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An Overview of Logic in Computer Science: Foundations and Applications

This course provides a comprehensive introduction to the principles of logic as they relate to computer science. It covers the evolution of logical theories from Aristotle to modern computation, exploring propositional logic, syllogisms, first-order logic, and proof theory. Students will learn about important figures like Frege and Russell, as well as significant concepts such as Gödel's incompleteness theorem and Turing machines. The course emphasizes constructive proof theory and algorithmic reasoning, equipping students with essential tools and methodologies for logical reasoning in computational contexts.

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An Overview of Logic in Computer Science: Foundations and Applications

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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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