From Propositional! to Predicate Logic" CSCI 2824, Fall 2011" classes/struct11/home.html "

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1 From Propositional! to Predicate Logic" CSCI 2824, Fall 2011" classes/struct11/home.html "

2 Assignments To read this week: Sections (Emsley/ Crawley) For next week: Chapter "!!!!

3 Playing with Logic: Mastermind

4 Let s See if We Can Represent Mastermind Ideas in Propositional Logic First, let s make some proposition symbols. We ll say that BLUE1 stands for the proposition In the correct code, the blue peg is in position 1. There are a bunch of these propositions: BLUE2, BLUE3, BLUE4, ORANGE1, ORANGE2

5 More Mastermind In the correct code, there is one peg (of some color) in position 1 : BLUE1 OR ORANGE1 OR RED1 OR WHITE1 OR YELLOW1 OR PURPLE1 But that s not quite enough The statement above is consistent with the (incorrect) possibility that both a blue and orange (say) are in position 1.

6 So we need some additional statements: NOT (ORANGE1 AND BLUE1) NOT (ORANGE1 AND RED1) NOT (ORANGE1 AND PURPLE1) etc. These statements say It is not the case that there are two distinct colors in position 1

7 What do we learn from a guess? My guess: RED BLUE WHITE ORANGE Answer: X,0 From the X, I get: RED1 OR BLUE2 OR WHITE3 OR ORANGE4 NOT (RED1 AND BLUE2) NOT (RED1 AND WHITE3) NOT (RED1 AND ORANGE4) NOT (BLUE2 AND WHITE3) NOT (BLUE2 AND ORANGE4) NOT (WHITE3 AND ORANGE4)

8 What do we learn from a guess? My guess: RED BLUE WHITE ORANGE Answer: X,0 From the 0, I get: RED2 OR RED3 OR RED4 OR BLUE1 OR BLUE3 OR BLUE4 OR WHITE1 OR WHITE2 OR WHITE4 OR ORANGE1 OR ORANGE2 OR ORANGE3 NOT (RED2 AND BLUE1) many others of this form

9 How do we solve a code? Suppose we know: ORANGE1 OR BLUE1 NOT ORANGE1 Then we can conclude: BLUE1

10 What have we learned about propositional logic from this? One thing that we ve learned on the good side is that propositions do allow us to represent situations like Mastermind. On the not-so-good side, we ve found that doing real tasks with propositional logic (if you want to call Mastermind a real task) can require a surprising amount of infrastructure.

11 Why (or why not?) Propositional Logic It s simple to understand But it s rather inexpressive And simple to understand does not mean that all problems are simple: (x1 OR ~x5 OR x3) AND (~x1 OR ~x2 OR x6) AND (x3 OR x5 OR ~x6) AND.

12 A Brief Aside: Illogical Humans? Problems with Modus Ponens and Modus Tollens When are we logical?

13 K E 4 7 If a card has a vowel on one side, then it has an even number on the other side.

14 Drinking Coke Drinking Beer 21 years old 18 years old If a person is drinking an alcoholic beverage, then that person is at least twenty years old.

15 Propositional Logic: Where We ve Come So Far Basic objects are sentences with T/F values Connectors (AND, OR, NOT, etc.) are used to make compound sentences Basic rules of inference (Modus Ponens, etc.) are used to derive new sentences from a knowledge base of existing sentences

16 First-Order Predicate Logic We introduce a world of objects. Our logical sentences will refer to these objects. In many computer programs, the objects might refer to elements of a finite set (think about a database for, say, a library, or a business payroll) For our initial examples, we ll mainly use numbers as our objects (and we ll specify whether we mean a finite set, integers, rationals, reals, etc.)

17 Some Sample Predicates Prime (n) is the predicate n is a prime number (defined over the infinite set of positive integers) Even (n) is the predicate n is an even number Rational (x) is the predicate x is a rational number (defined over the infinite set of real numbers)

18 Some statements about positive integers (note: we still have the usual connectors): Prime (19) Even (7) OR Even (8) Prime(23) AND NOT(Even (23))

19 First-Order Predicate Logic: A Bit More Terminology We allow predicates of two or more arguments: Greater-Than(5, 3) Relatively-Prime (25, 6) Equal (2, 2) [often we just write 2 = 2 for simplicity] We have two quantifiers: THERE-EXISTS and FOR-ALL FOR-ALL(n) [(Prime(n) AND Greater-than(n, 2)) -> NOT (Even (n))] THERE-EXISTS(n) [Even(n) AND Prime(n)]

20 Predicate Logic Classics FOR-ALL(n)[ Prime(n) à THERE-EXISTS(q) [Greater-than(q, n) AND Prime(q)]]

21 Predicate Logic Classics FOR-ALL(n) [(Even(n) AND Greater-than(n, 2)) > THERE-EXISTS (p, q) [Prime(p) AND Prime(q) AND n = (p + q)]] The Goldbach Conjecture

22 Quantifiers in English: An Example You can fool all of the people some of the time, and some of the people all of the time, but you can t fool all of the people all of the time. FOR-ALL (x) [Person(x) --> THERE-EXISTS(t) [Time(t) AND Fool-at-time(x, t)]] THERE-EXISTS(x) [Person(x) AND (FOR-ALL(t) [Time(t) --> Fool-at-time(x, t)]] NOT(FOR-ALL(x, t)[(person(x) AND Time(t)) --> Fool-at-time(x,t)]

23 Quantifiers and Dean Martin Everybody loves somebody sometime. FOR-ALL(x)[Person(x) --> THERE-EXISTS(y, t) [Person(y) and Time(t) and Loves-at-time(x,y,t)]] THERE-EXISTS(t,y)[Time(t) AND Person(y) AND FOR-ALL(x)[Person(x) --> Loves-at-time(x,y, t)]]

24 THERE-EXISTS(t)[Time(t) AND FOR-ALL(x) [Person(x) --> THERE-EXISTS(y) THERE-EXISTS(y)[Person(y) AND [Person(y) AND Loves-at-time(x,y,t)]} FOR-ALL(x) [Person (x) --> THERE-EXISTS(t)[Time(t) AND Loves-at-time(x, y, t)]]

25 Mastermind Challenge Problem (due 9/6) Colors: Blue, Orange, Red, Yellow, Purple, White Repeat colors in the code are allowed. Clues: x = a peg in the right position o = a peg that has to be moved to be in the right position W O B R!!!xo! P Y R B!!!xo! R R O Y!!!ooo! Y R R W!!!xx!