Lecturer(s)


Vychodil Vilém, doc. RNDr. Ph.D.

Course content

Structures of truth degrees: Residuated lattices and their properties. Subclasses of residuated lattices given by identities: MTLalgebras, BLalgebras, MValgebras, Galgebras, Pialgebras, and others. Filters on residuated lattices. Subdirect representation of MTL and BLalgebras. Propositional BLlogic and its schematic extensions: Language of BLlogic, formulas, axiomatic systems. Derived logical connectives. Provability, deduction theorem. Soundness and completeness of BLlogic. Schematic extensions, Lukasiewicz logic, Goedel logic, product logic, and their standard completeness. Pavelka's abstract logic: Logic with truthweighted syntax. Theories as fuzzy sets of formulas. Truthweighted proofs and provability degrees. General concepts of soundness and completeness in Pavelkastyle logics. Examples of Pavelkacomplete calculi: propositional Pavelka Rational Logic (RPL) and its completeness (via BL). Predicate BLlogic and further logics: Fuzzy structures and safe interpretations. Completeness of predicate BLlogic. Propositional and predicate MTLlogic. 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.

Learning activities and teaching methods

Lecture, Demonstration
 Preparation for the Exam
 120 hours per semester

Learning outcomes

The students become familiar with basic concepts of fuzzy logic.
1. Knowledge Describe and understand comprehensively principles and methods of fuzzy logic.

Prerequisites

unspecified

Assessment methods and criteria

Oral exam, Written exam
Active participation in class. Completion of assigned homeworks. Passing the oral (or written) exam.

Recommended literature


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).

Bělohlávek R., Vychodil V. (2005). Fuzzy Equational Logic. SpringerVerlag.

Gerla G. (2001). Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht.

Gottwald S. (2001). A Treatise on ManyValued Logics. Taylor & Francis Group.

Hájek P. (1998). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht.

Klement E. P., Mesiar R., Pap E. (2000). Triangular Norms. Kluwer, Dordrecht.

Klir G. J., Yuan B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. PrenticeHall.
