computing fundamentals 1 lecture 4 quantification lecturer: patrick browne room k308 based on...

Computing Fundamentals 1 Lecture 4

Quantification

Lecturer: Patrick Brownehttp://www.comp.dit.ie/pbrowne/

Room K308Based on Chapter 8.

A Logical approach to Discrete Math By David Gries and Fred B. Schneider

• We now study quantifiers of operators. These can be logical (e.g. ˄) or arithmetic (e.g. +) operators that act on a range of objects named by ‘dummy variables’. We use the symbols and ∃ ∀ to indicate that a variable refers to some object(s) or all objects.

• ∀ is called the universal quantifier.• ∃ is called the existential quantifier.• Quantifiers can be applied to Symmetric and

Associative operations that have an Identity operation.

Quantification

• We have already used expressions with constants and variables. We briefly mentioned functions application.

• A function is a rule for computing a result-value v from argument-value w, e.g.

• Definition of: g(z) = 3 ∙ z + 6 • The argument is designated in a function

application, which is a form of expression.

Functions

• We apply function to argument (function application)• g(5)• Gives the value of 3 ∙ 5+6• To reduce brackets we can write function.argument. • We evaluate this function• g.5• = < Apply function>• 3 ∙ 5 + 6• = < Arithmetic>• 21• Function application can be defined as textual

substitution.• g.z:E

Functions

• Function application can be defined as textual substitution. If

• g.z:E• Defines a function g then function

application g.X for any argument X is defined by

• g.X = E[z := X]

Functions

• This close correspondence between function application and textual substitution suggests that Leibniz (1.5) links equality and function application. So, we can reformulate Leibniz for functions.

g.Yg.XYX

Functions & Types

• Up to now we have used the Boolean type. • In Computer Science many types are

needed. – Simple types like integers and natural numbers– Complex types like sets, lists, sequences.

• Each function has a type which describes the types of its parameters and the type of its result:

• (8.1)1 f:arg1 arg⨯ 2 ... arg⨯ n r⟶

Functions & Types

• Some function signatures:• plus: (plus(1,3)or 1+3)ℤ⨯ℤ ⟶ℤ• not : ⟶ (not(true) or true)• less: ℤ⨯ℤ ⟶ (less(1,3)or 1<3)

Functions & Types

• Type and type correctness are syntactic notions. Type correctness depends only on the sequence of symbols in the proposed expression, and not on evaluation of the expression (in a state). For example1, (1/(x: )): ℤ ℝ is an expression even though its evaluation is undefined if x=0.

Functions & Types

• Certain restrictions are need to insure expression are type correct.

• During textual substitution E[x := F], x and F must have the same type.

• Equality b=c is defined only if b and c have the same type. Treating equality as an infix function:

• _=_ :t t ⨯ ⟶ . For any type t.• See exercise 13 in lab1. Checking the sort (or

type) of a term. This same-type restriction is relaxed in Order Sorted Algebras, where we have sub-types and super-types.

Syntax & interpretation of quantification

• Summation notation:• ∑n

i=1 expression • ∑3

i=1(i2) = 12+22+32

• In linear notation:• (∑i | 1 ≤ i ≤ n : e)• Or• (+i | 1 ≤ i ≤ n : e)• Where e is some expression (aka body) like i2

General form of Quantification

Type separators

Body Separators

(*x : type | R : P)

Binary operator

Dummy or quantified variable

Universal Quantifier

• The declaration introduces a typical term (or body) that may be optionally constrained or restricted in some way: For example:

( i: | i<10 : i∀ ℕ 2 < 100 )

• Which states that for all natural numbers less than 10 their square is less than 100.

• We use linear notation:• (+i | 1 ≤ i ≤ n : e)• The scope of the quantified variable i is

confined to expressions within the brackets. In this case quantified variable is know as a dummy, it does not obtain a value from the state in which the expression is evaluated. The dummy is used to represents a range of values.

• The range in linear notation can include any Boolean expression:

(+i | 0 ≤ i ≤ 7 ˄ even.i : i)

= 0 + 2 + 4 + 6• The above sums only even numbers

between 1 and 7 inclusive.

• See sumev in next slide.

Sum even numbers 0-n• mod SUM {• pr(INT)• op sumev : Int -> Int• op even_ : Int -> Int• var n : Int• **> Even returns n if n is even otherwise returns 0.• ceq even n = n if (n rem 2 == 0) .• ceq even n = 0 if (n rem 2 =/= 0) .• **> Sum of even numbers between 0 and n• -- base case• eq sumev(0) = 0 .• -- the general case• ceq sumev(n) = (even n) + sumev(n - 1) if (n =/= 0) .• }• red in SUM : sumev(7) . – gives 12

• We can have more than one dummy variable:

(+i,j | (1 ≤ i ≤ 2) ˄ (3 ≤ j ≤ 4): ij)

= 13 + 14 + 23 + 24 • The above sums ij over the given range

of i and j.

• Moving from summation we let * represent any binary operator (not just +).

• We consider symmetric associative binary operators that have an identity u.

• Symmetry: b * c = c * b• Associative: (b*c) * d = b * (c*d)• Identity u: u * b = b = b* u

• (8.6) (*x:t1,y:t2 | R : P)• The bound or dummy or quantified variables x

and y are distinct.• t1 and t2 are the types of the variables.• R is a Boolean expression; the range of the

quantification. x and y must satisfy R.• The expression P is the body of the

quantification.• The type (or sort in CafeOBJ terms) of the result

is the type of P.

• Specific instances of the general (8.6)1 (+i | 0 ≤ i < 4 : i∙8) = 0∙8 + 1∙8 + 2∙8 + 3∙8

(˄i | 0 ≤ i < 2 : i∙d≠6) = 0∙d≠6 ˄ 1∙d≠6 (⋁i | 0 ≤ i < 21 : b[i] = 0) = b[0]=0 ⋁ … ⋁ b[20]=0

Dot ∙ denote multiplication.

The variable d is not quantified.

b[i] is an array

Different representations of common quantifications

(+x | R : P) (∑x | R : P)

(∙x | R : P) (∏x | R : P)

( x | R : P)⋁ ( x | R : P)∃

(˄x | R : P) ( x | R : P)∀

Scope, Free, Bound

• There are two ways in which a variable can occur in a formula: free or bound.

• A variable is said to occur free in a expression E if and only if it is not within the “scope” of a quantifier.

• If a variable never occurs free in E, then we say the variable is bound.

• (8.7) (˄i | :x∙i = 0 )• (8.7) asserts that x multiplied by any integer

equals 0. This is only true for x=0. Expression depends on value of x not i.

• The occurrence of x in (8.7) is free.• The scope of the dummy i is the range and the

body of (8.7)• Occurrences of i in the scope of the dummy i

are said to be bound to dummy i.

Avoid Confusing Scope

• (8.8) i>0 ⋁ (˄i | 0≤i : x∙i=0)

• Two distinct variables both named i.

• In (8.8) the leftmost iis free.

• The second quantified i is bound.

• Best avoid this situation by renaming.

• Scope in quantified expressions can be considered similar to the concept of scope in programming languages.

Scope in a procedure1

procedure p(x : integer);var i:integerbegin i := x * y; x := 2 * I;end;• If variable is not local or an argument then it is free in a

function or procedure. Above, the scope of i is the procedure body, and the argument x is bound to the procedure p. In contrast y is free (it comes from outside the procedure, out of the blue, aka global). Note that a procedure differs from a function, it does not have to return a value, rather it can perform operations (e.g. open a file).

def p(x): global y i = x * y x = 2 * 1 return

Quantifiers bind variables

• The quantifiers and are the binding ∀ ∃operators.

• The formal definition of free: • (8.3) The occurrence of i in expression i

is free.• (8.4) If i is free in E then that same

occurrence is free in (E) , f(..,E,..), (*x | E:F), and (*x | F:E) provided that iis not on one of the dummies in list x. It is best to keep variable names distinct.

Occurs

• We define the predicate: • occurs(‘v’,’e’)• To mean that at least one variable in the list ‘v’ of

variables occurs free in at least one expression in the expression list ‘e’.

• The quotes mean we are considering a list of variables and a list expressions rather than values.

• Here we are interested in the occurrence of the variable that may be bound or free not the variable itself. An occurrence is in an expression. We are just talking about scope.

• (8.10) Let an occurrence of i be free in expression E. That occurrence of i is bound (to dummy i) in the expression (*x | E:F) or (*x | F:E) provided that i is one of the dummies in list x.

• Suppose an occurrence of i is bound in expression E. Then it is also bound to the same dummy) in (E) , f(..,E,..), (*x | E :F), and (*x | F:E).

Example

i + j + (∑i | 1≤ i ≤ 10 : b[i]j)+(∑i | 1≤ i ≤ 10 : (∑j | 1≤ j ≤ 10 : c[i,j]))On first line i and j are freeOn the second line j is free.All other occurrences of i & j are boundThere are two distinct dummies called i.

Textual Substitution

• Textual substitution that worked for propositions can be extended to cover quantification.

• (8.11) provided occurs(‘y’,’x,F’) (*y|R : P)[x:=F] = (*y|R[x:=F] : P[x:=F])

• The caveat in (8.11) means that the dummy of list y will have to be replaced by a fresh variable if that dummy occurs free in x or F. A fresh variable is a variable that does not already occur in the expression under consideration.

Textual Substitution =>

(+x|1 ≤ x ≤2:y)[y:=y+z] (+x|1 ≤ x ≤2:y+z)

(+i| 0≤i<n:b[i]=n)[n:=m] (+i| 0≤i<m:b[i]=m)

(+y| 0≤y<n:b[y]=n)[n:=y](+j| 0≤j<n:b[j]=n)[n:=y]

(+j| 0≤j<y:b[j]=y)To avoid clash use variable j instead of y.

(+y| 0≤y<n:b[y]=n)[y:=m] (+j| 0≤j<n:b[j]=n)Fresh variable j

Textual Substitution

=>(*x | 0≤x+r<n : x+ k)[k:=6] (*x | 0 ≤ x + r < n : x+6)

(+y | 0≤y<m : b[y]=m)[m:=y] (+k | 0 ≤ k < y : b[k] = y)

Quantified Textual Substitution Example

• (*x | 0 ≤ x + r < n : x + v)[v := 3]

• Solution• (*x | 0 ≤ x + r < n : x + 3)

• (+y | 0 ≤ y < m : b[y]=m)[m:=y]

• Solution: needs renaming of quantified variable y to k.

• (+k | 0 ≤ k < y : b[k] = y)

Rules about quantification

• Recall, for quantification we are interested in symmetric associative binary operators that have an identity u.

• Symmetry: b * c = c * b• Associative: (b*c) * d = b * (c*d)• Identity u: u * b = b = b* u• We need additional inference rules for

quantified expressions.

Inference Rules about quantification (8.12)

):]:[|(*):]:[|(* SQzExSPzEx

]):[:|(*]):[:|(*

QzERxPzERx

• 8.22 allows the change of dummy variable.

• 8.22 provided occurs(‘y’,’R,P’) (*x|R:P) = (*y| R[x:=f.y] : P[x:=f.y])

Where f is a function with an inverse i.e.

(x = f.y) (y = f-1.x)

(4 = squared(2)) (2 = sqroot(4))

Splitting ranges

• In computer science ranges are often split for operations that involve repetitions or loops, especially when determining the invariant of such a repetition1.

• An invariant of a loop is a predicate that that holds true for every repetition step and describes a kind of overall property of the whole loop.

Manipulating ranges

• Manipulating ranges allows us to split a range into two distinct ranges. The first example below takes one right hand (n+1) term out of the range. The second example below takes one left hand (0) term out of the range.

Manipulating ranges

(∑i | 0≤i<n+1:b[i])sum

(∑i| 0≤i<n: b[i]) + b[n]

(∏i| 0≤i<n+1:b[i])product

b[0]∙(∏ i | 0≤i<n : b[i])

(∀i| 0≤i≤n:b[i]=0)For all

(∀i| 0≤i<n:b[i]=0) ˄ b[n]=0

(∏i|5≤i≤10:i2)

product

52 ∙(∏i|5<i≤10:i2)

Methods for describing ranges

A 2 ≤ i ≤ 15

B 2 ≤ i < 16

C 1 < i ≤ 15

D 1 < i < 16

Various ways to describe i for values 2..15

For B and C the number of the number of values of i in the range is equal to the upper bound minus the lower bound (16-2 or 15-1).

Collapse or Split ranges

A 2 ≤ i ≤ 15 ⋁ 16 ≤ i ≤ 20 2 ≤ i ≤ 20

B 2 ≤ i < 16 ⋁ 16 ≤ i < 21 2 ≤ i < 21

C 1 < i ≤ 15 ⋁ 15 < i ≤ 20 1 < i ≤ 20

D 2 < i < 16 ⋁ 15 < i < 21 1 < i < 21

Substitutions in Quantification

• (*x | 0 ≤ x + r < n : x + v)[v := 3]

• Solution• (*x | 0 ≤ x + r < n : x + 3)

• (+y | 0 ≤ y < n : b[y]=n)[n:=y]• Solution• (+j | 0 ≤ j < y : b[j] = y)

English Quantification

• All even integers that are multiples of 2 • ( x∀ : ℤ | even.x:multiple(x,2))

• Express this set in English• {x:ℤ+ | 0 ≤ x < 4}• The set of non-negative integers less than

• The set of positive even integers that are less than 26.

{x: + ℤ | 0<x<26 ˄ (x mod 2 = 0) : x}

• The set of odd integers.• {x: ℤ | (x mod 2 = 1) : x}• OR• {x: ℤ | : 2*x-1}

Modulus or remainder function.

red in INT : 3 rem 2 .

red in INT : 2 rem 2 .

red in INT : 2 quo 2 .

red in INT : 2 divides 2 .

• Express this in English• {z | ( x,y ∃ | 0≤x ˄ 2≤y≤ 3 : z = xy}

• The set of squares and cubes of all natural numbers.

• All even integers are multiples of 2.• ( x: ∀ ℤ | even.x : multiple(x,2))• Where multiple(x,2)is a predicate that returns

true if x is a multiple of 2 (or 2 goes into x evenly)

English Quantification Example

• Some integer between 50 and m is a multiple of y.

• (∃i: ℤ | 50 ≤ i ≤ m : mutiple(i,y))

• Every integer is larger than one integer and smaller than another integer.

• ( z:∀ |:ℤ ( w,v∃ : | w≠v : w < z < ℤv))

Quantification and Programming Language C.

void P(int x) { int i; i = x*x; x = i*2;}• The scope rules for quantification are similar to

local variables in a C like language. The scope of i is P in the C function, in maths:

• (*i | R : P)• We ignore call by value and call by reference

issues.

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