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Spectrum problem
Poláková, Kristýna ; Krajíček, Jan (advisor) ; Jeřábek, Emil (referee)
In the present work we study the spectrum problem that was introduced by H. Scholz in 1952. We define the basic concepts associated with this problem. We follow its further development, especially context with sets from complexity class NE. We defined a generalized spectra. We introduce examples of sets of natural numbers, which are spectra.
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Definability in mathematical structures
Paták, Pavel ; Krajíček, Jan (advisor) ; Jeřábek, Emil (referee)
Nazev pram: Defiuovatelnost v matrnnatickych struktnrneh Auiur: Pave! Patak Kat.odra: Katedra algebry Vedouci bakalafske prace: Prof. R.XDr. Jan Krajicek, DrSc. e-mail vedouciho: krajicek'Q'math.cas.cz AbytrakL: V pfedlo/.ene praci so zabyvame popisem definovatelnych nmozin v ruznych matematickych st.rukturaeh. Ukazujerne, zo defiuovatelne mnoziny v pfirozenych, celych a racionalnich cfslcch inohon byt volico kompliko- vann. naproti toinn dnfiiiovatolnr mnoziny ve .striiktnrach s {'liininari kvanti- fikatoru (reaina, komplexni cfsla,. - . ) JHOU jcdnoduclic. Vciinjoino se i pojinu modolovo I'iplnosti. S poinoci zfskanych poznatku a vo.ty o nplno.sti pak snadno dokazeme nektere obtizne vety jinych disciplin - alji,ebraickou Xnll- stollcnsatz a Artinovn charaktorizaci pozitivnr dofinitnicb racionalnfcli fimkoi, geometrickou Tarski-Seidenbergovn vetn a ninohc dalyi. Klicova slova: nmtematicke sl.rnktnry, dcliiiuvatelnost, eliminace kvantilika- toru Title: DcfinnViility in inatlioinatica.l structures Author: Pavel Patak Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajirek, DrSc. Supervisor's e-mail address: kra.jicokv'iJina.lb.cas.cz Abstract: In t,be present work we study the description of definable sets in various mathematical structures. We show that, the definable sets in natural, integer...
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Intuitionistic logic and axiomatic theories
Brablec, Vladimír ; Švejdar, Vítězslav (advisor) ; Jeřábek, Emil (referee)
This thesis explores some properties of elementary intuitionistic theories. We focus on the following theories: the theory of equality, linear order, dense linear order, the theory of a successor function, Robinson arithmetic and the theory of rational numbers with addition; moreover, we usually deal with two dierent formulations of the theories. As for the properties, our main interest is in the following four: coincidence with the classical version of a theory, saturation, De Jongh's theorem and decidability. The thesis draws especially from the results of C. Smorynski and D. de Jongh and tries to develop them. Some results known for Heyting arithmetic are proved for other theories. We also try to answer the question of what is the eect of replacing an axiom by a dierent (classically equivalent) axiom, or which properties a \good" intuitionistic theory should have.
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Cognitive aspects of survey methodology: the method of immediate validation
Vinopal, Jiří ; Buriánek, Jiří (advisor) ; Jeřábek, Emil (referee) ; Tuček, Milan (referee)
The dissertation summarizes outputs of authors` research of cognitive aspects of survey methodology at the level of their theory as well as of their practical usage. On the one hand is the field of cognitive approaches elaborated systematically on the other the contribution to them is made by own Immediate validation technique. As a lead-in cognitive approaches are put into the frame of survey error that links the standardized questioning theory with general context of survey quality. Following chapters inform about historical sources and basis of cognitive approaches, their fundamental theoretical conceptions and methodological tools. Finally their whole standing in contemporary survey methodology is evaluated. The main attention is devoted to the situation of interviewing that is understood as dualistic: cognitive aspects [information processing and answering process] as well as interactional aspects [social encounter of interviewer/researcher and respondent] are defined and characterized. The course of cognitive processes is defined in accord with prevailing psychological concepts and explication of effects influencing mental processes of respondents and their answering proceeds with the logic: interpretation of a question, retrieval of information from memory, formulating the judgement and formatting...
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Cognitive aspects of survey methodology: the method of immediate validation
Vinopal, Jiří ; Buriánek, Jiří (advisor) ; Jeřábek, Emil (referee) ; Tuček, Milan (referee)
The dissertation summarizes outputs of authors` research of cognitive aspects of survey methodology at the level of their theory as well as of their practical usage. On the one hand is the field of cognitive approaches elaborated systematically on the other the contribution to them is made by own Immediate validation technique. As a lead-in cognitive approaches are put into the frame of survey error that links the standardized questioning theory with general context of survey quality. Following chapters inform about historical sources and basis of cognitive approaches, their fundamental theoretical conceptions and methodological tools. Finally their whole standing in contemporary survey methodology is evaluated. The main attention is devoted to the situation of interviewing that is understood as dualistic: cognitive aspects [information processing and answering process] as well as interactional aspects [social encounter of interviewer/researcher and respondent] are defined and characterized. The course of cognitive processes is defined in accord with prevailing psychological concepts and explication of effects influencing mental processes of respondents and their answering proceeds with the logic: interpretation of a question, retrieval of information from memory, formulating the judgement and formatting...
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An abstract study of completeness in infinitary logics
Lávička, Tomáš ; Noguera, Carles (advisor) ; Jeřábek, Emil (referee) ; Moraschini, Tommaso (referee)
In this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negation
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Spectrum problem
Poláková, Kristýna ; Krajíček, Jan (advisor) ; Jeřábek, Emil (referee)
In the present work we study the spectrum problem that was introduced by H. Scholz in 1952. We define the basic concepts associated with this problem. We follow its further development, especially context with sets from complexity class NE. We defined a generalized spectra. We introduce examples of sets of natural numbers, which are spectra.
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|
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Intuitionistic logic and axiomatic theories
Brablec, Vladimír ; Švejdar, Vítězslav (advisor) ; Jeřábek, Emil (referee)
This thesis explores some properties of elementary intuitionistic theories. We focus on the following theories: the theory of equality, linear order, dense linear order, the theory of a successor function, Robinson arithmetic and the theory of rational numbers with addition; moreover, we usually deal with two dierent formulations of the theories. As for the properties, our main interest is in the following four: coincidence with the classical version of a theory, saturation, De Jongh's theorem and decidability. The thesis draws especially from the results of C. Smorynski and D. de Jongh and tries to develop them. Some results known for Heyting arithmetic are proved for other theories. We also try to answer the question of what is the eect of replacing an axiom by a dierent (classically equivalent) axiom, or which properties a \good" intuitionistic theory should have.
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Definability in mathematical structures
Paták, Pavel ; Jeřábek, Emil (referee) ; Krajíček, Jan (advisor)
Nazev pram: Defiuovatelnost v matrnnatickych struktnrneh Auiur: Pave! Patak Kat.odra: Katedra algebry Vedouci bakalafske prace: Prof. R.XDr. Jan Krajicek, DrSc. e-mail vedouciho: krajicek'Q'math.cas.cz AbytrakL: V pfedlo/.ene praci so zabyvame popisem definovatelnych nmozin v ruznych matematickych st.rukturaeh. Ukazujerne, zo defiuovatelne mnoziny v pfirozenych, celych a racionalnich cfslcch inohon byt volico kompliko- vann. naproti toinn dnfiiiovatolnr mnoziny ve .striiktnrach s {'liininari kvanti- fikatoru (reaina, komplexni cfsla,. - . ) JHOU jcdnoduclic. Vciinjoino se i pojinu modolovo I'iplnosti. S poinoci zfskanych poznatku a vo.ty o nplno.sti pak snadno dokazeme nektere obtizne vety jinych disciplin - alji,ebraickou Xnll- stollcnsatz a Artinovn charaktorizaci pozitivnr dofinitnicb racionalnfcli fimkoi, geometrickou Tarski-Seidenbergovn vetn a ninohc dalyi. Klicova slova: nmtematicke sl.rnktnry, dcliiiuvatelnost, eliminace kvantilika- toru Title: DcfinnViility in inatlioinatica.l structures Author: Pavel Patak Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajirek, DrSc. Supervisor's e-mail address: kra.jicokv'iJina.lb.cas.cz Abstract: In t,be present work we study the description of definable sets in various mathematical structures. We show that, the definable sets in natural, integer...
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