**Previous years:** ZS 2023/24 EN, ZS 2022/23 EN, ZS 2021/22 EN (by Jakub Bulín), ZS 2018/19 EN (by Martin Pilát), ZS 2015/16 EN, ZS 2014/15 EN.

**Lecture in Czech:** ZS 2024/24 CZ (Jakub Bulín).

**Schedule:** Thursday 10:40-12:10, room S3.

**Lecturer:** Petr Gregor (gregor(at)ktiml.mff.cuni.cz)

**Information on the lecture:** 2/2 Ex/C NAIL062 in SIS

**Plan of the lecture**

The plan is only preliminary. I will try to synchronize with the parallel lecture in Czech.

*03.10.*Lecture 1: A brief history, paradoxes, the language of mathematics, relation of syntax and semantics. Basic syntax and semantics of propositional logic. Appendix.*10.10.*Lecture 2: Semantics in a theory, adequacy of logical connectives, normal forms, satisfiability problem, SAT-solvers.*17.10.*Lecture 3: 2-SAT, Horn-SAT, DPLL. Properties and extensions of theories. Algebra of propositions. Tableau method: introduction.*24.10.*Lecture 4: Tableau method: proofs, systematic tableau. Soundness and completeness.*31.10.*Lecture 5: Compactness. Resolution method. Linear resolution, resolution in Prolog.*07.11.*Lecture 6: Hilbert's calculus in propositional logic. Predicate logic: basic syntax, instances and variants. Semantics, models, theories.*14.11.*Lecture 7: Substructures, expansion, reduct. Theorem on constants.*22.11.*Lecture 8: Extensions by definitions. Definability. Tableau method in predicate logic.*28.11.*Lecture 9: Systematic tableau, equality. Soundness. Canonical model, completeness.*05.12.*Lecture 10: Löwenheim-Skolem theorem, compactness. Prenex form, Skolemisation, Herbrand's theorem. Resolution in predicate logic.*12.12.*Lecture 11: Resolution, unification, soundness and completeness. LI-resolution.*19.12.*Lecture 12: Hilbert's calculus in predicate logic. Elementary equivalence, completeness. Isomorphisms, categoricity. Axiomatizability.*09.01.*Lecture 13: Decidable theories, recursive axiomatizations. Undecidability of predicate logic. Fixed-point theorem, undefinability of truth. Incompleteness theorems, corollaries.

**Recommended literature**

- J. Bulín,
*Lecture Notes on Propositional and Predicate Logic*, 2024. - M. Pilát,
*Lecture Notes on Propositional and Predicate Logic*, 2020. - A. Nerode, R. A. Shore,
*Logic for Applications*, Springer, 2nd edition, 1997. - P. Pudlák,
*Logical Foundations of Mathematics and Computational Complexity - A Gentle Introduction*, Springer, 2013. - J. R. Shoenfield,
*Mathematical Logic*, A. K. Peters, 2001. - W. Hodges,
*Shorter Model Theory*, Cambridge University Press, 1997. - W. Rautenberg,
*A concise introduction to mathematical logic*, Springer, 2009.

The lecture is accompanied by tutorials on Thursday 12:20-13:50, room S11 (taught by me), or on Monday 10:40-12:10, room S11 (taught by Martin Pilát).

To obtain a credit from tutorials you need at least 140 points out of max 250 points. There will be two tests during the semester (each for 45 minutes). The first one (in the mid of semester) will cover propositional logic, the other (at the end of semester) will cover predicate logic. For each test you can obtain max 100 points, and in each test you must obtain at least 40 points. Apart from that you can obtain max 40 points for SAT solver project and 2x5 points for two homeworks. At the end of semester it will be possible to write a remedial test for each test (during the first week of the exam period). If you did not score at least 40 points, you will need to take these remedial test. There will be no other chances for remedy.

*03.10.*Tutorial 1: Statements expressed in formulas of various orders.

**Exam**

The exam is oral with written preparation. Requirements for the exam correspond to the syllabus of the course in the extent that has been covered in the lecture. A prerequisite for the exam is the credit from the tutorials. Exam dates will be available in SIS. Details about the exam (questions, format, grading) will be specified later, but it will be similar as in the previous year.

**Consultation hours**

*Thursday from 14:30 or by an (email) appointment.*