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Lecturer(s)
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Konečný Jan, doc. RNDr. Ph.D.
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Course content
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Structures of truth degrees: Residuated lattices and their properties. Subclasses of residuated lattices given by identities: MTL-algebras, BL-algebras, MV-algebras, G-algebras, Pi-algebras, and others. Filters on residuated lattices. Subdirect representation of MTL and BL-algebras. Propositional BL-logic and its schematic extensions: Language of BL-logic, formulas, axiomatic systems. Derived logical connectives. Provability, deduction theorem. Soundness and completeness of BL-logic. Schematic extensions, Lukasiewicz logic, Goedel logic, product logic, and their standard completeness. Pavelka's abstract logic: Logic with truth-weighted syntax. Theories as fuzzy sets of formulas. Truth-weighted proofs and provability degrees. General concepts of soundness and completeness in Pavelka-style logics. Examples of Pavelka-complete calculi: propositional Pavelka Rational Logic (RPL) and its completeness (via BL). Predicate BL-logic and further logics: Fuzzy structures and safe interpretations. Completeness of predicate BL-logic. Propositional and predicate MTL-logic. Extension of fuzzy logics by unary connectives (Baaz's delta connective). Fuzzy logic vs. modalities and generalized quantifiers. Fuzzy logic calculi over restricted types of formulas: fuzzy equational logic, fuzzy horn logic, logic of fuzzy attribute implications. Fuzzy structures and their properties: Fuzzy sets and fuzzy relations (in naive sense) as particular fuzzy structures. Properties of fuzzy structures. Representation of fuzzy structures by classical sets (cutlike representation). Special fuzzy relations: similarity and fuzzy equality. Compatibility and similarity preservation. Cutlike semantics.
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Learning activities and teaching methods
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Lecture, Demonstration
- Preparation for the Exam
- 120 hours per semester
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Learning outcomes
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The students become familiar with basic concepts of fuzzy logic.
1. Knowledge Describe and understand comprehensively principles and methods of fuzzy logic.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Active participation in class. Completion of assigned homeworks. Passing the oral (or written) exam.
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Recommended literature
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Bělohlávek R. (2002). Fuzzy Relational Systems: Foundations and Principles. NY: Kluwer Academic/Plenum Press (Vol.20 of IFSR Int. Series on Systems Science and Engineering).
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Bělohlávek R., Vychodil V. (2005). Fuzzy Equational Logic. Springer-Verlag.
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Gerla G. (2001). Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht.
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Gottwald S. (2001). A Treatise on Many-Valued Logics. Taylor & Francis Group.
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Hájek P. (1998). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht.
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Klement E. P., Mesiar R., Pap E. (2000). Triangular Norms. Kluwer, Dordrecht.
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Klir G. J., Yuan B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall.
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