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Local operators in topos theory and separation of semi-classical axioms in intuitionistic arithmetic / トポス理論における局所作用素と直観主義算術における準古典的公理の分離Nakata, Satoshi 25 March 2024 (has links)
京都大学 / 新制・課程博士 / 博士(理学) / 甲第25096号 / 理博第5003号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 照井 一成, 教授 大木谷 耕司, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM
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Contributions for Handling Big Data Heterogeneity. Using Intuitionistic Fuzzy Set Theory and Similarity Measures for Classifying Heterogeneous DataAli, Najat January 2019 (has links)
A huge amount of data is generated daily by digital technologies such as
social media, web logs, traffic sensors, on-line transactions, tracking data,
videos, and so on. This has led to the archiving and storage of larger and
larger datasets, many of which are multi-modal, or contain different types
of data which contribute to the problem that is now known as “Big Data”.
In the area of Big Data, volume, variety and velocity problems remain difficult
to solve. The work presented in this thesis focuses on the variety
aspect of Big Data. For example, data can come in various and mixed formats
for the same feature(attribute) or different features and can be identified
mainly by one of the following data types: real-valued, crisp and
linguistic values. The increasing variety and ambiguity of such data are
particularly challenging to process and to build accurate machine learning
models. Therefore, data heterogeneity requires new methods of analysis
and modelling techniques to enable useful information extraction and the
modelling of achievable tasks. In this thesis, new approaches are proposed
for handling heterogeneous Big Data. these include two techniques for filtering
heterogeneous data objects are proposed. The two techniques called
Two-Dimensional Similarity Space(2DSS) for data described by numeric
and categorical features, and Three-Dimensional Similarity Space(3DSS)
for real-valued, crisp and linguistic data are proposed for filtering such data. Both filtering techniques are used in this research to reduce the noise
from the initial dataset and make the dataset more homogeneous. Furthermore,
a new similarity measure based on intuitionistic fuzzy set theory
is proposed. The proposed measure is used to handle the heterogeneity
and ambiguity within crisp and linguistic data. In addition, new combine
similarity models are proposed which allow for a comparison between the
heterogeneous data objects represented by a combination of crisp and linguistic
values. Diverse examples are used to illustrate and discuss the efficiency
of the proposed similarity models. The thesis also presents modification
of the k-Nearest Neighbour classifier, called k-Nearest Neighbour
Weighted Average (k-NNWA), to classify the heterogeneous dataset described
by real-valued, crisp and linguistic data. Finally, the thesis also
introduces a novel classification model, called FCCM (Filter Combined
Classification Model), for heterogeneous data classification. The proposed
model combines the advantages of the 3DSS and k-NNWA classifier and
outperforms the latter algorithm. All the proposed models and techniques
have been applied to weather datasets and evaluated using accuracy, Fscore
and ROC area measures. The experiments revealed that the proposed
filtering techniques are an efficient approach for removing noise from heterogeneous
data and improving the performance of classification models.
Moreover, the experiments showed that the proposed similarity measure
for intuitionistic fuzzy data is capable of handling the fuzziness of heterogeneous
data and the intuitionistic fuzzy set theory offers some promise
in solving some Big Data problems by handling the uncertainties, and the
heterogeneity of the data.
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Mrežno vrednosne intuicionističke preferencijske strukture i primene / Lattice-valued intuitionistic preference structures and applicationsMarija Đukić 24 September 2018 (has links)
<p>Intuicionistički rasplinuti skupovi su već proučavani i definisani u kontekstu mrežnovrednosnih struktura, ali svaka od postojećih definicija imala je odgovarajuće nedostatke. U ovom radu razvijena je definicija intuicionističkog poset-vrednosnog rasplinutog skupa, kojom se poset predstavlja kao podskup distributivne mreže. Na ovaj način možemo ispitivati funkcije pripadanja i nepripadanja i njihove odnose bez upotrebe komplementiranja na posetu. Takođe, u ovako postavljenim okvirima, svaki poset (a samim tim i mreža) može biti kodomen intuicionističkog rasplinutog skupa (čime se isključuje uslov ograničenosti poseta). Primenom uvedene definicije razmatrane su IP-vrednosne rasplinute relacije, x-blokovi ovih relacija i familije<br />njihovih nivoa.Razvijene su jake poset vrednosne relacije reciprociteta koje predstavljaju uopštenje relacija reciprociteta sa intervala [0,1]. Pokazano je da ovakve relacije imaju svojstva slična poset-vrednosnim relacijama preferencije. Međutim, postoje velika ograničenja za primenu ovakvih relacija jer su zahtevi dosta jaki.<br />Uvedene su IP-vrednosne relacije reciprociteta koje se mogu definisati za veliku klasu poseta.Ovakve relacije pogodne su za opisivanje preferencija. Posmatrana je intuicionistička poset-vrednosna relacija preferencije, koja je refleksivna rasplinuta relacija, nad skupom alternativa. U samom procesu višekriterijumskog odlučivanja<br />može se pojaviti situacija kada alternative nisu međusobno uporedive u odnosu na relaciju preferencije, kao i nedovoljna određenost samih alternativa. Da bi se prevazišli ovakvi problemi uvodi se intuicionistička poset-vrednosna relacija preferencije kao intuicionistička rasplinuta relacija na skupu alternativa sa vrednostima u uređenom skupu. Analizirana su neka njena svojstva. Ovakav model pogodan je za upoređivanje alternativa koje nisu, nužno, u linearnom poretku. Dato je nekoliko opravdanja za uvodjenje oba tipa definisanih relacija. Jedna od mogućnosti jeste preko mreže intervala elemenata iz konačnog lanca S, a koji predstavljaju ocene određene alternative. Relacije preferencije mogu uzimati vrednosti sa ove mreže i time se može prevazići nedostatak informacija ili neodlučnost donosioca odluke.</p> / <p>Intuitionistic fuzzy sets have already been explored in depth and defined in the context of lattice-valued intuitionistic fuzzy sets, however, every existing definition has certain drawbacks. In this thesis, a definition of poset-valued intuitionistic fuzzy sets is developed, which introduces a poset as a subset of a distributive lattice. In this manner, functions of membership and non-membership can be examined as well as their relations without using complement in the poset. Also, in such framework, each poset (and the lattice) can be a co-domain of an intuitionistic fuzzy set (which excludes the condition of the bounded poset). Introduced definition defines IP-valued fuzzy relations, x-blocks of these relations andfamilies of their levels. Strong IP-valued reciprocialy relations have been developed as a generalization of reciprocal relations from interval [0,1]. It has been shown that these relations have properties similar to the P-valued preferences relations. However, there are great constraints on the application of these relations because the requirements are quite strong.IP- valued reciprocial relations have been introduced, which can be defined for a large class of posets. Such relations are suitable for describing preferences.An intuitionistic poset-valued preference relation, which is a reflexive fuzzy relation, over the set of alternatives, has been examined. In the process of a multi-criteria decision making, a situation can occur that the alternatives cannot be compared by the preference relation, as well as insufficient determination of the mentioned alternatives. In order to overcome similar problems, we have introduced an intuitionistic poset-valued preference relation as an intuitionistic fuzzy set over the set of alternatives with values in a certain poset. We have analyzed some its performances. This model is suitable for comparing alternatives which are not necessarily linearly ordered. There are several justifications for the introduction of both types of defined relations. One of the possibilities is via the lattice of the intervals of elements from the finite chain S, which represent the preference of a particular alternative. Preferences relations can take values from this lattice and this can overcome the lack of informations or the decisiveness of the decision maker.</p>
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Probabilidades imprecisas: intervalar, fuzzy e fuzzy intuicionistaCosta, Claudilene Gomes da 20 August 2012 (has links)
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Previous issue date: 2012-08-20 / The idea of considering imprecision in probabilities is old, beginning with the Booles
George work, who in 1854 wanted to reconcile the classical logic, which allows the modeling
of complete ignorance, with probabilities. In 1921, John Maynard Keynes in his
book made explicit use of intervals to represent the imprecision in probabilities. But only
from the work ofWalley in 1991 that were established principles that should be respected
by a probability theory that deals with inaccuracies.
With the emergence of the theory of fuzzy sets by Lotfi Zadeh in 1965, there is another
way of dealing with uncertainty and imprecision of concepts. Quickly, they began to propose
several ways to consider the ideas of Zadeh in probabilities, to deal with inaccuracies,
either in the events associated with the probabilities or in the values of probabilities.
In particular, James Buckley, from 2003 begins to develop a probability theory in which
the fuzzy values of the probabilities are fuzzy numbers. This fuzzy probability, follows
analogous principles to Walley imprecise probabilities.
On the other hand, the uses of real numbers between 0 and 1 as truth degrees, as
originally proposed by Zadeh, has the drawback to use very precise values for dealing with
uncertainties (as one can distinguish a fairly element satisfies a property with a 0.423 level
of something that meets with grade 0.424?). This motivated the development of several
extensions of fuzzy set theory which includes some kind of inaccuracy.
This work consider the Krassimir Atanassov extension proposed in 1983, which add
an extra degree of uncertainty to model the moment of hesitation to assign the membership
degree, and therefore a value indicate the degree to which the object belongs to the set
while the other, the degree to which it not belongs to the set. In the Zadeh fuzzy set
theory, this non membership degree is, by default, the complement of the membership
degree. Thus, in this approach the non-membership degree is somehow independent of
the membership degree, and this difference between the non-membership degree and the
complement of the membership degree reveals the hesitation at the moment to assign a
membership degree. This new extension today is called of Atanassov s intuitionistic fuzzy
sets theory. It is worth noting that the term intuitionistic here has no relation to the term
intuitionistic as known in the context of intuitionistic logic.
In this work, will be developed two proposals for interval probability: the restricted
interval probability and the unrestricted interval probability, are also introduced two notions
of fuzzy probability: the constrained fuzzy probability and the unconstrained fuzzy
probability and will eventually be introduced two notions of intuitionistic fuzzy probability:
the restricted intuitionistic fuzzy probability and the unrestricted intuitionistic fuzzy
probability / A id?ia de considerar imprecis?o em probabilidades ? antiga, remontando aos trabalhos
de George Booles, que em 1854 pretendia conciliar a l?gica cl?ssica, que permite
modelar ignor?ncia completa, com probabilidades. Em 1921, John Maynard Keynes em
seu livro fez uso expl?cito de intervalos para representar a imprecis?o nas probabilidades.
Por?m, apenas a partir dos trabalhos de Walley em 1991 que foram estabelecidos
princ?pios que deveriam ser respeitados por uma teoria de probabilidades que lide com
imprecis?es.
Com o surgimento da teoria dos conjuntos fuzzy em 1965 por Lotfi Zadeh, surge uma
outra forma de lidar com incertezas e imprecis?es de conceitos. Rapidamente, come?aram
a se propor diversas formas de considerar as id?ias de Zadeh em probabilidades, para
lidar com imprecis?es, seja nos eventos associados ?s probabilidades como aos valores
das probabilidades.
Em particular, James Buckley, a partir de 2003 come?a a desenvolver uma teoria de
probabilidade fuzzy em que os valores das probabilidades sejam n?meros fuzzy. Esta probabilidade
fuzzy segue princ?pios an?logos ao das probabilidades imprecisas de Walley.
Por outro lado, usar como graus de verdade n?meros reais entre 0 e 1, como proposto
originalmente por Zadeh, tem o inconveniente de usar valores muito precisos para lidar
com incertezas (como algu?m pode diferenciar de forma justa que um elemento satisfaz
uma propriedade com um grau 0.423 de algo que satisfaz com grau 0.424?). Isto motivou
o surgimento de diversas extens?es da teoria dos conjuntos fuzzy pelo fato de incorporar
algum tipo de imprecis?o.
Neste trabalho ? considerada a extens?o proposta por Krassimir Atanassov em 1983,
que adicionou um grau extra de incerteza para modelar a hesita??o ao momento de se
atribuir o grau de pertin?ncia, e portanto, um valor indicaria o grau com o qual o objeto
pertence ao conjunto, enquanto o outro, o grau com o qual n?o pertence. Na teoria dos
conjuntos fuzzy de Zadeh, esse grau de n?o-pertin?ncia por defeito ? o complemento do
grau de pertin?ncia. Assim, nessa abordagem o grau de n?o-pertin?ncia ? de alguma
forma independente do grau de pertin?ncia, e nessa diferencia entre essa n?o-pertin?ncia
e o complemento do grau de pertin?ncia revela a hesita??o presente ao momento de se
atribuir o grau de pertin?ncia. Esta nova extens?o hoje em dia ? chamada de teoria dos
conjuntos fuzzy intuicionistas de Atanassov. Vale salientar, que o termo intuicionista
aqui n?o tem rela??o com o termo intuicionista como conhecido no contexto de l?gica
intuicionista.
Neste trabalho ser? desenvolvida duas propostas de probabilidade intervalar: a probabilidade
intervalar restrita e a probabilidade intervalar irrestrita; tamb?m ser?o introduzidas
duas no??es de probabilidade fuzzy: a probabilidade fuzzy restrita e a probabilidade
fuzzy irrestrita e por fim ser?o introduzidas duas no??es de probabilidade fuzzy intuicionista:
a probabilidade fuzzy intuicionista restrita e a probabilidade fuzzy intuicionista
irrestrita
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Runtime Service Composition via Logic-Based Program SynthesisLämmermann, Sven January 2002 (has links)
No description available.
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Runtime Service Composition via Logic-Based Program SynthesisLämmermann, Sven January 2002 (has links)
No description available.
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Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logicsWallen, Lincoln A. January 1987 (has links)
In this thesis we develop efficient methods for automated proof search within an important class of mathematical logics. The logics considered are the varying, cumulative and constant domain versions of the first-order modal logics K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of these non-classical logics is commonplace within Computing Science and Artificial Intelligence in applications in which efficient machine assisted proof search is essential. Traditional techniques for the design of efficient proof methods for classical logic prove to be of limited use in this context due to their dependence on properties of classical logic not shared by most of the logics under consideration. One major contribution of this thesis is to reformulate and abstract some of these classical techniques to facilitate their application to a wider class of mathematical logics. We begin with Bibel's Connection Calculus: a matrix proof method for classical logic comparable in efficiency with most machine orientated proof methods for that logic. We reformulate this method to support its decomposition into a collection of individual techniques for improving the efficiency of proof search within a standard cut-free sequent calculus for classical logic. Each technique is presented as a means of alleviating a particular form of redundancy manifest within sequent-based proof search. One important result that arises from this anaylsis is an appreciation of the role of unification as a tool for removing certain proof-theoretic complexities of specific sequent rules; in the case of classical logic: the interaction of the quantifier rules. All of the non-classical logics under consideration admit complete sequent calculi. We anaylse the search spaces induced by these sequent proof systems and apply the techniques identified previously to remove specific redundancies found therein. Significantly, our proof-theoretic analysis of the role of unification renders it useful even within the propositional fragments of modal and intuitionistic logic.
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Undecidability of intuitionistic theoriesBrierley, William. January 1985 (has links)
No description available.
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Du fini à l'infini: esquisse d'analyse phénoménologique de l'intuitionisme en mathématiquesLanciani, Albino January 1997 (has links)
Doctorat en philosophie et lettres / info:eu-repo/semantics/nonPublished
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[en] EXTRACTION OF COMPUTATIONAL CONTENTS FROM INTUITIONIST PROOFS / [pt] EXTRAÇÃO DE CONTEÚDO COMPUTACIONAL DE PROVAS INTUICIONISTASGEIZA MARIA HAMAZAKI DA SILVA 10 September 2004 (has links)
[pt] Garantir que programas são implementados de forma a
cumprir uma especificação é uma questão fundamental em
computação, por isso, têm sido propostos vários métodos
que almejam provar a correção dos programas. Este
trabalho apresenta um método, baseado no isomorfismo de
Curry-Howard, que extrai conteúdos computacionais de
provas intuicionistas, conhecido como síntese construtiva
ou proofs-as-programs. É proposto um processo de síntese
construtiva de programas, onde a extração do conteúdo
computacional gera um programa em linguagem imperativa a
partir de uma prova em lógica intuicionista poli-sortida,
cujos axiomas definem os tipos abstratos de dados, sendo
utilizado como sistema dedutivo a Dedução Natural. Também
é apresentada uma prova de correção, bem como uma prova
de completude do método atráves do uso de um sistema com
regra ômega (computacional) para a aritmética de Heyting,
concluindo com uma demonstração da relação entre o uso da
indução finita no lugar da regra ômega computacional no
processo de síntese. / [en] One of the main problems in computer science is to assure
that programs are implemented in such a way that they
satisfy a given specification. There are many studies about
methods to prove correctness of programs. This work
presents a method, belonging to the constructive synthesis
or proofs-as-programs paradigm, that comes from the Curry-
Howard isomorphism and extracts the computational contents
of intuitionist proofs. The synthesis process proposed
produces a program in an imperative language from a proof
in many-sorted intuitionist logic, where the axioms define
the abstract data types using Natural Deduction as
deductive system. It is proved the correctness, as well as
the completeness of the method regarding the Heyting
arithmetic with ômega-rule(in its computational version). A
discussion about the use of the finitary induction instead
of computational ômega-rule concludes the work.
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