Theory of Computation CS3102 Spring 2014

Transcription

1 Theory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia

2 Today: Formal Logic Review of discrete math Notation will be used heavily Unambiguous All answers require argument Goals: Develop intuition Practice fluency Note: today we ll be more formal than I expect on the problems

3 Propositional Logic Sentence which is true/false E.g. It is snowing New propositions created by combining others,,,, E.g. let p= It is snowing. p = It isn t snowing = p DeMorgan s Laws: p q = p q p q = p q What about every dog has his day?

4 First Order Logic Predicates can also have variables E.g. CS3102(x) = x is taking CS3102 Taking (x, y) = x is taking y Taking (x, CS3102) = CS3102(x) Quantifiers, Every Dog has his day has(a, b) = a has his b a b has(a, b)

5 First Order Logic Not every CS Student takes both Theory of computation and Operating Systems Consider the opposite: Every CS Student takes both Theory of computation and Operating Systems Formally: CS(x) = x is a CS student Takes(x, y) = x takes y x CS x takes (x, Theory of computation ) takes (x, Operating Systems) Now negate: [ x CS x takes (x, Theory of computation ) takes (x, Operating Systems)] x CS x [ takes (x, Theory of computation ) takes (x, Operating Systems)]

6 First Order Logic Only one CS student got in A in Operating Systems CS x = x is a CS student gota x, y = x got and A in y! x CS x gota(x, Operating Systems) x [CS x gota(x, Operating Systems) ( y CS y gota(x, Operating Systems) y = x)] x congressman x [ y represents(x, y) like x, y ] No one likes a congressman, unless that congressman is her representative.

7 First Order Logic Inference This is how we write proofs! Universal Elimination If a statement is true for all x, it must be true for a particular x. x hasmass x becomes hasmass(nathan) Existential Elimination If there exists something which satisfies the predicate, then you may consider an arbitrary thing which does. x haslostcar x becomes haslostcar(dude) Existential Introduction If something is true for one item, you may conclude that it s safe for some item haslostcar(dude) becomes x haslostcar x Universal Generalization If something is true for an arbitrary item you may conclude it s true for any item For arbitrary x hasmass x becomes y hasmass(y)

8 Simple Proof: First Order Logic It is against the law for any grocery store to sell alcohol to individuals under the age of 21. Timmy, who is age 12, comes home one day with a six pack of beer and a receipt from Kroger (a grocery store) showing his mother s credit card number, which had gone missing from her purse the day before. Beer contains alcohol. Did Kroger break the law? 1. x y z Grocer x alcohol y underage z sells x, y, z criminal(x) 2. Grocer Kroger 3. underage Timmy 4. Receipt(Kroger, beer, Timmy) 5. Alcoholic(beer) 6. x y z receipt x, y, z sells(x, y, z)

9 First Order Logic 1. x y z Grocer x alcohol y underage z sells x, y, z criminal(x) 2. Grocer Kroger 3. underage Timmy 4. Receipt(Kroger, beer, Timmy) 5. Alcoholic(beer) 6. x y z receipt x, y, z sells(x, y, z) 7. [4] + [6] with Universal elimination gives sells Kroger, beer, Timmy 8. [2]+[1] with Universal Elimination gives y z Grocer Kroger alcohol y underage z sells Kroger, y, z criminal(kroger) 9. [3]+[5]+[7]+[8] with Existential introduction gives criminal Kroger

10 General Proof Guidelines No need to be super formal Shorter is always better! A picture is worth 1000 words Don t try to surprise people Consider all corner cases

11 Set Theory Unordered collection of objects Notation: {}: Set notation : Union : Intersection : Cross product : Complement : Set membership : Proper Subset : Subset 2 S : Powerset

12 Set Theory Notation: {}: Set notation {1, 2, 3} is a set {1, 2, 3, } is the set of all positive whole numbers

13 Set Theory Unordered collection of objects Notation: : Union {1, 2, 3} {3, 4, 5} = {1, 2, 3, 4, 5}

14 Set Theory Unordered collection of objects Notation: : Intersection {1, 2, 3} {3, 4, 5} = {3}

15 Set Theory Unordered collection of objects Notation: : Cross product {1, 2, 3} {3, 4, 5} = {(1, 3), (1, 4), (1, 5) (2, 3), (2, 4), (4,5) (3, 3), (3, 4), (3, 5)} Note that S 1 S 2 = S 1 S 2

16 Set Theory Unordered collection of objects Notation: : Complement, also overbar {even numbers} = {odd numbers}

17 Set Theory Unordered collection of objects Notation: : Set membership 2 1, 2, 3 12 {1, 2, 3}

18 Set Theory Unordered collection of objects Notation: : Proper Subset 1,2 1, 2, 3 1, 2, 3 1, 2, 3 1, 2, 4 {1, 2, 3} : Subset 1, 2, 3 {1, 2, 3}

19 Set Theory Unordered collection of objects Notation: 2 S : Powerset 2 {1,2,3} = {{}, 1, 2, 3, 1,2, {1,3} 2, 3, {1, 2, 3}}

20 Useful Sets N = natural numbers = {0, 1, 2, 3,... } Z= integers = {..., 3, 2, 1, 0, 1, 2, 3,... } Z + = positive integers = {1, 2, 3,... } R= set of real numbers R + = R >0 = set of positive real numbers C = set of complex numbers Q = set of rational numbers

21 Problem: (1/4) + (1/4) 2 + (1/4) 3 + (1/4) 4 + =? Find a short, geometric, induction-free proof. 1 i 1 4 i

22 Problem: (1/8) + (1/8) 2 + (1/8) 3 + (1/8) 4 + =? Find a short, geometric, induction-free proof. i i 1 7

23 Binary Relations A relation between sets S and T is a subset of S T consider the relation R for x has eaten y for dinner P= {all people}, F = {all food}, R P F To say Nathan has eaten ice cream for dinner (Nathan, ice cream) Nathan ~ ice cream Without loss of generality we can assume S = T Why? If S T then let A = S T