456-0349/01 – Mathematical Logic (ML)
| Gurantor department | Department of Computer Science | Credits | 8 |
| Subject guarantor | prof. RNDr. Marie Duží, CSc. | Subject version guarantor | prof. RNDr. Marie Duží, CSc. |
| Study level | undergraduate or graduate | Requirement | Choice-compulsory |
| Year | 1 | Semester | winter |
| | Study language | Czech |
| Year of introduction | 2006/2007 | Year of cancellation | 2009/2010 |
| Intended for the faculties | FEI | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
The goal of the subject is to provide basic principles of logical proof calculi and axiomatic theories, and their application in the area of algebras and theory of lattices. A student should be able to exactly formulate and solve particular problems of computer science and applied mathematics.
Teaching methods
Lectures
Individual consultations
Tutorials
Summary
The course deals with fundamentals of mathematical logic and formal proof calculi. The following main topics are covered: propositional logic, 1st-order predicate logic, 1st-order proof calculi of Gentzen and Hilbert style and general resolution method. These methods are used in many areas of informatics in order to achieve a rigorous formalisation of intuitive theories (automatic theorem proving and deduction, artificial intelligence, and many others).
Compulsory literature:
M.Duží: Mathematical logic.
http://www.cs.vsb.cz/duzi/Mat-logika.html
Z. Manna: Mathematical theory of Computer Science. McGraw-Hill, 1974.
Recommended literature:
Brown, J.R.: Philosophy of Mathematics. Routledge, 1999.
Thayse, A.: From Standard Logic to Logic Programming, John Wiley & Sons, 1988
Nerode, Anil - Shore, Richard A. Logic for applications. New York : Springer-Verlag, 1993. Texts and Monographs in Computer Science.
Richards, T.: Clausal Form Logic. An Introduction to the Logic of Computer Reasoning. Adison-Wesley, 1989.
Bibel, W.: Deduction (Automated Logic). Academia Press, 1993.
Fitting, Melvin. First order logic and automated theorem proving [1996]. 2nd ed. New York : Springer, 1996. Graduate texts in computer science.
Additional study materials
Way of continuous check of knowledge in the course of semester
Conditions for credit:
Three written tests, each of which max. 10 grades.
Maximum grades is thus 30.
Minimum to obtain accreditation 15 grades.
E-learning
Other requirements
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
Lectures:
The notion of relation: Homogeneous and heterogeneous relations. Binary relations and their types. Reflexivity, ireflexivity, symmetry, antisymmetry, asymmetry, tranzitivity.
Mapping as a special type of relation. Complete, total, partial mapping. Surjection, injection, bijection. Semantic exposition of propositional and 1st-order predicate logic.
Basic principles of logical calculi and theories. The notion of a proof, axiom and theorem.
Rezolution method of proving logical validity and validity of an argument. Robinson's unification algorithm. Logical programming.
The natural deduction system (Gentzen). Proof in the system. Soundness and completeness.
Hilbert-like proof calculus. The notion of a proof in the calculus. Theorem of deduction, soundness and completeness.
Axiomatic theories and their properties.
The set theory, relational and algebraic theories. Robinson and Peano arithmetic. (In)completeness, decidability, models.
Gödel theorems and their importance in computer science.
Closure of a relation, equivalence, factor set.
Ordering Relations (partial, complete, quasi-ordering, linear).
General notion of an operation. Algebras and their morphisms.
Fundamentals of the lattice theory.
Exercises:
Deductively valid arguments
Naive theory of sets
Propositional logic, language and semantics
Resolution method in propositional logic
First-order predicate logic, language and semantics
Relation, function, countable and uncountable sets
Semantic tableau
Aristotelova logika
Resolution method in 1st-order predicate logic
Proof calculi: natural deduction and Hilbert calculus
Theory of relations, functions, algebras
Projects:
Solving a given problem by natural deduction and resolution methods
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction