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Shadowing and function-application
hw09: B4

Due Nov.18 23:59 for a 10% bonus. I'll accept through Nov.20 (Sun) 23:59 w/o bonus, and I'll feel bad about casting a shadow over the start of your break :-(.

• Please add a comment “;>>> B3” or “;>>> B4” next to places you add/update for this homework.
• You can add all new tests to the list “all-tests” in the B4-test.rkt (or, ExprTest.java).
• On D2L, submit all files except those provided with B0. (E.g. either B4,-test.rkt, or if using Java then {Expr{,Test},BinOp,IfPos,Id,Let,Func,FuncApply}.java. No .zip/.jar).

We continue to build on the language implementation of B1/B2. You can implement this homework in either Java or Racket. You may use the B2-soln if you want. Link is usuable once you D2L-submit B2; just let me know if you need to bypass that.

B3 is just like B2, except we now allow one variable to shadow another.

For example, we might have a hrrm … in …, x is which in turn contains another hrrm … in …, x is somewhere inside of it. In that case the inner x should shadow the outer one:
⇒ hrrm 7 in hrrm 5 in ^x 3/, x is, x is
⇒ hrrm 5 in ^x 3/, x is¹
⇒ ^5 3/
⇒ 8.
And of course, shadowing may occur between non-adjacent scopes: hrrm 3 in hrrm 4 in hrrm 5 in …, x is, y is, x is.

So what does our interpreter need to do? Well, when eval does substitution in a LetExpr, we just need to be a bit cautious: If we're substituting every x in an Expr, and we encounter a hrrm Expr_rhs in Expr_body, x is then we shouldn't substitute any xs inside the Expr_body. Though of course, if we encounter a hrrm Expr_rhs in Expr_body, y is, then this doesn't affect our substitution.

Using our programming-language vocabulary: when substituting, only substitute “free occurrences” of an Id in E₁, not any “bound occurrences”².

As an example, for the expression hrrm 3 in dark hrrm 4 in ^x dark hrrm 5 in ^x y/, x is light/, y is light, x is, we have the syntax tree drawn at the right albeit with / written as boii, with a dotted-arrow from each binding-occurrence to its bound occurrence(s). This corresponds to some runnable test-cases³:

(check-expect (eval (string->expr "hrrm 3 in dark hrrm 4 in ^x dark hrrm 5 in ^x y/, x is light/, y is light, x is"))
              (+ 3 (+ 5 4)))
(check-expect (subst "x" 3        (string->expr "dark hrrm 4 in ^x dark hrrm 5 in ^x y/, x is light/, y is light"))
              (string->expr                     "dark hrrm 4 in ^3 dark hrrm 5 in ^x y/, x is light/, y is light"))

hint/spoiler: in hrrm Expr_initialize in Expr_body, zed is, we know that

zed can never occur free in Expr_body. We don't even need to look inside it!

0. (0pts) Change the purpose-statement of subst to be “substitute any free occurrences of …”. Note that you will not substitute other binding occurrences, either.
1. For each of the following, fill in the blanks by replacing the outermost LetExpr with just its body, but substituting its variable respecting shadowing. You don't need to submit this, but filling it out is meant to help you understand the problem/issue. If you don't get the code working, but do have a comment with these answers, I can use them to justify partial credit!
   For example (using the program/image above):
   ⇒ hrrm 3 in dark hrrm 4 in ^x dark hrrm 5 in ^x y/, x is light/, y is light, x is
   ⇒ dark hrrm 4 in ^3 dark hrrm 5 in ^x y/, x is light/, y is light
   ⇒ dark ^3 dark hrrm 5 in ^x 4/, x is light/ light
   ⇒ dark ^3 dark ^5 4/ light/ light
   ⇒ 12
   Complete the blanks in a similar way, below:
   1. ⇒ hrrm 3 in hrrm 5 in ^x y/, x is, y is
      ⇒                                             
      ⇒               
      ⇒ 8
   2. ⇒ hrrm 3 in hrrm y in ^x y/, x is, y is
      ⇒                                                               
      ⇒               
      ⇒ 6
   3. ⇒ hrrm 5 in hrrm 3 in ^hrrm y in ^x y/, x is x/, y is, x is
      ⇒                                                                                                                           
      ⇒                                                                           
      ⇒                           
      ⇒               
      ⇒ 11
   4. ⇒ hrrm 5 in hrrm ^x 1/ in ^x 2/, x is, x is
      ⇒ hrrm ^     1/ in ^     2/,      is
      ⇒ hrrm      in ^     2/,      is
      ⇒ ^     2/
      ⇒     

2. Update B2 to B3, by the necessary changes to enable shadowing.
   You are encouraged to build on your own previous solution, but you may also use the B2-soln (.rkt)

   • The change should be quite small, but is surgically precise.
   • Recall that B2's subst is essentially the same code as change-name in AncTrees.
     Hint: What is needed for B3's (and B4's) subst is essentially the same as hw07's change-eyes-to-until.
   • Please label each section of lines you change with a comment “;>>>B3”.

Expr ::= … | FuncExpr | FuncApplyExpr  ;>>>B4

FuncExpr ::= grab Expr <= Id            ;>>>B4   Interpretation: function-value; the Id is the parameter, and the Expr is the function's body.
FuncApplyExpr ::= push Expr => Expr    ;>>>B4   Interpretation: pass the argument (first Expr) to the function (2nd Expr).

Here is the function (λ (x) (+ (* 3 x) 1)) written in B4: grab ^^x 3! 1/ <= x.
And, here is the (uniterated) collatz function, (λ(x) (if (even? n) (* n 1/2) (+ (* 3 x) 1))), written in B4:

grab mace ^n 0.5! ^^3 n! 1/ windu ^n 2# <= n

Just as numbers are self-evaluating, so are FuncExprs. Evaluating (an internal representation of) a function results in that same (internal representation of the) function. We won’t actually evaluate the body until the function is applied. (This is exactly how Java, racket, python, javascript, etc. treat functions.)

A FuncApplyExpr represents calling a function. Here are two expressions, both evaluating to 5·3+1 = 16:

hrrm grab ^^x 3! 1/ <= x       For this function…
in   push 5 => tripleAndInc     …call tripleAndInc(5)…
,    tripleAndInc  is           …where “tripleAndInc” is the name for that function 2 lines above.

push 5 => grab ^^x 3! 1/ <= x   Equivalently: apply a function-literal (w/o bothering to give it a name)

3. First, write the following four functions as B4 programs not as racket programs -- you'll write keywords like grab and ^. They'll just be in comments since they're not racket code. Or hey, you can put them in a string, and then pass those strings to parse as test-cases!.
   1. A constant function that always returns (say) 17.
   2. the function sqr, which squares its input; give it a name with a B4 LetExpr whose body we’ll just leave as “…”. I.e. write the B4 equivalent of racket (let {[sqr (lambda (x) (* x x))]} …).
   3. the factorial function, written in B4 using a LetExpr with body “…” as in the previous example.

      Note: You won’t be able to evaluate function-applications for recursive functions yet (see B5), but we can still write the test cases! (You can comment out that one test case for now, since it’ll trigger a run-time exception otherwise.)

   and
   4. The B4 equivalent of the following racket defining and calling make-adder:

      (define (make-adder n) (lambda (m) (+ n m))) ((make-adder 3) 4) ; evals to 7 ; Note that `(make-adder 3)` evals to `(lambda (m) (+ 3 m))` ; the `((` means we have *two* function-applications: ; we first call `make-adder` (getting back a lambda-value), ; then we call that result we got back.

4. Then, upgrade B3 so that it allows functions to be represented; label each section of lines you change with a comment “;>>>B4”.
   1. Add a struct/class for representing FuncExprs internally.
   2. parse! (and tests).

      heads-up: The provided racket and java parsers each return a punctuation-character as a single token. So when parsing “<=” you'll need to pop twice to consume both punctuation-characters.

   3. expr->string (and tests)
   4. eval (and tests)
5. Implement function-application.
   1. Add a struct/class for representing FuncApplyExprs internally.
   2. parse! (and tests)
   3. expr->string (and tests)
   4. eval (and tests). Here, more than half the points are for tests, since you want to try several situations involving shadowing variables. (You don’t need to test eval’ing your factorial function, though.)

      The semantics of eval’ing the function-application push Expr₁ => Expr₀:

      1. Evaluate Expr₀; let’s call the result actual-arg.
      2. Evaluate Expr₁; let’s call the result f. (f had better be a function-value!)
      3. Substitute f’s parameter with actual-arg in f’s body; call this new expression E′.
      4. Evaluate E′ and return that value.
      Hey, those semantics are practically the same as LetExpr’s! Indeed, it’s not very different; the function holds the identifier and body; when you eval a function-application then we do the same substitution.

The interpreter project is based on the first chapters of Programming Languages and Interpretation, by Shriram Krishnamurthi. As a result, this homework assignment is covered by the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. Although we’re using a different dialect of racket than that book, you might find it helpful to skim it.

  eval(string->expr("hrrm 5 in ^x 3/, x is"))
= eval(string->expr("^5 3/"))
= eval(string->expr("8"))

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