2.2 Data Abstraction
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1 ITP Programming Languages Chapter 2 Building Abstractions with Data 2.1 Introduction Functional Abstraction vs. Data Abstraction 2.2 Data Abstraction Example: Rational Numbers Major references: 1. Structure and Interpretation of Computer Programs(SICP) by Abelson and Sussman, MIT Composing Programs by John DeNero, Google 2. MITx OPENCOURSEWARE(OCW) 6.00 SC Lecture 3. UC Berkley CS61A Lecture 4. Python, Wikipedia and many on-line resources. Prof. Youngsup Kim, idebtor@handong.edu, 2014 Programming Languages, CSEE Dept., Handong Global University
2 Function 2
3 Functional Abstraction: 3
4 Functional Abstraction: using primitive data and operations 4
5 Functional Abstraction: using primitive data and operations forming functions through composition and control 5
6 Functional Abstraction: using primitive data and operations forming functions through composition and control giving names to processes 6
7 Functional Abstraction: using primitive data and operations forming functions through composition and control giving names to processes using higher-order functions 7
8 Functional Abstraction: using primitive data and operations forming functions through composition and control giving names to processes using higher-order functions These abstractions enhance the power of our programming language that is much of the essence of programming Functional Abstraction 8
9 Functional Abstraction: using primitive data and operations forming functions through composition and control giving names to processes using higher-order functions These abstractions enhance the power of our programming language that is much of the essence of programming Functional Abstraction Data Abstraction 9
10 Data 10
11 2. What is Data? Data 11
12 2. What is Data? Data: the things that programs fiddle with 12
13 2. What is Data? Data: the things that programs fiddle with Primitive values are the simplest type of data Integers: 2, 3, 2013, Floating point (decimal) values: 4.5, 98.6 Booleans: True, False 13
14 2. What is Data? Data: the things that programs fiddle with Primitive values are the simplest type of data Integers: 2, 3, 2013, Floating point (decimal) values: 4.5, 98.6 Booleans: True, False How do we represent more complex data? 14
15 2. What is Data? Data: the things that programs fiddle with Primitive values are the simplest type of data Integers: 2, 3, 2013, Floating point (decimal) values: 4.5, 98.6 Booleans: True, False How do we represent more complex data? Example: 15
16 2. What is Data? Data: the things that programs fiddle with Primitive values are the simplest type of data Integers: 2, 3, 2013, Floating point (decimal) values: 4.5, 98.6 Booleans: True, False How do we represent more complex data? Example: Creation Sin Sacrifice Sky 16
17 2. What is Data? Data: the things that programs fiddle with Primitive values are the simplest type of data Integers: 2, 3, 2013, Floating point (decimal) values: 4.5, 98.6 Booleans: True, False How do we represent more complex data? Example: Creation Sin Sacrifice Sky We need data abstraction! 17
18 2. Data Abstraction 18
19 2. Data Abstraction Compound Data combines smaller pieces of data together. A date: a year, month, and day A coordinate: x, y, z 19
20 2. Data Abstraction Compound Data combines smaller pieces of data together. A date: a year, month, and day A coordinate: x, y, z An abstract data type(adt) lets us manipulate compound data as a unit. This isolates any program that deals with into two parts: - How data are represented (as parts) - How data are manipulated (as units) This is the data abstraction - a powerful design methodology. 20
21 2. Data Abstraction Compound Data combines smaller pieces of data together. A date: a year, month, and day A coordinate: x, y, z All Programs, All Programmers An abstract data type(adt) lets us manipulate compound data as a unit. This isolates any program that deals with into two parts: - How data are represented (as parts) - How data are manipulated (as units) This is the data abstraction - a powerful design methodology. 21
22 2. Data Abstraction Compound Data combines smaller pieces of data together. A date: a year, month, and day A coordinate: x, y, z All Programs, All Programmers Great Programs, Great Programmers! An abstract data type(adt) lets us manipulate compound data as a unit. This isolates any program that deals with into two parts: - How data are represented (as parts) - How data are manipulated (as units) This is the data abstraction - a powerful design methodology. 22
23 2. Data Abstraction A methodology by which functions enforce an abstraction barrier between representation and use. 23
24 2. Data Abstraction A methodology by which functions enforce an abstraction barrier between representation and use. makes programs much easier to design, maintain, and modify. isolates how a compound data value is used from the details of how it is constructed. 24
25 2.2 Example: Rational Number see Rational Numbers: 25
26 2.2 Example: Rational Number see Rational Numbers: numerator denominator 26
27 2.2 Example: Rational Number see Rational Numbers: numerator denominator Exact representation of fractions A pair of integers As soon as division occurs, the exact representation is lost! Assume we can compose and decompose rational numbers: 27
28 2.2 Example: Rational Number see Rational Numbers: numerator denominator Exact representation of fractions A pair of integers As soon as division occurs, the exact representation is lost! Assume we can compose and decompose rational numbers: rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 28
29 2.2 Example: Rational Number see Rational Numbers: numerator denominator Exact representation of fractions A pair of integers As soon as division occurs, the exact representation is lost! Assume we can compose and decompose rational numbers: Constructor Selectors rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 29
30 2.2 Example: Rational Number see To facilitate data abstraction, we create two types of functions: Constructors are functions that build the abstract data type. Selectors are functions that retrieve information from the data type Constructor Selectors rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 30
31 2.2 Example: Rational Number Arithmetic Example: General Form: = 9 10 nx dx ny dy = nx ny dx dy = nx dx + ny dy = nx ny + ny dx dx dy 31
32 2.2 Example: Rational Number Arithmetic Example: General Form: = 9 10 nx dx ny dy = nx ny dx dy = nx dx + ny dy = nx ny + ny dx dx dy 32
33 2.2 Example: Rational Number Code 33
34 2.2 Example: Rational Number Code rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 34
35 2.2 Example: Rational Number Code Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 35
36 2.2 Example: Rational Number Code Wishful thinking: a powerful strategy for designing programs. The following three functions have not defined yet; even so, if we define them, we could then add, multiply, and test equality of rational numbers. Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 36
37 2.2 Example: Rational Number Code def mul_rationals(x, y): return rational(numer(x) * numer(y), denom(x) * denom(y)) Constructor Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 37
38 2.2 Example: Rational Number Code def mul_rationals(x, y): return rational(numer(x) * numer(y), denom(x) * denom(y)) Constructor Selector Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 38
39 2.2 Example: Rational Number Code def mul_rationals(x, y): return rational(numer(x) * numer(y), denom(x) * denom(y)) def add_rationals(x, y): nx, dx = numer(x), denom(x) ny, dy = numer(y), denom(y) return rational(nx * dy + ny * dx, dx * dy) Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 39
40 2.2 Example: Rational Number Code def print_rational(r): print (numer(r), '/', denom(r)) def eq_rationals(x, y): return numer(x) * denom(y) == numer(y) * denom(x) Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 40
41 2.2 Example: Rational Number Code Wishful thinking: a powerful strategy for designing programs. Now, let s think about these three functions: Wishful thinking rational(n, d) returns a rational number x. numer(x) returns the numerator of x. denom(x) returns the denominator of x. 41
42 pair 42
43 Representing Pairs Using Lists >>> pair = [1, 2] >>> pair [1, 2] >>> x, y = pair >>> x 1 >>> y 2 >>> pair[0] 1 >>> pair[1] 2 >>> from operator import getitem >>> getitem(pair, 0) 1 >>> getitem(pair, 1) 2 A list literal: Comma-separated expressions in brackets "Unpacking" a list Element selection using the selection operator Element selection function tuple( ): fixed-length, immutable sequence Stay tuned for more tuple and list later 43
44 2.2 Example: Representing Rational Number rational(n, d): It is a pair of integer. 44
45 2.2 Example: Representing Rational Number def rational(n, d): It is a pair of integer. """Construct a rational number x that represents n/d.""" return [n, d] 45
46 2.2 Example: Representing Rational Number def rational(n, d): """Construct a rational number x that represents n/d.""" return [n, d] 46
47 2.2 Example: Representing Rational Number def rational(n, d): """Construct a rational number x that represents n/d.""" return [n, d] Construct a list 47
48 2.2 Example: Representing Rational Number def rational(n, d): """Construct a rational number x that represents n/d.""" return [n, d] Construct a list from operator import getitem def numer(x): """Return the numerator of rational number x.""" return getitem(x, 0) # return x[0] def denom(x): """Return the denominator of rational number x.""" return getitem(x, 1) # return x[1] Select from a list 48
49 2.2 Example: Rational Number Demo Demo: >>> half = rational(1, 2) >>> third = rational(1, 3) >>> print_rational(half) 1 / 2 >>> print_rational(third) 1 / 3 >>> print_rational(mul_rationals(half, third)) 1 / 6 >>> print_rational(add_rationals(half, thrid)) 5 / 6 Work or not? >>> >>> print_rational(add_rationals( [1,2], [1,4] )) 6 / 8 >>> print_rational(mul_rationals( (1,2), (1,4) )) 1 / 8 >>> 49
50 2.2 Example: Rational Number Challenge: reducing the rational numbers to lowest terms Example: = = /3 1/3 = /25 1/25 =
51 2.2 Example: Rational Number Challenge: reducing the rational numbers to lowest terms Example: = = /3 1/3 = /25 1/25 = 1 2 from fractions import gcd def rational(n, d): """Construct a rational number x that represents n/d.""" g = gcd(n, d) return [n//g, d//g] 51
52 2.2 Example: Rational Number Challenge: reducing the rational numbers to lowest terms Example: = = /3 1/3 = /25 1/25 = 1 2 Greatest common divisor from fractions import gcd def rational(n, d): """Construct a rational number x that represents n/d.""" g = gcd(n, d) return [n//g, d//g] 52
53 2.2 Data Abstraction: Abstraction Barriers see
54 2.2 Data Abstraction: Abstraction Barriers see Rational numbers as whole data values add_rationals mul_rationals eq_rationals print_rational Rational numbers as numerators & denominators rational numer denom Rational numbers as lists (or tuples) list getitem However list or tuple is implemented in Python. 54
55 2.2 Data Abstraction: Abstraction Barriers see Example: Breaking Abstraction Barriers Does not use constructors add_rationals( [1, cvcv 2], [1, 4] ) 55
56 2.2 Data Abstraction: Abstraction Barriers Example: Breaking Abstraction Barriers Does not use constructors Twice! add_rationals( [1, cvcv 2], [1, cvcv 4] ) 56
57 2.2 Data Abstraction: Abstraction Barriers Example: Breaking Abstraction Barriers Does not use constructors Twice! add_rationals( [1, cvcv 2], [1, cvcv 4] ) def divide_rationals(x, y): return (x[0] * y[1], x[1] * y[0]) 57
58 2.2 Data Abstraction: Abstraction Barriers Example: Breaking Abstraction Barriers Does not use constructors Twice! add_rationals( [1, cvcv 2], [1, cvcv 4] ) def divide_rationals(x, y): return (x[0] * y[1], x[1] * y[0]) cvcv No selectors! 58
59 2.2 Data Abstraction: Abstraction Barriers Example: Breaking Abstraction Barriers Does not use constructors Twice! add_rationals( [1, cvcv2], [1, cvcv4] ) def divide_rationals(x, y): return (x[0] * y[1], x[1] * y[0]) cvcv No selectors! cvcv And no constructor! 59
60 2.2 Data Abstraction: Abstraction Barriers see Example: Breaking Abstraction Barriers Does not use constructors Twice! add_rationals( [1, cvcv 2], [1, cvcv 4] ) def divide_rationals(x, y): return (x[0] * y[1], x[1] * y[0]) cvcv No selectors! cvcv And no constructor! These implementations would require rewriting whenever the implementation of rational number functions changed. 60
61 2.2 Example: Rational Number Demo Demo: >>> half = rational(1, 2) >>> third = rational(1, 3) >>> print_rational(half) 1 / 2 >>> print_rational(third) 1 / 3 >>> print_rational(mul_rationals(half, third)) 1 / 6 >>> print_rational(add_rationals(half, thrid)) 5 / 6 Work or not? >>> >>> print_rational(add_rationals( [1,2], [1,4] )) 6 / 8 >>> print_rational(mul_rationals( (1,2), (1,4) )) 1 / 8 >>> 61
62 2.2 Example: Rational Number Demo Demo: >>> half = rational(1, 2) >>> third = rational(1, 3) >>> print_rational(half) 1 / 2 >>> print_rational(third) 1 / 3 >>> print_rational(mul_rationals(half, third)) 1 / 6 >>> print_rational(add_rationals(half, thrid)) 5 / 6 Work or not? >>> >>> print_rational(add_rationals( [1,2], [1,4] )) 6 / 8 >>> print_rational(mul_rationals( (1,2), (1,4) )) 1 / 8 >>> It worked, but broke abstraction barriers. 62
63 2.2 What is an Abstract Data Type? We need to guarantee that constructor and selector functions together specify the right behavior. Behavior condition: If we construct rational number x from numerator n and denominator d, then numer(x)/denom(x) must equal n/d. An abstract data type is a collection of constructors and selector and, together with some behavior conditions. If behavior conditions are met, the representation is valid. You can recognize data types by behavior, not by bits. 63
64 2.2 Behavior Conditions of a Pair see To implement our rational number abstract data type, we used a two element list (called as a pair, here). What is a pair? Constructors, selectors, and behavior conditions: If a pair p was constructed from elements x and y, then select(p, 0) returns x, and select(p, 1) returns y. Together, selectors are the inverse of the constructor. Generally true of container types. Not true for rational numbers because of GCD 64
65 2.2 Functional Pairs Implementation see 2.2.2~4 We don t need to use the Python list or tuple type if we can implement two functions pair and select that fulfills behavior conditions of a pair. 65
66 2.2 Functional Pairs Implementation see 2.2.2~4 We don t need to use the Python list or tuple type if we can implement two functions pair and select that fulfills behavior conditions of a pair. def pair(x, y): """Return a functional pair.""" def get(m): if m == 0: cvcv return x elif m == 1: return y return get This function represents a pair. not using list. Constructor is a higher-order function 66
67 2.2 Functional Pairs Implementation see 2.2.2~4 We don t need to use the Python list or tuple type if we can implement two functions pair and select that fulfills behavior conditions of a pair. def pair(x, y): """Return a functional pair.""" def get(m): if m == 0: cvcv return x elif m == 1: return y return get This function represents a pair. Constructor is a higher-order function def select(p, i): """Return the element at index i of pair p.""" return p(i) 67
68 2.2 Functional Pairs Implementation see 2.2.2~4 We don t need to use the Python list or tuple type if we can implement two functions pair and select that fulfills behavior conditions of a pair. def pair(x, y): """Return a functional pair.""" def get(m): if m == 0: cvcv return x elif m == 1: return y return get This function represents a pair. Constructor is a higher-order function def select(p, i): """Return the element at index i of pair p.""" return p(i) cvcv Selector defers to the functional pair. 68
69 2.2 Functional Pairs Implementation see 2.2.2~4 Demo: >>> p = pair(1, 2) >>> select(p, 0) 1 >>> select(p, 1) 2 69
70 2.2 Functional Pairs Implementation see 2.2.2~4 Demo: >>> p = pair(1, 2) >>> select(p, 0) 1 As long as we do not violate the abstraction barrier, we don't need to know that pairs are just simple functions. >>> select(p, 1) 2 70
71 2.2 Functional Pairs Implementation see 2.2.2~4 Demo: >>> p = pair(1, 2) >>> select(p, 0) 1 As long as we do not violate the abstraction barrier, we don't need to know that pairs are just simple functions. >>> select(p, 1) 2 If a pair p was constructed from elements x and y, then select(p, 0) returns x, and select(p, 1) returns y. 71
72 2.2 Functional Pairs Implementation see 2.2.2~4 Demo: >>> p = pair(1, 2) >>> select(p, 0) 1 As long as we do not violate the abstraction barrier, we don't need to know that pairs are just simple functions. >>> select(p, 1) 2 If a pair p was constructed from elements x and y, then select(p, 0) returns x, and select(p, 1) returns y. Therefore, this pair representation is valid! 72
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