1. Logic in Computer Science - Overview Sep 1, 2011 POSTECH 박성우

2. Introduction to Logic [ inspired by The Universal Computer, Martin Davis]

3. 3 Logic Study of propositions and their use in argumentation (Encyclopædia Britannica) –Propositions (A Æ B) ¾ (B Æ A) A Ç : A –Argumentation (A Æ B) ¾ (B Æ A) is true or false? (A Æ B) ¾ (B Æ A) is provable or not provable? Use of a system of symbols for reasoning or deduction

4. 4 Aristotle [384 BC - 322 BC] Syllogisms –inferences from premises to a conclusion –sentences All X are Y.No X are Y. Some X are Y.Some X are not Y. –valid: All X are Y All Y are Z -------------- All X are Z All students are humans All humans are animals -------------------------------------- All students are animals

5. 5 Gottfried Leibniz [1646 - 1716] Inventor of differential and integral calculus –mathematics reduces to manipulating symbols Dream: Calculus ratiocinator –bring human reasoning under mathematical laws –use symbols

6. 6 George Boole [1815 - 1864] Turns logic into (Boolean) algebra L = Joe left his checkbook at the supermarket F = Joe's checkbook was found at the supermarket W = Joe wrote a check at the restaurant last night P = After writing the check last night, Joe put his checkbook in his jacket pocket H = Joe hasn't used his checkbook since last night S = Joe's checkbook is still in his jacket pocket Premises: - If L, then F.- Not F. - W and P.- If W and P and H, then S. - H. Conclusions: - Not L.- S.

7. 7 Gottlob Frege [1848 - 1925] Breakthrough: first-order logic –formal syntax –universal (for all) and existential (some) quantifiers A ::= P(x) | A ¾ A | A Æ A | A Ç A | : A | 8 x.A | 9 x.A... shown to be self-contradictory by Russell

8. 8 Bertrand Russell [1872 - 1970] Coauthored Principia Mathematica Russell's paradox –shows that the set theory by Georg Cantor is inconsistent "a set containing all sets that are not members of themselves" Proposes type theory

9. 9 Further Story Georg Cantor [1845 - 1918] –set theory, diagonal argument David Hilbert [1862 - 1943] –Hilbert's program Kurt Gödel [1906 - 1978] –incompleteness theorem, undecidability Alan Turing [1912 - 1954] –Turing machine, algorithmic unsolvability John von Neumann [1903 - 1957] –von Neumann architecture

10. 10 Outline Methodology –Model theory ( 모델이론 ) –Proof theory ( 증명이론 ) Philosophy –Classical logic –Constructive logic

11. 11 Model Theory vs. Proof Theory Model theory Model ¼ assignment of truth values Semantic consequence A 1, ¢¢¢, A n ² C Proof theory Inference rules –use premises to obtain the conclusion Syntactic entailment A 1, ¢¢¢, A n ` C

12. 12 Disjunction & Implication

13. 13 ) Truth of A is not affected by truth of B.

14. 14 Inference Rules in Proof Theory With premises Axioms

15. 15 Three Types of Systems 1.Hilbert-type system (Axiomatic system) 2.Natural deduction system 3.Sequent calculus

16. 16 1. Hilbert-type System Consists of axioms and Modus Ponens Axioms I : A ¾ A K : A ¾ (B ¾ A) S : (A ¾ (B ¾ C)) ¾ ((A ¾ B) ¾ (A ¾ C)) Inference rule

17. 17 2. Natural Deduction System Introduced by Gentzen, 1934 For each connective Æ, Ç, ¾,... –introduction rule(s) –elimination rule(s)

18. 18 Implication

19. 19 Outline Methodology –Model theory –Proof theory Philosophy –Classical logic ( 고전 논리 ) –Constructive logic ( 건설적 논리, 직관 논리 ) ( ¼ intuitionistic logic)

20. 20 Tautology Intuitive interpretation of ) Truth of A is not affected by truth of B.

21. 21 Tautology But what is an intuitive interpretation of

22. 22 Classical Logic Concerned with: –"whether a given proposition is true or not." Logic from God's point of view –Every proposition is either true or false. Tautologies in classical logic ¼ Logic for mathematics

23. 23 Constructive Logic Concerned with: –"how a given proposition becomes true." Logic from a human's point of view –we know only what we can prove. Not true in constructive logic (for all A and B) ¼ Logic for computer science

24. 24 Example Theorem: There are two irrational numbers a and b such that a b is rational. Proof in classical logic: –Let c = p 2 p 2 If c is rational, we take a = b = p 2. If c is not rational, we take a = c and b = p 2. Proof in constructive logic: –a lot more involved, but presents a procedure for computing a and b.

25. This course is about Constructive Proof Theory. Natural deduction Curry-Howard isomorphism First-order logic Sequent calculus Classical logic Automated theorem proving

26. Welcome to the world of logic!