You must include a signature, purpose, check- expects and complete function definition.

Transcription

1 Page 1 of 12 Problem 1: Self- Referential Data Natural Numbers Given the data definition below, design a function that consumes a natural number n and produces a list of the squares of the natural numbers from 1 to n. So, for example, given the natural number 4, the function must produce (list ). If n is 0, the function must produce the empty list. ;; Natural is one of: ;; - 0 ;; - (add1 Natural) ;; interp. a natural number (define N0 0) (define N1 1) (define N2 2) #; (define (fn-for-natural n) (cond [(zero? n) (...)] (... n (fn-for-natural (sub1 n)))])) You must include a signature, purpose, check- expects and complete function definition. ;; Natural -> (listof Natural) ;; produce list of squares of naturals from 1 to n, empty if n is 0 (check-expect (list-sqrs 0) empty) (check-expect (list-sqrs 4) (list )) (define (list-sqrs n) (cond [(zero? n) empty] (append (list-sqrs (sub1 n)) (list (* n n)))])) Or: /1 Signature (no part marks) /1 Purpose /2 check-expects (base, n at least 2 each must be correct) /1 function has reasonable name /1 use of template is evident /1 base case result /2 combination /1 for (cons (list-sqrs (sub1 n)) (list (* n n)) (no other part marks) (define (list-sqrs n) (build-list n (λ(n) (sqr (add1 n)))))

2 Page 2 of 12 Problem 2: Built- in Abstract List Processing Functions a) Given the following partial function design: ;; (listof Image) Natural -> Boolean ;; produce true if all images in loi have a height greater than h (check-expect (all-taller-than? empty 10) true) (check-expect (all-taller-than? (list (square 20 "solid" "blue") (circle 40 "outline" "yellow")) 10) true) (check-expect (all-taller-than? (list (square 3 "solid" "green") (square 100 "outline" "red")) 10) false) (define (all-taller-than? loi h) true) ;stub i) Which abstract built- in function is most appropriate to use in the body of all-taller-than? andmap ii) What's the signature of the function you'll pass to the built- in abstract function? Image -> Boolean iii) Does the function argument to the built- in abstract function need to be a closure? Yes /2 Function /2 Signature /1 Closure?

3 Page 3 of 12 b) Given the following partial function design: ;; (listof String) -> (listof String) ;; produce list of only strings in los that start with "p" (check-expect (start-with-p-only empty) empty) (check-expect (start-with-p-only (list "red" "green" "purple")) (list "purple")) (check-expect (start-with-p-only (list "cat" "dog" "horse")) empty) (check-expect (start-with-p-only (list "purple" "pink")) (list "purple" "pink")) (define (start-with-p-only los) los) ;stub i) Which abstract built- in function is most appropriate to use in the body of start-with-p-only? filter ii) What's the signature of the function you'll pass to the built- in abstract function? String -> Boolean iii) Does the function argument to the built- in abstract function need to be a closure? No /2 Function /2 Signature /1 Closure?

4 Page 4 of 12 c) Given the following partial function design: ;; (listof Image) -> Image ;; overlay all images in loi (check-expect (overlay-all empty) empty-image) (check-expect (overlay-all (list (square 8 "solid" "black") (circle 12 "solid" "white") (triangle 18 "solid" "black"))) (overlay (square 8 "solid" "black") (circle 12 "solid" "white") (triangle 18 "solid" "black") empty-image)) (define (overlay-all loi) empty-image) i) Which abstract built- in function is most appropriate to use in the body of overlay-all? foldr ii) What's the signature of the function you'll pass to the built- in abstract function? Image Image -> Image iii) Does the function argument to the built- in abstract function need to be a closure? No /2 Function /2 Signature /1 Closure?

5 Page 5 of 12 Problem 3: Two One- Ofs The data definition for a person having a name, a biological mother and biological father is provided: (define-struct person (name mother father)) ;; Person is one of: ;; - false ;; - (make-person String Person Person) ;; interp. a person with a name, biological mother ;; and biological father; false means person unknown (define P1 (make-person "Jing" false false)) (define P2 (make-person "Mack" false false)) (define P3 (make-person "Mae" P1 P2)) (define P4 (make-person "Pat" false false)) (define P5 (make-person "Chris" false false)) (define P6 (make-person "Jack" P4 P5)) (define P7 (make-person "Tina" P3 P6)) #; (define (fn-for-person p) (cond [(false? p) (...)] (... (person-name p) (fn-for-person (person-mother p)) (fn-for-person (person-father p)))])) Design a function that consumes a person and a list of names. The function must produce true if the list of names represents a path through the ancestors of the given person and false otherwise. The path must start with the name of the given person. So, in the context of person P7 defined above and pictured here: Jing Mack Pat Chris \ / \ / Mae Jack \ / Tina the following are paths through the ancestor tree: (list "Tina") (list "Tina" "Mae" "Mack") (list "Tina" "Jack" "Chris") - because Tina is at the root of the tree - because Tina is at the root, has a parent named Mae and Mae has a parent named Mack. - because Tina is at the root, has a parent named Jack and Jack has a parent named Chris.

6 Page 6 of 12 The following, however, are not paths through the ancestor tree: (list "Jack") (list "Tina" "Pat") - because Jack is not at the root of the tree - because Tina does not have a parent named Pat Your solution must include signature, purpose, a correctly labeled cross- product of types comments table, check- expects and a complete function definition. The function and check- expects that you design must be consistent with the table! p false (make-person String Person Person) empty true true los (cons String false (... ListOfString) (valid-path? (person-mother p) (rest los)) (valid-path? (person-father p) (rest los))) /1 parameters labeled (consistent with function definition) /1 two correct cases for ListOfString /1 two correct cases for Person /3 three correct base case answers (1 for each) /2 general case (1 each natural recursion w/ correct arguments)

7 Page 7 of 12 ;; Person ListOfString -> Boolean ;; produce true if los is a valid path through p (check-expect (valid-path? false empty) true) (check-expect (valid-path? P7 empty) true) (check-expect (valid-path? false (list "Jane")) false) (check-expect (valid-path? P7 (list "Tina" "Pat")) false) (check-expect (valid-path? P7 (list "Tina" "Mae" "Mack")) true) (define (valid-path? p los) (cond [(empty? los) true] [(false? p) false] (and (string=? (person-name p) (first los)) (or (valid-path? (person-mother p) (rest los)) (valid-path? (person-father p) (rest los))))])) /1 signature /1 purpose /3 check-expect (/1 for each base case) /3 check-expects for general case /1 produces false /1 produces true /1 involves mother and father tree (OK if last two are combined) /2 cond has three cases (no marks for 4) /2 base case questions and answers are correct AND consistent with table (/1 for each) /3 correct combination in general case

8 Page 8 of 12 Problem 4: Designing Abstract Functions from Templates Given the data definition that follows: (define-struct node (name next)) ;; Path is one of: ;; - false ;; - (make-node String Path) ;; interp. a path is either empty (false) or ;; is a node with a name and a next Path (define P1 false) (define P2 (make-node "a" false)) (define P3 (make-node "b" (make-node "a" false))) (define (fn-for-path p) (cond [(false? p) (...)] (... (node-name p) (fn-for-path (node-next p)))])) Design an abstract fold function for Path. You must include the signature, purpose, two check- expects as described below, and complete function definition. You must include two check-expects where each fulfills one of the following requirements: 1) creates a copy of the argument 2) produces a list of the names of all the nodes in the path Please write your answer to this question on the following page.

9 Page 9 of 12 ;;(String X -> X) X Path -> X ;; abstract fold function for Path (check-expect (fold-path make-node false P3) P3) (check-expect (fold-path cons empty P3) (list "b" "a")) (define (fold-path c b p) (cond [(false? p) b] (c (node-name p) (fold-path c b (node-next p)))])) /5 signature (/2 function argument, /1 each other piece) /1 purpose /3 correct check-expect to copy argument (/1 make-node, /1 false, /1 rest) /3 correct check-expect to produce list of names of nodes (/1 cons, /1 empty, /1 rest) /2 correct parameterization of combination function (/1 adding parameter, /1 for passing it in the NR) /1 correct use of combination function parameter /2 correct parameterization of base case (/1 adding parameter, /1 for passing it in the NR) /1 correct use of base case parameter

10 Page 10 of 12 Problem 5: Designing Data Definitions In this problem we imagine that we are working in the domain of human geography and are interested in geographical regions, their sizes and populations. A geographical region has a name, an arbitrary number of sub- regions (which are themselves geographical regions) and an arbitrary number of cities. A city has a name, an area in square kilometres (measured to the nearest square kilometre) and a population (again, the number of people who live there). 1) Design data definitions to represent the information described above. Your data definitions must include an example that represents the geographical region shown in the following diagram: 2) For each data definition, be sure to include struct definitions (if needed), types comments, interpretation statements, examples and template functions.

11 Page 11 of 12 Note: It is not necessary to encapsulate the template functions in a local expression. Do not make use of the built- in data definition (listof ). You must provide a complete data definition for any ListOf (define-struct city (name area pop)) ;; City is (make-city String Natural Natural) ;; interp. a city with a name, geographical area in sq kms and population (define C1 (make-city "Vancouver" )) (define C2 (make-city "Calgary" )) (define C3 (make-city "Edmonton" )) (define C4 (make-city "Whitehorse" )) #; (define (fn-for-city c) (... (city-name c) (city-area c) (city-pop c))) ;; ListOfCity is one of: ;; - empty ;; - (cons City ListOfCity) ;; interp. a list of cities (define LOC1 empty) (define LOC2 (list C1 C2)) #; (define (fn-for-loc loc) (cond [(empty? loc) (...)] (... (fn-for-city (first loc)) (fn-for-loc (rest loc)))]))

12 Page 12 of 12 (define-struct geo-area (name subs loc)) ;; GeoArea is (make-geo-area String ListOfGeoArea ListOfCity) ;; interp. a geographical area with a name, list of sub-regions and list of cities (define G1 (make-geo-area "British Columbia" empty (list C1))) (define G2 (make-geo-area "Alberta" empty (list C2 C3))) (define G3 (make-geo-area "Western Canada" (list G1 G2) (list C4))) #; (define (fn-for-ga ga) (... (geo-area-name ga) (fn-for-loga (geo-area-subs ga)) (fn-for-loc (geo-area-loc ga)))) ;; ListOfGeoArea is one of: ;; - empty ;; - (cons GeoArea ListOfGeoArea) ;; interp. a list of geographical areas (define LOG1 empty) (define LOG2 (list G1 G2)) #; (define (fn-for-loga loga) (cond [(empty? loga) (...)] (... (fn-for-ga (first loga)) (fn-for-loga (rest loga)))])) In general, -1 for each distinct error, unless otherwise noted: /4 arbitrary-arity tree data is represented (/2 for arbitrariness in each of two directions) /2 GeoArea with name, subs, and cities /2 City with name, area, and population /2 differentiating GeoArea and City (either by having separate data definitions or an itemization, or...) (no part marks) /4 types comments are well formed /2 interpretations are clear /3 example that matches "Western Canada" in given diagram In following section, minimum -2 for each distinct error (must be consistent with types comments): /3 NR in templates is correct /3 NMR in templates is correct /2 NH in templates is correct (give these marks if NH not needed)