CITS2211 Discrete Structures Logic

Transcription

1 CITS2211 Discrete Structures Logic Unit coordinator: Rachel Cardell-Oliver August 6, 2017

2 Highlights All men are mortal, Socrates is a man, therefore Socrates is mortal.

3 Reading Chapter 3: Logical Formulas Mathematics for Computer Science by Lehman, Leighton and Meyer

4 Why Optimisation 0 ( x <= 0 && y > 100 ) ) x = 0; See Verification Hardware (post Pentium Division Bug 1995) Software e.g. sel4 microkernel

5 Lectures 3 and 4 will address the following questions: 1 What is a predicate? 2 What are well formed formulae of predicate logic? 3 Introduce some common rules of predicate logic 4 When is a predicate valid? 5 When is a predicate satisfiable?

6 Definition: A predicate is a proposition whose truth depends on the value of one or more variables. Example: Less than: 3 < 5 is true 5 < 3 is false Example: Likes(mark, coffee) Likes(mark, fanta)

7 Like other propositions, predicates are often named with a letter. Furthermore, a function-like notation is used to denote a predicate supplied with specific variable values. Example: An object is green in colour, G. G(grass) is true. G(sky) is false. G(jumbliehands) is false. G(jumblieheads) is true. Far and few, far and few, Are the lands where the Jumblies live Their heads are green, and their hands are blue, And they went to sea in a sieve.

8 For All Definition: The universal quantifier symbol is read for all. So the expression x D.P(x) is read For all x in D, P of x is true. Remember that upside-down A stands for All. Example: x N. 0 x All natural numbers x are greater than or equal to 0. (In CITS2211 we define N to include 0) Example: o. G(o) All objects are green (this statement is false)

9 Exists Definition: The existential quantifier symbol, is read there exists. The expression x D.P(x) is read, There exists an x in D such that P of x is true. Remember that backward E stands for Exists. Example: x Z.0 x Some integers x are greater than or equal to 0. Example: o.g(o) Some objects are green (this statement is true)

10 Predicate Logic 1 The alphabet of predicate logic contains symbols for denoting predicates called predicate symbols, for example, P, Q and R, each one with a given fixed arity (number of arguments needed). It also includes, constants, variables, function symbols, punctuation symbols, that is brackets ( ), quantifiers ( and ) and the propositional connectives,,,,. 2 The syntax of predicate logic is defined by the following rules for construction of well formed formulae (wff), often just called formulae: A predicate with the right number of arguments is a formula. If A and B are formulae then so are x.a, x.a, A, A B, etc. And build recursively. 3 A slightly complicated definition tells us when a variable in a formula is free or bound. Formulae with no free variables are called sentences.

11 Sometimes true 1 The semantics of a sentence (its truth value) is derived from the possible truth values of its sub-parts and the truth tables for connectives. (Can be done in a formal way but we will not.) 2 A sentence is either true or false (in a particular situation).

12 Precedence and Scope of Quantifiers Definition: The quantifiers have higher precedence than all the propositional connectives. That is, the precedence order is,,,,,,. You will find some differences in interpretation of the precedence of quantifiers in notes and text books, so I recommend always using brackets for quantifiers to make clear what is intended, rather than relying on the above precedence rule. Example: x. (child(x) clever(x) y. loves(y, x) )

13 Definition: A bound variable is a formula is an occurrence of a variable which has been introduced by a quantifier in that formula and lies within the scope of that quantifier. Bound variables can be replaced so long as the same pattern of bound variables is maintained. Definition: A free variable in a formula is an occurrence of a variable which does not lie within the scope of any of the quantifiers appearing in that formula. Free variables denote unknowns, therefore their replacement alters the meaning of a formula. Example: ( x.child(x)) clever(x) The first x is bound and the second is free. That is the x for clever is not the same as the x of child. ( y.child(y)) clever(x) means the same.

14 Types in Predicate Logic So far we have assumed nothing about the type of quantified variables. A predicate may be typed, in that its variables are specified to come from a certain set of values. Example: y : person. mortal(y) Example: x : N. x 0 y : Z. y < 0 But these mean just the same as Example: y person. mortal(y) Example: x N. x 0 y Z. y < 0

15 Mixing quantifiers Example: Every even integer greater than 2 is the sum of two primes (Goldbach s conjecture) Let Evens be the set of even integers greater than 2 and Primes be the set of primes. n Evens. p Primes. q Primes. n = p + q

16 Swapping the order of quantifiers usually changes the meaning of a proposition. Example: x. y. x < y is true, since for any x we could choose y=x+1 to make x < y true Example: y. x. x < y is false, since there is not some y that is greater than every x

17 Negating Quantifiers Example: It is not the case that everything is green: ( x. G(x)) Example: There is something that is not green: x. (G(x)) Example: It is not the case that some number is both greater and less than 0: x. (x > 0 x < 0) Example: Every number is not greater and less than 0: x. ( x > 0 x < 0) Example: Every number is either not greater or not less than 0 (De Morgan) x. ( (x > 0) (x < 0))

18 Rules of Predicate Logic From Can Derive Name Restrictions x. P(x) P(t) where t is a Universal Instantiation variable or constant symbol x. P(x) P(t) where t is a variable or constant Existential Instantiation Must be the first rule used that introduces symbol not previously t used in the proof sequence

19 Rules of Predicate Logic From Can Derive Name Restrictions P(c) x. P(x) Universal Generalizatioduced by E.I.. P(a) x. P(x) Existential Generalization c must be completely arbitrary, especially not intro- a is any constant.

20 Definition: A propositional formula is called valid when it evaluates to true no matter what truth values are assigned to the individual propositional variables. For propositional formulas, validity can be checked using a proof table. For predicate formulas, the formula must evaluate to true no matter what values its variables may take and no matter what interpretation a predicate variable may be given. Definition: A proposition is satisfiable if some setting of the variables makes the proposition true.

21 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold

22 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x)

23 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x) Example: Some countries are warm and are holiday resorts

24 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x) Example: Some countries are warm and are holiday resorts W (x) for x is warm (or C(x))

25 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x) Example: Some countries are warm and are holiday resorts W (x) for x is warm (or C(x)) H(x) for x is a holiday resort

26 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x) Example: Some countries are warm and are holiday resorts W (x) for x is warm (or C(x)) H(x) for x is a holiday resort Example: Northern European countries are cold but beautiful

27 What is a predicate? Identify the predicates needed for expressing the following sentences. Just identify the predicates - don t worry about the logical operators yet. Example: Iceland is cold but not all countries are cold C(x) for x is cold eg. C(Iceland) or x. C(x) Example: Some countries are warm and are holiday resorts W (x) for x is warm (or C(x)) H(x) for x is a holiday resort Example: Northern European countries are cold but beautiful N(x) for x is a NE country C(x) for x is a cold country B(x) for x is a beautiful country

28 Quantifiers Express the following in colloquial English as precisely as possible. Example: x. (adult(x) child(x)) Example: x. adult(x) x. child(x) Example: x. adult(x) x.child(x)

29 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever

30 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s))

31 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking

32 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking s.(c(s) H(s))

33 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking s.(c(s) H(s)) Example: Clever students work hard

34 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking s.(c(s) H(s)) Example: Clever students work hard s.(c(s) H(s)) or it could be s.(c(s) H(s))

35 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking s.(c(s) H(s)) Example: Clever students work hard s.(c(s) H(s)) or it could be s.(c(s) H(s)) Example: Those who do not work hard are lazy

36 Quantifiers Express the following colloquial English statements using predicate logic. Example: All first-year students are clever s.(f (s) C(s)) Example: No one can be clever without being hardworking s.(c(s) H(s)) Example: Clever students work hard s.(c(s) H(s)) or it could be s.(c(s) H(s)) Example: Those who do not work hard are lazy s.( H(s) L(s))

37 Hoare Logic for Reasoning about Programs Example: If the value of program variable foo is 3 and the statement foo := foo+1 is executed then the value of foo afterwards is 4. That is foo = 3 exec(foo := foo + 1; ) foo = 4 More generally, we write {P}S{Q} to mean if S is executed in a state satisfying P then the final state satisfies Q. foo. exec(foo := foo + 2; ) foo = foo + 2 foo. exec(foo := foo + 2; foo := foo + 3; ) foo = foo + 5