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CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science

CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science. Slides adapted from Michael P. Frank ' s course based on the text Discrete Mathematics & Its Applications (5 th Edition) by Kenneth H. Rosen. Proof – Natural Deduction. Topics.

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CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science

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  1. CS2013Mathematics for Computing ScienceAdam WynerUniversity of AberdeenComputing Science Slides adapted from Michael P. Frank's course based on the textDiscrete Mathematics & Its Applications(5th Edition)by Kenneth H. Rosen

  2. Proof – Natural Deduction

  3. Topics • What is proofand why? • How with rules and examples • Proof strategies – direct, contrapositive, and contradiction. Frank / van Deemter / Wyner

  4. Nature of Proofs • In mathematics and logic, a proof is: • An argument (sequence of statements) that rigorously (systematically, formally) establishes the truth of a statement given premises and rules. Frank / van Deemter / Wyner

  5. Importance of Proofs • Given a specification of some domain (facts and rule) • What can be inferred? • Are there any contradictions? • Are there undesirable inferences? • Do we have all the consequences? Frank / van Deemter / Wyner

  6. Symbolic Reasoning • Start with some logical formulas that you want to use in your proof (premises and rules) • Identify what you want to prove (a conclusion) • Use reasoning templates and equivalences to transform formulas from your start formulas till you get what you want to prove. • Skill in knowing the templates and equivalences. Frank / van Deemter / Wyner

  7. Proofs in Programming • Applies in program verification, computer security, automated reasoning systems, parsing, etc. • Allows us to be confident about the correctness of a specification. • Discovers flaws (e.g., a reason why the program is not correct or not accurate). • Not doing proofs of programming (yet). • Oracle Policy Modelling proves determinations from input information. Frank / van Deemter / Wyner

  8. Deductive Calculi • There exist various precise calculi for proving theorems in logic. For example • Natural Deduction • Axiomatic approaches • Semantic tableaus ("proof in trees") • Look at Natural Deduction, which is characterised by the use of inference rules. • Look at Axiomatic proof, which is characterised by the use of axioms to substitute expressions. Frank / van Deemter / Wyner

  9. Proof Terminology • Premises • statements that are often unproven and assumed. • Conclusion • a statement that follows from premises and an inference rule • Rules of inference • Patterns of reasoning from premises to conclusions. • Theorem • A statement that has been proven to be true. Frank / van Deemter / Wyner

  10. More Proof Terminology • Lemma • a minor theorem used as a stepping-stone to proving a major theorem. • Corollary • a minor theorem proved as an easy consequence of a major theorem. • Conjecture • a statement whose truth value has not been proven, but may be believed to be true. Frank / van Deemter / Wyner

  11. Inference Rules - General Form • An Inference Rule is • A reasoning pattern (template) such that if we know (accept, agree, believe) that a set of premises are all true, then we deduce (infer) that a certain conclusion statement must also be true. • premise 1 premise 2 …  conclusion “” means “therefore” Different forms, names, etc to present this.... Frank / van Deemter / Wyner

  12. Inference Rules 1 • Double negative elimination (DNE) • From ¬ ¬ φ, we infer φ • From "It is not the case that Bill is not happy", we infer "Bill is happy". • Conjunction introduction (CI) • From φand ψ, we infer ( φ∧ψ ). • From "Bill is happy" and "Jill is happy", we infer "Bill is happy and Jill is happy". order of conjunction and disjunction does not matter. Frank / van Deemter / Wyner

  13. Inference Rules 2 • Conjunction elimination (CE) • From ( φ∧ψ ), we infer φ and ψ • From "Bill is happy and Jill is happy", we infer "Bill is happy" (and also "Jill is happy"). • Disjunction introduction (DI) • From φ, we infer (φ∨ψ). • From "Bill is happy", we infer "Bill is happy or Jill is happy". Frank / van Deemter / Wyner

  14. Inference Rules 3 • Disjunction elimination (DE) • From ¬ φ and (φ∨ψ), we infer ψ • From "Bill is not happy" and "Bill is happy or Jill is happy", we infer "Jill is happy". • Implication elimination (Modus ponens – MP) • From φ and ( φψ ), we infer ψ. • From "Bill is happy" and "If Bill is happy, then Jill is happy", we infer "Jill is happy". Frank / van Deemter / Wyner

  15. Inference Rules 4 • Implication elimination (Modus tollens- MT) • From ¬ ψ and ( φψ ), we infer ¬ φ. • From "Bill is not happy" and "If Bill is happy, then Jill is happy", we infer "Jill is not happy". • Hypothetical syllogism (HS) • ( φψ ) and (ψ β ), we infer (φ β ) • From "If Bill is happy, then Jill is happy" and "If Jill is happy, then Mary is happy", we infer "If Bill is happy, then Mary is happy". Frank / van Deemter / Wyner

  16. Inference Rules - Tautologies • Each valid logical inference rule corresponds to an implication that is a tautology. • From premise 1, premise 2 …, it follows conclusion • Corresponding tautology: ((premise 1)  (premise 2)  …) conclusion • Demonstrate with a T-table. Frank / van Deemter / Wyner

  17. Modus Ponens T-table Proof that the reasoning template is a tautology. Other reasoning templates can be demonstrated similarly. Frank / van Deemter / Wyner

  18. Validity and truth • We say that a proof method is valid if it can never lead from true premises to a false conclusion. • You see a valid proof, one of whose premises is false.  Conclusion may be true of false. • You see an invalid proof.  Conclusion may be true of false. • You see a valid proof, whose premises are true  Conclusion must be true Frank / van Deemter / Wyner

  19. Fallacies • Afallacy is an inference rule or other proof method that may yield a false conclusion. • Fallacy of affirming the conclusion: • “pq is true, and q is true, so p must be true.” (No, because FT is true.) • Fallacy of denying the hypothesis: • “pq is true, and p is false, so q must be false.” (No, again because FT is true.) Frank / van Deemter / Wyner

  20. "Invalid" Reasoning Patterns • Argumentation templates used in everyday reasoning: • Bill is in a position to know whether or not Jill is happy. • Bill asserts "Jill is happy". • Therefore, Jill is happy. • Problem is that being in a position to know something and asserting it is so does not make it so. Bill might be mistaken. Frank / van Deemter / Wyner

  21. Completeness of inference rules • See handout for a complete set of rules that can prove all theorems. • However, there may be different systems that are not complete. There are issues similar to the expressivity of the logical connectives and quantifiers. Frank / van Deemter / Wyner

  22. Formal Proofs • A formal proof of a conclusion C, given premises p1, p2,…,pnconsists of a finite sequence of steps, each of which is either a premise or applies some inference rule to premises or previously-proven statements to yield a new statement (the conclusion). Frank / van Deemter / Wyner

  23. Method of Proof • Write down premises. • Write down what is to be shown. • Use a proof strategy. • Apply natural deduction rules. • Write down the result of applying the rule to the premise(s). Make a note of what rule is applied and what premises are used. • Reapply 2-5 until have shown the result. • Record result on line 2. Frank / van Deemter / Wyner

  24. Super Simple Example Problem: Prove that p implies p ∨ q • p Premise. • Show: p ∨ qDirect derivation, 3 • p ∨ q Disjunction introduction, 1 Frank / van Deemter / Wyner

  25. Pretty Simple Example Problem: Prove that p and (p ∨q)  s imply s • p Premise • (p ∨ q)  s Premise • Show: s Direct derivation, 5 • p ∨ q Disjunction introduction, 1 • s Implication elimination, 2 and 3 Have to think ahead. Tricky with long chains of reasoning. Frank / van Deemter / Wyner

  26. Longer Example 1. (p ∧ q)  r Premise 2. Show: (p (q  r)) CD 3,4 3. p Assumption 4. Show: q  r CD 4,5 • q Assumption 6. Show: r ID 10 7. ¬ r Assumption 8 (p ∧ q) CI 3,5 9. r MP 1,8 10. ¬ r ∧ r ContraI 7,9 Frank / van Deemter / Wyner

  27. A Direct Proof 1. ((A ∨ ¬ B) ∨ C)(D (E F)) 2. (A ∨ ¬ B)((F G)H) 3. A  ((EF)(FG)) 4. A 5. Show: D  H 6. A ∨ ¬ B 7. (A ∨ ¬ B) ∨ C 8. (D  (E F)) 9. (E F) (F G) 10. D (F G) 11. (F G) H 12. DH Frank / van Deemter / Wyner

  28. A Conditional Proof 1. (A ∨ B)(C ∧ D) 2. (D ∨ E)F 3. Show: A  F 4. A 5. Show: F 6. A ∨ B 7. C ∧ D 8. D 9. (D ∨ E) 10. F Frank / van Deemter / Wyner

  29. An Indirect Proof 1. A (B ∧ C) 2. (B ∨ D)E 3. (D ∨ A) 3. Show: E 4. ¬ E 5. ¬ (B ∨ D) 6. ¬ B ∧¬ D 7. ¬ D 8. A 9. B ∧C 10. B 11. ¬ B 12. B ∧¬ B Frank / van Deemter / Wyner

  30. Next • Proofs using logical equivalences • Quantifier proof rules • Other proof strategies • contrapositive • cases Frank / van Deemter / Wyner

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