CSE 311: Foundations of Computing. Lecture 7: Logical Inference

Transcription

1 CSE 311: Foundations of Computing Lecture 7: Logical Inference

2 Logical Inference So far we ve considered: How to understand and expressthings using propositional and predicate logic How to compute using Boolean (propositional) logic How to show that different ways of expressing or computing them are equivalent to each other Logic also has methods that let us inferimplied properties from ones that we know Equivalence is a small part of this

3 Applications of Logical Inference Software Engineering Express desired properties of program as set of logical constraints Use inference rules to show that program implies that those constraints are satisfied Artificial Intelligence Automated reasoning Algorithm design and analysis e.g., Correctness, Loop invariants. Logic Programming, e.g. Prolog Express desired outcome as set of constraints Automatically apply logic inference to derive solution

4 Proofs Start with hypotheses and facts Use rules of inference to extend set of facts Result is proved when it is included in the set

5 An inference rule: Modus Ponens If Aand A Bare both true then Bmust be true Write this rule as A ; A B B Given: If it is Wednesday then you have a 311 class today. It is Wednesday. Therefore, by Modus Ponens: You have a 311 class today.

6 My First Proof! Show that rfollows from p, p q, and q r 1. Given 2. Given 3. Given Modus Ponens

7 My First Proof! Show that rfollows from p, p q, and q r 1. Given 2. Given 3. Given 4. MP: 1, 2 5. MP: 3, 4 Modus Ponens

8 Proofs can use equivalences too Show that pfollows fromp q and q 1. Given 2. Given 3. Contrapositive: 1 4. MP: 2, 3 Modus Ponens

9 Inference Rules If Ais true and Bis true. Requirements: Conclusions: A ; B C, D Then, Cmust be true Then D must be true Example (Modus Ponens): A ; A B B If I have Aand A Bboth true, Then Bmust be true.

10 Axioms: Special inference rules If I have nothing Requirements: Conclusions: C, D Then, Cmust be true Then D must be true Example (Excluded Middle): A A A A must be true.

11 Simple Propositional Inference Rules Two inference rules per binary connective, one to eliminate it and one to introduce it Elim A B A, B Intro A ; B A B Elim A B ; A B Intro A x A B, B A Modus Ponens A ; A B B Direct Proof Rule A B A B Not like other rules

12 Proofs Show that rfollows fromp,p qand(p q) r How To Start: We have givens, find the ones that go together and use them. Now, treat new things as givens, and repeat. A ; A B B A B A, B A ; B A B

13 Proofs Show that follows from,,and Two visuals of the same proof. We will use the top one, but if the bottom one helps you think about it, that s great! ; ; ; MP Intro MP 1. Given 2. Given 3. MP:1, 2 4. Intro : 1, 3 5. Given 6. MP: 4, 5

14 Important: Applications of Inference Rules You can use equivalences to make substitutions of any sub-formula. Inference rules onlycan be applied to whole formulas(not correct otherwise). e.g. 1. given 2. ( ) intro from 1. Does not follow! e.g. p=f, q=t, r=f

15 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given First: Write down givens and goal 20. Idea: Work backwards! Idea: Work backwards!

16 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given Idea: Work backwards! We want to eventually get. How? We can use to get there. The justification between 2 and 20 looks like elim which is MP. 20. MP: 2,

17 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given Idea: Work backwards! We want to eventually get. How? Now, we have a new hole We need to prove Notice that at this point, if we prove, we ve proven MP: 2, 19

18 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given This looks like or-elimination MP: 2, 19

19 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given Elim: 3, MP: 2, 19 doesn t show up in the givens but does and we can use equivalences

20 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given Double Negation: Elim: 3, MP: 2, 19

21 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given No holes left! We just need to clean up a bit. 17. Elim: Double Negation: Elim: 3, MP: 2, 19

22 Proofs Prove that rfollows from p s, q r,and s q. 1. Given 2. Given 3. Given 4. Elim: 1 5. Double Negation: 4 6. Elim: 3,5 7. MP: 2, 6

23 To Prove An Implication: We use the direct proof rule The pre-requisite A Bfor the direct proof rule is a proof that Given GivenA, we can prove B. The direct proof rule: If you have such a proof then you can conclude that A Bis true Example: Prove p (p q). Indent proof subroutine proof subroutine 1. Assumption 2. Intro : 1 3. ( ) Direct Proof Rule

24 Proofs using the direct proof rule Show that p r follows from qand (p q) r This is a proof of 1. Given 2. ( ) Given 3.1. Assumption 3.2. Intro : 1, MP: 2, Direct Proof Rule Ifwe know is true Then, we ve shown ris true

25 Example Prove: (p q) (p q) There MUST be an application of the Direct Proof Rule (or an equivalence) to prove this implication. Where do we start? We have no givens

26 Example Prove: (p q) (p q)

27 Example Prove: (p q) (p q) 1.1. Assumption 1.2. Elim : Intro : ( ) ( ) Direct Proof Rule

28 Example Prove: ((p q) (q r)) (p r)

29 Example Prove: ((p q) (q r)) (p r) 1.1. ( ) Assumption 1.2. Elim: Elim: Assumption MP: 1.2, MP: 1.3, Direct Proof Rule 1. ( ) Direct Proof Rule

30 One General Proof Strategy 1. Look at the rules for introducing connectives to see how you would build up the formula you want to prove from pieces of what is given 2. Use the rules for eliminating connectives to break down the given formulas so that you get the pieces you need to do Write the proof beginning with what you figured out for 2 followed by 1.