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:umu:diva-70127 (URN)978-91-7459-606-9 (ISBN)978-91-7459-605-2 (ISBN)
Public defence
2013-05-28, Mit-huset, MA121, Umeå Universitet, Umeå, 13:00 (English)
Opponent
Supervisors
Note

 

 

Available from: 2013-05-07 Created: 2013-05-06 Last updated: 2018-06-08Bibliographically approved
Eklund, P., Helgesson, R., Galán, M. Á. & Kortelainen, J. (2011). Paradigms for many-sorted non-classical substitutions. In: 2011 41st IEEE International Symposium on Multiple-Valued Logic (ISMVL 2011). Paper presented at 41st IEEE International Symposium on Multiple-Valued Logic (ISMVL), 23-25 May 2011, Tuusula, Finland (pp. 318-321). IEEE Computer Society
Open this publication in new window or tab >>Paradigms for many-sorted non-classical substitutions
2011 (English)In: 2011 41st IEEE International Symposium on Multiple-Valued Logic (ISMVL 2011), IEEE Computer Society, 2011, p. 318-321Conference paper, Published paper (Refereed)
Abstract [en]

We present three paradigms for non-classical substitution in a many-sorted context. Such an exposition has previously been demonstrated in the unsorted case but its extension is far from trivial. The first paradigm, classical many-sorted substitution taking variables to terms, is traditionally presented in a rather informal and "verbal" manner but we find that a strict categorical formulation is necessary to pave the way for non-classical extensions. The second paradigm provides substitution of variables for many-valued sets of terms and relies heavily on functors and monads over the category of indexed sets. Finally, in the third paradigm, we establish full non-classical substitution of many-valued sets of variables by many-valued sets of terms. The third paradigm has the category of many-valued indexed sets as its underlying category. These paradigms ensures transparency of the underlying categories and also makes a clear distinction between set-theoretic operation in the meta language and operations on sets and many-valued sets as found within respective underlying categories.

Place, publisher, year, edition, pages
IEEE Computer Society, 2011
National Category
Computer and Information Sciences
Identifiers
urn:nbn:se:umu:diva-52248 (URN)10.1109/ISMVL.2011.10 (DOI)978-1-4577-0112-2 (ISBN)978-0-7695-4405-2 (ISBN)
Conference
41st IEEE International Symposium on Multiple-Valued Logic (ISMVL), 23-25 May 2011, Tuusula, Finland
Available from: 2012-02-14 Created: 2012-02-14 Last updated: 2018-06-08Bibliographically approved
Eklund, P. & Helgesson, R. (2010). Monadic extensions of institutions. Fuzzy sets and systems (Print), 161(18 (Special issue)), 2354-2368
Open this publication in new window or tab >>Monadic extensions of institutions
2010 (English)In: Fuzzy sets and systems (Print), ISSN 0165-0114, E-ISSN 1872-6801, Vol. 161, no 18 (Special issue), p. 2354-2368Article in journal (Refereed) Published
Abstract [en]

In this paper we show how partially ordered monads can be used to provide a non-classical extension of institutions and entailment systems as appearing in the framework of general logics. General logics is an axiomatization of a general framework for logics. This framework builds upon traditional power sets of sentences as appearing in satisfaction relations and entailment. The underlying power set monad is implicit, and the utility of a monadic machinery was therefore not explored. Making the use of the power set monad more explicit in the satisfaction and entailment relations opens up possibilities to experiment with various non-classical representations of sentences as appearing in inference. These representations are enabled by partially ordered monads, where the partial order attached to the underlying set functor is essential for extending the axiomatization.

Keywords
entailment system, institution, (generalized) general logic, partially ordered monad
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-40007 (URN)10.1016/j.fss.2010.03.002 (DOI)000281273900002 ()
Available from: 2011-02-14 Created: 2011-02-14 Last updated: 2018-06-08Bibliographically approved
Eklund, P. & Helgesson, R. (2009). Composing partially ordered monads. In: Rudolf Berghammer, Ali Mohamed Jaoua, Bernhard Möller (Ed.), Relations and Kleene Algebra in Computer Science: 11th International Conference on Relational Methods in Computer Science, RelMiCS 2009, and 6th International Conference on Applications of Kleene Algebra, AKA 2009, Doha, Qatar, November 1-5, 2009. Proceedings. Paper presented at RelMiCS (pp. 88-102). Springer, 5827
Open this publication in new window or tab >>Composing partially ordered monads
2009 (English)In: Relations and Kleene Algebra in Computer Science: 11th International Conference on Relational Methods in Computer Science, RelMiCS 2009, and 6th International Conference on Applications of Kleene Algebra, AKA 2009, Doha, Qatar, November 1-5, 2009. Proceedings / [ed] Rudolf Berghammer, Ali Mohamed Jaoua, Bernhard Möller, Springer, 2009, Vol. 5827, p. 88-102Conference paper, Published paper (Refereed)
Abstract [en]

Composition of the many-valued powerset partially ordered monad with the term monad provides extensions to non-classical relations and also new examples for Kleene algebras.

Place, publisher, year, edition, pages
Springer, 2009
Series
Lecture Notes in Computer Science, ISSN 0302-9743 ; 5827
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-40003 (URN)10.1007/978-3-642-04639-1_7 (DOI)978-3-642-04638-4 (ISBN)978-3-642-04639-1 (ISBN)
Conference
RelMiCS
Available from: 2011-02-14 Created: 2011-02-14 Last updated: 2018-06-08Bibliographically approved
Eklund, P., Helgesson, R. & Lindgren, H. (2008). Towards refining clinical evidence using general logics. In: Artificial Intelligence and Soft Computing – ICAISC 2008: 9th International Conference Zakopane, Poland, June 22-26, 2008, Proceedings. Paper presented at 9th International Conference Zakopane, Poland, June 22-26, 2008 (pp. 1029-1040). Springer
Open this publication in new window or tab >>Towards refining clinical evidence using general logics
2008 (English)In: Artificial Intelligence and Soft Computing – ICAISC 2008: 9th International Conference Zakopane, Poland, June 22-26, 2008, Proceedings, Springer , 2008, p. 1029-1040Conference paper, Published paper (Refereed)
Abstract [en]

Clinical knowledge building upon evidence-based medicine is typically represented in textual guidelines, thus providing a rather informal description from a logical point of view. Further, the context which provides utility of these guidelines is not specified in any detail with respect to workflow and underlying motivations for decision-making. In addition, the level of detail is mostly static in the sense that measurements and decision values are fixed and intended for specific user groups. There is thus a lack of flexibility which disables knowledge to be shifted coherently between user levels in the entire workflow and decision process. In this paper, we will discuss formalizations of the underlying logical structures of guidelines from the viewpoint of being represented appropriately at each user level. Further, to establish a formal correctness criterion, the shift from one level of representation to another is required to be morphic in the categorical sense. General logics [6] is the selected generalized, and categorical, framework for our approach to flexible guideline representation. Our medical scope is dementia differential diagnosis based on consensus guidelines [2], and we focus on types of cognitive disorders as a prerequisite for further diagnostic tasks.

Place, publisher, year, edition, pages
Springer, 2008
Series
Lecture Notes in Computer Science, ISSN 0302-9743 ; 5097
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:umu:diva-20044 (URN)978-3-540-69572-1 (ISBN)
Conference
9th International Conference Zakopane, Poland, June 22-26, 2008
Available from: 2009-03-15 Created: 2009-03-15 Last updated: 2018-06-09Bibliographically approved
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