Logic Propositional logic; First order/predicate logic

Transcription

1 Logic Propositional logic; First order/predicate logic School of Athens Fresco by Raphael Wikimedia Commons Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois

2 Mathematical logic (symbolic logic) Study of inference using abstract rules that does not assume any particular knowledge of things or of properties. E.g.: All men are mortal Socrates is a man Inference: Socrates is mortal. E.g. All pigs are boisterous Alfred is a pig. Inference: Alfred is boisterous

3 All snarks are frabjous Yeti is a snark. Inference: Yeti is frabjous Key idea: Inference is independent of the subjects (men, pigs, snarks) and properties (mortality, boisterousness, frabjousness). Inference follows simply from language!

4 All p s are q. h is a p. Inference: h is q.. Inference: q(h)

5 But inference rules needn t hold in natural language! quirks of English Sam and Sally are programmers. Inference: Sam is a programmer Sam and Sally are together. Inference: Sam is together! So we need a formal language. logic!

6 Propositional logic A proposition is a statement that is either true or false. Examples: Socrates is a man This car is purple 43 is prime Non examples: Trucks Hello! Trkjkjugirtu How are you? This red car

7 Propositional logic Propositional logic talks about Boolean combinations of propositions and inferences we can make about them. E.g., If it is raining, then it is cloudy. It is not cloudy. Inference: It is not raining. Abstraction: p: it is raining q: it is cloudy Inference: ~

8 Propositional logic Propositions: p, q, r, s,. Constants: T, F Operators (boolean): and :, ~ : bi implication; iff Syntax: Any formula that combines propositions and constants using these operators

9 Sample formulas Sample translations from English to logic p: it is raining q: it is cloudy If it is raining then it is cloudy It is not cloudy

10 Propositional logic: Semantics A formula f, in general, doesn t have a truth value associated to it. Model or valuation: v Assigns truth/falsehood to each proposition Any formula f evaluates to true/false under such a valuation.

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12 Implication can be non intuitive says if p is true then q is true If the model sets p to true, and q to true, then evaluates to true. If the model sets p to true, and q to false, then evaluates to false. If the model sets p to false and q to true, then evaluates to true. If the model sets p to false and q to false, then evaluates to true! (vacuosly) The only way can evaluate to false is when p is true and q is false

13 Implication So is really the same as ~ If p then q is same as either p is false or q is true

14 Tautology A formula is a tautology (or called valid) if it evaluates to true in every valuation. E.g. ~ If model sets p to true, then formula is true. If model sets p to false, then formula is true. E.g., ( Why? Non example:, Friend to mathematician: Congratulations. Was it a boy or a girl? Mathematician: Yes!

15 Equivalence Formulas f and g are equivalent ( ) if in every model M, either both f and g evaluate to true in M or both evaluate to false in M. E.g., ~ If then you can take any formula A where f occurs and replace it by g to get a formula equivalent to A. E.g., ~

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17 Some important equivalences ~~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ De Morgan s laws

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19 Some important equivalences Distributive laws: Commutativity Associativity

20 Contrapositive, converse, negation If the sky is green, then I m a monkey s uncle. Converse If I m a monkey s uncle, then the sky is green. Contrapositive If I m not a monkey s uncle, then the sky is not green. Negation The sky is green and I am not a monkey s uncle

21 More manipulation examples Show that these are tautologies: ~ ~ ~

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23 Negation normal form

24 The satisfiability problem (SAT) Recall: f is a tautology if f evaluates to true in all models f is satisfiable if there is some model M under which f evaluates to true E.g., Non example: In fact, for any formula f, is satisfiable iff is not a tautology! Important problem in theoretical CS NP complete The classical hard problem that no one knows how to solve in polynomial time (P=?NP question)

25 Applications of SAT solvers: model checking Recent advent of extremely fast SAT solvers (despite theoretical hardness of problem) See Z3 online: Circuits/hardware checking Pentium division error half a billion dollars to Intel Can we check whether a circuit is correct? Idea: model circuit as a Boolean formula Each wire can be carrying hi/lo voltage 1/0 Each gate is a propositional operator (or, xor, and, not..) So equivalence of circuits is a satisfiability problem! (<<c>> <<d>>)

26 First order logic or Predicate logic All men are mortal Socrates is a man Inference: Socrates is mortal.. Inference: q(h) Can t say this is propositional logic. We need variables (like x)

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28 Predicate logic (first order logic) Example: Consider the universe of integers, with constants 0, 1, 2,, functions +,, *, and relations, =,, etc. Every number n added to 0 gives n. There is some number which when multiplied by itself is 0:. There are two numbers whose sum is 1..

29 Quantifiers For some or there exists some x: Some creatures are greyhounds that run fast., 0, 0 For all : For all creatures, if it is fat, then it does not run well., 0 For exactly one :! There is exactly one fat creature than runs well.!, 0

30 Binding and scope binding,, 0 scope Similar to local variables in programs: int f(x) { int y; y := x+2; }

31 Manipulating quantifiers Negation,,,, Not all dogs are fat is equivalent to At least one dog is not fat. There does not exist one fat dog is equivalent to All dogs are not fat. Contrapositive,,

32 Manipulating quantifiers Negation,,,, Not every number is even is equivalent to There is some number that is not even There is some number that is prime is equivalent to All numbers are not composite. Contrapositive,, Every number that is a square is a composite number. Every number that is a prime number is not a square.

33 Quantifiers with two variables For all integers and, false,, or,, There are two integers whose sum is 12 (true),. +b = 12) For every real, there exists an integer, such that (true),,

34 Nested quantifiers (will do more later)

35 To do There will be a Mini HW due next Thursday Reading quizzes for Mon and Wed next week. Discussion sections start this week (Friday) Did you buy your discussion handbook and workbook? Did you enroll on Piazza? Only 115/192 enrolled! Next class: Strategies for proving and disproving different types of claims.