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Telecooperation/RBG Technische Universitt Darmstadt Copyrighted material; for TUD student use only Introduction to Computer Science I Topic 2: Structured Data Types Data abstraction Prof. Dr. Max Mhlhuser Dr. Guido Rling Slide 2 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Structures The input/output of a function is seldom an atomic value (number, boolean, symbol), but frequently a data object with many different attributes. i.e. CD: title and price We need mechanisms to put compounding data together One of these mechanisms is the structure A structure definition has the following form for example 2 (define-struct s (field1 fieldn)) (define-struct point (x y)) Slide 3 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Structures This definition creates a series of procedures: make-s a constructor procedure, which gets n arguments and returns a structure-value e.g. (define p (make-point 3 4)) creates a new point s? a predicate procedure, which returns true for a value that is generated by make-s and false for every other value e.g. (point? p) true s-field for every field a selector, which gets a structure as an argument and extracts the value of the field e.g. (point-y p) 4 3 (define-struct s (field1 fieldn)) (define-struct point (x y)) Slide 4 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Design of procedures for compound data When do we need structures? If the description of an object consists of many different pieces of information How does our design recipe change? Data analysis: Search the problem statement for descriptions of relevant objects, then generate corresponding data types; describe the contract of the data type Definition of a contract can use the new defined type names, i.e. ;; grant-qualified : Student bool Template: Header + Body, which contains all possible selectors Implementation of the bodies: Design an expression that uses primitive operations, other functions, selector expressions and the variables 4 Slide 5 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Example 5 ;; Data Analysis & Definitions: (define-struct student (last first teacher)) ;; A student is a structure: (make-student l f t) ;; where f, l, and t are symbols. ;; Contract: subst-teacher : student symbol -> student ;; Purpose: to create a student structure with a new ;; teacher name if the teacher's name matches 'Fritz ;; Examples: ;;(subst-teacher (make-student 'Find 'Matthew 'Fritz) 'Elise) ;; = (make-student 'Find 'Matthew 'Elise) ;;(subst-teacher (make-student 'Smith 'John 'Bill) 'Elise) ;; = (make-student 'Smith 'John 'Bill) Slide 6 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Example (continued) 6 ;; Template: ;; (define (subst-teacher a-student a-teacher) ;;... (student-last a-student)... ;;... (student-first a-student)... ;;... (student-teacher a-student)...) ;; Definition: (define (subst-teacher a-student a-teacher) (cond [(symbol=? (student-teacher a-student) 'Fritz) (make-student (student-last a-student) (student-first a-student) a-teacher)] [else a-student])) ;; Test 1: (subst-teacher (make-student 'Find 'Matthew 'Fritz) 'Elise) ;; expected value: (make-student 'Find 'Matthew 'Elise) ;; Test 2: (subst-teacher (make-student 'Smith 'John 'Bill) 'Elise) ;; expected value: (make-student 'Smith 'John 'Bill) Slide 7 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 The meaning of structures in the substitution model (1/2) How does define-struct work in the substitutions model? This structure produces the following operations: make-c : a constructor c-f1 c-f2 : a series of selectors c? : a predicate 7 (define-struct c (f1 fn)) Slide 8 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 The meaning of structures in the substitution model We proceed like in every combination Evaluation of the operator and the operands The value of (make-c v1 vn) is (make-c v1 vn) this way constructors are self evaluating! The evaluation of (c-fi v) is vi if v = (make-c v1 vi vn) An error in all other cases The evaluation of (c? v) is true, if v = (make-c v1 vn) false, otherwise Try it with the DrScheme Stepper! 8 Slide 9 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Data abstraction For procedures we have primitive expressions ( +, -, and, or, ) means of combination (procedure implementation) means of abstraction (procedural abstraction) We have the same for data : primitive data (numbers, boolean, symbols) compounded data (i.e. structures) Data abstraction. 9 Slide 10 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Why do we need data abstraction? Example: Implementation of an operation for adding rational numbers It is composed of a numerator and a denominator, i.e. 1/2 or 7/9. The addition of two rational numbers produces two results: the resulting numerator and the resulting denominator. But a procedure can only return one value. Thats why we would need two procedures: One returns the resulting numerator, the other the resulting denominator. We have to remember which numerator is part of which denominator. Data abstraction is a method that combines several Data objects, so they can be used as a single object. How this works is hidden by means of data abstraction. 10 Slide 11 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Why do we need data abstraction? The new data objects are abstract data: They are used without making any assumptions about how they are implemented. Data abstraction helps to elevate the conceptual level at which programs are designed, increase the modularity of designs and enhance the expressive power of a programming language. A concrete data representation is defined independent of the programs using the data. The interface between the representation and a program using the abstract data is a set of procedures, called selectors and constructors. 11 Slide 12 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Language extensions for handling abstract data Constructor: a procedure that creates instances of abstract data from data that is passed to it Selector: a procedure that returns a data item that is in an abstract data object The component data item returned might be the value of an internal variable, or it might be computed. Constructors/Selectors generated by define-struct are a special case The component data returned by these selectors is never computed, but always one of the values that was passed during the constructor call 12 Slide 13 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Example: rational Numbers Mathematically represented by a pair of integers: 1/2, 7/9, 56/874, 78/23, etc. Constructor: (make-rat numerator denominator) Selectors: (numer rn) (denom rn) That's all a user needs to know! 13 Slide 14 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 User-defined Operation for Rational Numbers Multiplication of x = n x /d x and y = n y /d y (n x /d x ) * (n y /d y ) = (n x *n y ) / (d x *d y ) 14 ;; mul-rat: rat rat -> rat (define (mul-rat x y) (make-rat (* (numer x) (numer y)) (* (denom x) (denom y)))) Slide 15 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 User-defined Operations on Rational Numbers Addition of x = n x /d x and y = n y /d y n x /d x + n y /d y = (n x *d y + n y *d x ) / (d x *d y ) 15 ;; add-rat: rat rat -> rat (define (add-rat x y) (make-rat (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) Subtraction and division are defined similarly to addition and multiplication. Slide 16 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 A test Equality: n x /d x = n y /d y iff n x *d y = n y *d x 16 iff means: if and only if ;; equal-rat: rat rat -> bool (define (equal-rat? x y) (= (* (numer x) (denom y)) (* (numer y) (denom x)))) Slide 17 string (denom x)))) 17 To output rational numbers in a convenient form, we define an output procedure using data abstraction. This is your first example with string manipulation! string-append puts several strings together. number->string turns a number in a string. This is not possible with symbols."> Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 An output operation ;; print-rat: rat -> String (define (print-rat x) (string-append (number->string (numer x)) "/" (number->string (denom x)))) 17 To output rational numbers in a convenient form, we define an output procedure using data abstraction. This is your first example with string manipulation! string-append puts several strings together. number->string turns a number in a string. This is not possible with symbols. Slide 18 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Below the abstract data We implemented the operators add-rat, mul-rat and equal-rat using make-rat, denom, numer. Without implementing make-rat, denom, numer! Even without knowing, how they will be implemented We still need to define make-rat, denom, numer. Therefore, we have to glue together numerator and denominator. To achieve this, we create a Scheme structure for storing pairs: 18 (define-struct xy (x y)) Slide 19 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Representing rational numbers (define (make-rat n d) (make-xy n d)) (define (numer r) (xy-x r)) (define (denom r) (xy-y r)) 19 We can define the constructor and the selectors with the assistance of the xy structure. Slide 20 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Using operations on rational numbers (define one-third (make-rat 1 3)) (define four-fifths (make-rat 4 5)) (print-rat one-third) 1/3 (print-rat (mul-rat one-third four-fifths)) 4/15 (print-rat (add-rat four-fifths four-fifths)) 40/25 20 Slide 21 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Levels of abstraction Programs are built up as layers of language extensions. Each layer is a level of abstraction. Each abstraction hides some implementation details. There are four levels of abstraction in our rational numbers example. 21 add-rat mul-rat equal-rat make-rat numer denom Rational numbers as numerators and denominators Rational numbers as structures Rational numbers in problem domain make-xy xy-x xy-y Whatever way structures are implemented Programs that use rational numbers Slide 22 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Bottom level Level of pairs Procedures make-xy, xy-x and xy-y are already constructed by the interpreter due to the structure definition. The actual implementation of structures is hidden. 22 Rational numbers as structures make-xy xy-x xy-y Whatever way structures are implemented Slide 23 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Second level Level of rational numbers as data objects Procedures make-rat, numer and denom are defined at this level. The actual implementation of rational numbers is hidden at this level. 23 make-rat numer denom Rational numbers as numerators and denominators Rational numbers as structures Slide 24 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Third level Level of service procedures on rational numbers Procedures add-rat, mul-rat, equal-rat, etc. are defined at this level. Implementation of these procedures are hidden at this level. 24 add-rat mul-rat equal-rat Rational numbers as numerators and denominators Rational numbers in problem domain Slide 25 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Top level Program level Rational numbers are used in calculations as if they were ordinary numbers. 25 Rational numbers in problem domain Programs that use rational numbers Slide 26 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Abstraction barriers Each level is designed to hide implementation details from higher-level procedures. These levels act as abstraction barriers. 26 Slide 27 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Advantages of data abstraction Programs can be designed one level of abstraction at a time. Thereby data abstraction supports top-down design. We can gradually figure out representations, constructors, selectors and service procedures that we need, one level at a time. We do not have to be aware of implementation details below the level at which we are programming. This means there is less to keep in mind at any one time while programming. An implementation can be changed later without changing procedures written at higher levels. 27 Slide 28 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Change data representation A few slides ago we saw: Rational numbers are not always in reduced form. We decide that rational number should be represented in a reduced form. 40/25 and 8/5 are the same number. Thanks to data abstraction our service procedures do not care in which form the number is represented. The procedures like add-rat or equal-rat function correctly in either case. 28 (print-rat (add-rat four-fifths four-fifths)) "40/25" Slide 29 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Change data representation (define (make-rat n d) (make-xy (/ n (gcd n d)) (/ d (gcd n d)))) 29 We can change the constructor or the selectors. (define (numer x) (/ (xy-x x) (gcd (xy-x x) (xy-y x)))) (define (denom x) (/ (xy-y x) (gcd (xy-x x) (xy-y x)))) gcd is a built-in procedure that produces the greatest common divisor! Slide 30 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Up to this point, our procedures have handled only one type of data Numbers Booleans Symbols Types of special structures But we often want that procedures operate with different types of data We will also learn how to protect procedures from wrong use 30 Slide 31 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Example We have (define-struct point (x y)) for points Many points are on the x-axis In this case we want to represent these points just by a number A = (make-point 6 6), B= (make-point 1 2) C = 1, D = 2, E = 3 31 Slide 32 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data To document our representation of points, we make the following informal definition of data types Now the contract, the description and the header of a procedure distance-to-0 is easy: How can we differentiate between the data types? With the help of the predicates: number?, point? etc. 32 ;; a pixel-2 is either ;; 1. a number ;; 2. a point-structure ;; distance-to-0 : pixel-2 -> number ;; to compute the distance of a-pixel to the origin (define (distance-to-0 a-pixel)...) Slide 33 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Base structure: Procedure body with cond-expression that analyzes the type of the input We know that in the second case the input is composed of two coordinates 33 (define (distance-to-0 a-pixel) (cond [(number? a-pixel)...] [(point? a-pixel)...])) (define (distance-to-0 a-pixel) (cond [(number? a-pixel)...] [(point? a-pixel) (point-x a-pixel) (point-y a-pixel) ])) Slide 34 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Now it is easy to complete the function 34 (define (distance-to-0 a-pixel) (cond [(number? a-pixel) a-pixel] [(point? a-pixel) (sqrt (+ (sqr (point-x a-pixel)) (sqr (point-y a-pixel))))])) built-in procedures: (sqr x) : x square (sqrt x) : square root of x Slide 35 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Another Example: graphical objects Variants: squares, circles, procedures: calculating the perimeter, draw, 35 ;; A shape is either ;; a circle structure: ;; (make-circle p s) ;; where p is a point describing the center ;; and s is a number describing the radius; or ;; a square structure: ;; (make-square nw s) ;; where nw is the north-west corner point ;; and s is a number describing the side length. (define-struct square (nw length)) (define-struct circle (center radius)) ;; Examples: (make-square (make-point 20 5) 5) ;; (make-circle (make-point 5 9) 87) Slide 36 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Compute the perimenter: Using our design-recipe Using our design-recipe (continued) 36 ;; perimeter : shape -> number ;; to compute the perimeter of a-shape (define (perimeter a-shape) (cond [(square? a-shape)... ] [(circle? a-shape)... ])) ;; perimeter : shape -> number ;; to compute the perimeter of a-shape (define (perimeter a-shape) (cond [(square? a-shape)... (square-nw a-shape)..(square-length a-shape)...] [(circle? a-shape)... (circle-center a-shape)..(circle-radius a-shape)..])) Slide 37 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Designing Procedures for Mixed Data Compute perimeter: Final result 37 ;; perimeter : shape -> number ;; to compute the perimeter of a-shape (define (perimeter a-shape) (cond [(square? a-shape) (* (square-length a-shape) 4)] [(circle? a-shape) (* (* 2 (circle-radius a-shape)) pi)])) Slide 38 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 Program design and heterogeneous data The data analysis gets more important Which classes of objects exists and what are their attributes? Which meaningful groupings of classes are there? So called subclass creation Yields a hierarchy of data definitions in general Templates 1. step: cond -expression that analyzes the types of data inside a group 2. step: add selectors accordingly for every branch Alternative: call a procedure specific to the respective data type (i.e.: procedure perimeter-circle, perimeter-square ) Program body Combine the available information in every branch of the case, depending on the purpose Alternative: implement procedures specific to data types one by one using the normal design recipe For an overview of the new design process, see HTDP Fig. 18 38 Slide 39 Dr. G. Rling Prof. Dr. M. Mhlhuser RBG / Telekooperation Introduction to Computer Science I: T2 39 Get your data structures correct first, and the rest of the program will write itself. David Jones