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Principles of Programming Languages 2014.09.23 Notes • Total available time: 1h 30’. • Y ou may use any written material you need. • Y ou cannot use computers or phones during the exam. 1 Scheme 1.1 Duplicates (7 pts) Define a procedure called rep which takes a list L of elements and returns a list of the elements of L that are repeated at least twice. The procedure must have linear time complexity, and it can be imperative and use imperative data structures. E.g. (rep ’(3 2 "hi" 2 "hello" hello "hi")) is (2 "hi") . 1.2 Duplicates, functional version (6 pts) Define a purely functional version of rep without any limits of time complexity. 2 Haskell 2.1 Duplicates (5 pts) Define rep in Haskell, without any limits of time complexity, and declaring its type. 2.2 List comparison with duplicates (5 pts) Define a predicate comprep for comparing lists, declaring its type. The predicate must accept another predicate (e.g. <=) and use it to compare the lists. The lists are compared counting the number of duplicated elements in them: e.g. comprep((<=), [1,2,1,2], [0,0,1,0]) is false. 3 Prolog (8 pts) Define a Prolog predicate that, given a generic term t, returns a list containing all the atomic elements that appear in t at least twice. E.g. given a(1,b(2),1,a(3,b)), it must return [a,1,b]. 1 Solutions Scheme ;; linear complexity version ;; (under the standard hypotesis that the complexity of hash access is constant) (define (rep L) (let ((h (make-hash)) (out ’())) (for-each (lambda (x) (hash-set! h x (+ 1 (hash-ref h x 0))) ) L) (hash-for-each h (lambda (el n) (when (> n 1) (set! out (cons el out))))) out)) ;; functional, quadratic version (define (repff L) (define (rep0 lst out) (if (null? lst) out (let ((x (car lst)) (xs (cdr lst))) (rep0 xs (if (and (member x xs) (not (member x…