1.2 Project Speci cation

Update your github repository with a directory submit/prj2-sol containing two les prj2-sol.scm and arith-parser.scm.

The prj2-sol.scm le should contain de nitions for the following functions:

1. (mul-list list x) which when given a proper list list of numbers returns the list formed by multiplying each element of list by x. 5-points
2. (sum-lengths list) which when given a proper list list of proper lists returns the sum of the lengths of all the lists contained directly in list. This function should not be tail-recursive. 5-points
3. (poly-eval coeffs x) which evaluates the polyomial speci ed by list coeffs at x. Speci cally, given a list of n + 1 numeric polynomial coe cients coeffs (c[n] c[n-1] c[0]) and a number x, return the value of c[n]xⁿ+c[n1]xⁿ¹++c[1]x+c[0]. Your computation should evaluate each term as written above; i.e. each c[i]xⁱ term should be explicitly evaluated and added together.. This function should not be tail-recursive. 10-points
4. (poly-eval-horner coeffs x) which uses Horners method to evaluate the polynomial given by list coeffs at number x. The list coeffs is as speci ed in the previous exercise. 10-points
5. (count-occurrences s-exp x) which given a Scheme s-expression s-exp returns the number of sub-expressions which are equal? to x. 10-points
6. (arith-eval exp) which returns the result of evaluating arithmetic expression exp, where exp is either a Scheme number or one of ( op exp-1 exp-2 ) where op is one of add, sub, mul or div specifying respectively the binary arithmetic operators +, -, *, / and exp-1 and exp-2 are arithmetic expressions. 10-points
7. (sum-lengths-tr list) with the same speci cation as sum-lengths but with the requirement that all recursive calls must be tail-recursive. 10points
8. (poly-eval-horner-tr coeffs x) with the same speci cation as poly -eval-horner but with the requirement that all recursive calls must be tail-recursive. 10-points
9. (mul-list-2 list x) with the same speci cation as mul-list but which replaces all recursion with the use of one or more of map, foldl or foldr. 5-points
10. Write a function (sum-lengths-2 list) with the same speci cation as sum-lengths but which replaces all recursion with the use of one or more of map, foldl or foldr. 5-points

The arith-parse.scm le should contain a de nition of a function (arith- parse tokens) which should return an AST representing the structure of the proper list tokens as speci ed by the following EBNF grammar:

expr

: term ( + term )*

; term

: factor ( * factor )*

; factor

: NUMBER

| < expr >

;

Both the + and * operators should be left-associative.

The proper list tokens should contain Scheme numbers, +, * and < and > (the latter two symbols are used as parentheses). The AST for + and * should use tags add and mul repectively; i.e. the output of arith-parse should be suitable as input to arith-eval.

If tokens cannot be parsed according to the above grammar, then (arith- parse tokens) should return #f. 20-points

The project is subject to the following additional restrictions:

• You should not use any of Schemes mutation operators; i.e. no Scheme function with name ending in ! may be used.
• The functions de ned in prj2-sol.scm should not de ne any top-level auxiliary functions; i.e. any auxiliary functions needed for the operation of a required function should be de ned within that function. This restriction does not apply to arith-parse.scm.
• None of the functions may use map, foldl, or foldr unless explicitly speci ed.
• Some of the function speci cations give implementation restrictions. Those must be followed.

1.3 Example Log

The following provides a log of interaction with the code submitted with this project:

$ racket

Welcome to Racket v7.2. > (load prj2-sol.scm)

> (mul-list (3 4 5) 8)

(24 32 40)

> (mul-list () 8) ()

> (sum-lengths ( (1 2 3) (()) (() 2 (3 4 5))))

7

> (sum-lengths ( )) 0

> (poly-eval (5 4 3 2 1) 1)

15

> (poly-eval (5 4 3 2 1) 2)

129

> (poly-eval () 1) 0

> (poly-eval-horner (5 4 3 2 1) 1)

15

> (poly-eval-horner (5 4 3 2 1) 2)

129

> (poly-eval-horner () 1)

0

> (count-occurrences ( (+ 1 2) (a (+ 1 2) 3) ) 1)

2

> (count-occurrences ( (+ 1 2) (a (+ 1 2) 3) ) 3)

1

> (count-occurrences ( (+ 1 2) (a (+ 1 2) 3) ) (+ 1 2))

2

> (count-occurrences ( (+ 1 2) (a (+ 1 2) 3) ) (+ 1 3))

0

> (arith-eval (mul 3 (add 4 (mul 4 3))))

48

> (arith-eval 45) 45

> (sum-lengths-tr ( (1 2 3) (()) (() 2 (3 4 5))))

7

> (sum-lengths-tr ()) 0

> (poly-eval-horner-tr (5 4 3 2 1) 1)

15

> (poly-eval-horner-tr (5 4 3 2 1) 2)

129

> (poly-eval-horner-tr () 2) 0

> (mul-list-2 (4 5 9) 4)

(16 20 36)

> (mul-list-2 () 4) ()

> (sum-lengths-2 ( (1 2 3) (()) (() 2 (3 4 5))))

7

> (sum-lengths-2 ( )) 0

> (load arith-parse.scm) > (arith-parse ( 1 + 2 + 4 * 3 ))

(add (add 1 2) (mul 4 3))

> (arith-parse ( 1 + < 2 + 4 * 3 > ))

(add 1 (add 2 (mul 4 3)))

> (arith-parse ( 1 + < 2 + + 4 * 3 > ))

#f

> (arith-parse ( 1 + 2 + 4 * 3 >))

#f

> (arith-parse ( 1 + < 2 + 4 > * 3 ))

(add 1 (mul (add 2 4) 3))

> (arith-parse ( @ 1 + < 2 + 4 > * 3 ))

#f

> (arith-eval (arith-parse ( 1 + < 2 + 4 > * 3 )))

19

>

$

1.4 Provided Files

The prj2-sol directory contains the following:

README A template README; replace the XXX with your name, B-number and email. You may add any other information you believe is relevant to your project submission. In particular, you should document the datastructure used for your word-store.

prj2-sol.scm A le containing skeletons functions for all except the last exercise. arith-parse.scm A le containing partial code for the last exercise.

1.5 Hints

You may choose to work within the drracket GUI tool or simply use the racket

CLI. Documentation is available from within drracket or from the web site racket-lang.org.

Note that you can use trace to debug your code.

The following points are worth noting for the initial exercises:

• It may be a good idea to initially ignore the requirement of not creating any new top-level functions. Once you have the code for a particular exercise working, then you can squirrel the de nitions of any auxiliary functions into the body of the top-level function using let, let* or letrec as appropriate.
• Almost all the exercises require recursive solutions. Hence you need to clearly identify your base cases and recursive cases. For the former, you will need to provide a basic solution not involving any recursive calls; for the latter you will need to gure out how to combine solutions from one or more recursive calls into a solution to the current call.

In many cases, the recursive solutions will be based on the structure of the data: this is referred to as structural recursion.

• A list is either () or a pair.
• For the arith-eval function, an arith expression is either a Scheme number or (add exp-1 exp-2), , (div exp-1 exp-2) where exp-1 and exp-2 are themselves arith expressions. So for the base case, your evaluator function needs to return the evaluation of a Scheme number. For the recursive cases, all you need to do is combine the results of the recursive calls to the evaluator on the subexpressions appropriately.

The count-occurrences function is one which requires you to process a Scheme expression to an arbitrary depth. Make sure you clearly identify the base case (one of the examples provided in the log should help).

The arith-parser illustrates the techniques necessary when programming using pure functions without destructive assignment. If you understand the provided code in conjunction with the following comments, the code needed to complete the functionality should be straight-forward:

• Since we cannot use destructive assignment, we pass around the list of tokens which still need to be consumed by the parser. The lookahead token will be the head of the list.
• We represent the state of the parser as a parse-state which is either a non-empty list of remaining tokens, () representing end-of- le, or #f representing a syntax error on match failure.
• If a function returns multiple results it is necessary to package up those results into a single return value and then have the caller unpack the return value. An idiom which is used in the provided code:

(let* ([fn-result (fn )]

[val1 (access-val1 fn-result)] [val2 (access-val2 fn-result)]

[valn (access-valn fn-result)]) )

for accessor functions access-val1, access-val2, , access-valn.

[This can be done more conveniently in Scheme using (match ) or

(values ) in conjunction with (call-with-values ) but those have not been covered in class.]

• The (check? tok parse-state) function returns #t i the lookahead token in parse-state matches tok (which should be either NUMBER or a Scheme symbol).
• The (match tok parse-state) returns the new parse-state which returns from matching tok against the lookahead from parse-state and advancing to the next token if the match succeeds.
• We set up our parsing functions to take a single parse-state argument and return a compound parse-result which contains the constructed ast and the next parse-state. Note that the constructed ast is set to #f on error.
• Note that our recursive-descent parser recipe would create the parsing function for expr as:

expr() { Ast t = term(); while (lookahead == +) {

match(+); Ast t1 = term(); t = new Ast(add, t, t1);

} return t;

}

In the absence of destructive assignments, we would need to implement the while loop using a recursive auxiliary function like expr-loop.

• The code for (term) would be structured identically to that for (expr).
• Note that Scheme provides and and or functions with the usual shortcircuit semantics but the value returned is that of the last expression which is evaluated:

> (and 1 (a b))

(a b)

> (or 1 (a b))

1

> (or #f (a b))

(a b)