10/13/15. Proofs: what and why. Proposi<onal Logic Proofs. 1 st Proof Method: Truth Table. A sequence of logical arguments such that:

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1 Proofs: what and why Rules of Inference (Rosen, Section 1.6) TOPICS Logic Proofs ² via Truth Tables ² via Algebraic Simplification ² via Inference Rules A sequence of logical arguments such that: each one follows from one (or more) of the previous ones star<ng from some premise to a logical conclusion Why: reasoning about programs Games, puzzles, logical systems, correctness CS160 - Fall Semester Proposi<onal Logic Proofs An argument is a sequence of proposi<ons: ² Premises (Axioms) are the first n proposi<ons ² Conclusion is the final proposi<on. An argument is valid if ( p 1 p 2... p n ) q tautology, given that p i are the premises (axioms) and q is the conclusion. is a 1 st Proof Method: Truth Table If the conclusion is true in the truth table whenever the premises are true, it is proved Warning: when the premises are false, the conclusion my be true or false Problem: given n proposi<ons, the truth table has 2 n rows Proof by truth table quickly becomes infeasible, e.g needs ~4 billion rows CS160 - Fall Semester CS160 - Fall Semester

2 Truth Table Example Modus Ponens (p (p q)) q p q p q p (p q) (p (p q)) q nd Proof Method: Rules of Algebra You are probably familiar with solving algebraic equa<ons and showing your steps : Start with an equa<on Repeatedly apply rules of algebra on parts of the equa<on, thereby obtaining an equivalent one Stop when the equa<on is of the form, x=number What does it mean to apply a rule of algebra? Find a part of the equa<on that fits one side of a rule, and Systema<cally replace the part with the other side of the rule CS160 - Fall Semester CS160 - Fall Semester Algebraic Proofs of Proposi<ons Algebraic Proofs: example Use the rules in the resources tab Ignore the top third (rules of inference with weird La<n names Use these instead of the rules of algebra Apply the same methods, by payern matching and replacement Prove the Absorp<on Law x (x y) = x Using only Iden<ty, Domina<on, Nega<on, Idempotent, Associa<ve, and Commuta<ve Laws x (x y) (x T) (x y) Identity Law 1 (r l) x (T y) Distributive Law 2 (r l) x T Domination Law 1 (l r) x Identity Law 1 (l r) CS160 - Fall Semester CS160 - Fall Semester

3 Algebraic Proofs: comments Arbitrarily complicated sub- expressions can match one side of a rule At each step, the new equa<ons/expression is iden<cal to the previous one Inference rules allow you to go beyond that Start with a collec<on of proposi<ons that we assume are true, call them facts for now Repeatedly deduce new facts by making valid inferences Stop when the desired proposi<on is added to your collec<on of facts 3 rd Proof Method: Inference Rules A rule of inference is a pre- proved rela<on: any <me the le_ hand side (LHS) is true, the right hand side (RHS) is also true. Therefore, if we can match a premise to the LHS (by subs<tu<ng proposi<ons), we can assert the (subs<tuted) RHS CS160 - Fall Semester CS160 - Fall Semester Inference proper<es Inference rules are truth preserving If the LHS is true, so is the RHS Applied to true statements Axioms or statements proved from axioms Inference is syntac<c Subs<tute proposi<ons if p replaces q once, it replaces q everywhere If p replaces q, it only replaces q Apply rule CS160 - Fall Semester Inference Rule Example Modus Tollens ( q (p q)) p p q p q p q q (p q) ( q (p q)) p CS160 - Fall Semester

4 Rules of Inference Logical Equivalences CS160 - Fall Semester CS160 - Fall Semester Modus Ponens If p, and p implies q, then q p q, it is hot whenever it is sunny Given the above, if it is sunny, it must be hot. Modus Tollens If not q and p implies q, then not p p q, it is hot whenever it is sunny Given the above, if it is not hot, it cannot be sunny. CS160 - Fall Semester CS160 - Fall Semester

5 Hypothe<cal Syllogism If p implies q, and q implies r, then p implies r, r = it is dry p q, it is hot when it is sunny q r, it is dry when it is hot Given the above, it must be dry when it is sunny Disjunc<ve Syllogism If p or q, and not p, then q p q, it is hot or sunny Given the above, if it not sunny, but it is hot or sunny, then it is hot CS160 - Fall Semester CS160 - Fall Semester Resolu<on If p or q, and not p or r, then q or r, r = it is dry p q, it is sunny or hot p r, it is not hot or it is dry Given the above, if it is sunny or hot, but not sunny or dry, it must be hot or dry Not obvious! CS160 - Fall Semester Addi<on If p then p or q p q, it is hot or sunny Given the above, if it is sunny, it must be hot or sunny Of course! CS160 - Fall Semester

6 Simplifica<on If p and q, then p p q, it is hot and sunny Given the above, if it is hot and sunny, it must be hot Of course! Conjunc<on If p and q, then p and q p q, it is hot and sunny Given the above, if it is sunny and it is hot, it must be hot and sunny Of course! CS160 - Fall Semester CS160 - Fall Semester A Simple Proof Given x, x y, y z, z w, prove w Step Reason 1. x y Premise 2. y z Premise 3. x z Hypothetical Syllogism (1, 2) 4. x Premise 5. z Modus Ponens (3, 4) 6. z w Premise 7. w Disjunctive Syllogism (5, 6) CS160 - Fall Semester A Simple Proof In order to complete CS161, I must complete CS160 and either M160 or M161. I have not completed M160 but I have completed CS161. Prove that I have completed M161. STEP 1) Assign proposi<ons to each statement. A : I have completed CS161 B : I have completed CS160 C : I have completed M160 D : I have completed M161 CS160 - Fall Semester

7 Setup the proof STEP 2) Extract axioms and conclusion. Axioms: A B (C D) A C Conclusion: D Now do the Proof STEP 3) Use inference rules to prove conclusion. Step Reason 1. A B (C D) Premise 2. A Premise 3. B (C D) Modus Ponens (1, 2) 4. C D Simplification (3) 5. C Premise 6. D Disjunctive Syllogism (4, 5) CS160 - Fall Semester CS160 - Fall Semester Another Example Proof of Another Example Given: p q p r r s Conclude: q s Step Reason 1. p q Premise 2. q p Implication law (1) 3. p r Premise 4. q r Hypothetical syllogism (2, 3) 5. r s Premise 6. q s Hypothetical syllogism (4, 5) CS160 - Fall Semester CS160 - Fall Semester