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Logic in Computer Science - Overview

Logic in Computer Science - Overview. 박성우. Sep 1, 2011 POSTECH. Introduction to Logic [ inspired by The Universal Computer, Martin Davis]. Logic. Study of propositions and their use in argumentation (Encyclopædia Britannica) Propositions (A Æ B) ¾ (B Æ A) A Ç : A

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Logic in Computer Science - Overview

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

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

  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. 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. 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. 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. Gottlob Frege [1848 - 1925] • Breakthrough: first-order logic • formal syntax • universal (for all) and existential (some) quantifiers A ::= P(x) | A¾A | AÆA | AÇA | :A | 8x.A | 9x.A • ... shown to be self-contradictory by Russell

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

  11. Model theory Model¼ assignment of truth values Semantic consequenceA1, ¢¢¢, An² C Proof theory Inference rules use premisesto obtain the conclusion Syntactic entailment A1, ¢¢¢, An` C Model Theory vs. Proof Theory

  12. Disjunction & Implication

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

  14. Inference Rules in Proof Theory • With premises • Axioms

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

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

  18. Implication

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

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

  21. Tautology But what is an intuitive interpretation of

  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. 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. Example • Theorem:There are two irrational numbers a and b such that ab is rational. • Proof in classical logic: • Let c = p2p2If c is rational, we take a = b = p2.If c is not rational, we take a = c and b = p2. • Proof in constructive logic: • a lot more involved, but presents a procedure for computing a and b.

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

  26. Welcome tothe world of logic!

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