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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

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

topics
Topics
  • What is proofand why?
  • How with rules and examples
  • Proof strategies – direct, contrapositive, and contradiction.

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nature of proofs
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.

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importance of proofs
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?

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symbolic reasoning
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.

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proofs in programming
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.

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deductive calculi
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.

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proof terminology
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.

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more proof terminology
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.

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inference rules general form
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....

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inference rules 1
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.

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inference rules 2
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".

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inference rules 3
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".

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inference rules 4
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".

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inference rules tautologies
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.

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modus ponens t table
Modus Ponens T-table

Proof that the reasoning template is a tautology.

Other reasoning templates can be demonstrated similarly.

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validity and truth
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

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fallacies
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.)

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invalid reasoning patterns
"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.

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completeness of inference rules
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.

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formal proofs
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).

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method of proof
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.

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super simple example
Super Simple Example

Problem: Prove that p implies p ∨ q

  • p Premise.
  • Show: p ∨ qDirect derivation, 3
  • p ∨ q Disjunction introduction, 1

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pretty simple example
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.

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longer example
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

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a direct proof
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

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a conditional proof
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

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an indirect proof
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

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slide30
Next
  • Proofs using logical equivalences
  • Quantifier proof rules
  • Other proof strategies
    • contrapositive
    • cases

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