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Shadowing and function-application; prolog
hw09: Z4

Due Nov.30 (Tue) in class23:59.

• Please add a comment “;>>> Z3” or “;>>> Z4” next to places you add/update for this homework.
• You can add all new tests to the list “all-tests” in the Z4-test.rkt (or, ExprTest.java).
• On D2L, submit all files except those provided with Z0. (E.g. either Z4,-test.rkt, or if using Java then {Expr{,Test},BinOp,IfPos,Id,Let,Func,FuncApply}.java. No .zip/.jar).
• Your prolog queries can be inside a comment of a racket file.

We continue to build on the language implementation of Z1/Z2. You can implement this homework in either Java or Racket. You may use the Z2-soln if you want.

Z3 is just like Z2, except we now allow one variable to shadow another.

For example, we might have a let x := … in … which in turn contains another let x := … in … somewhere inside of it. In that case the inner x should shadow the outer one:
⇒ let x := 3 in let x := 5 in [add x 3]
⇒ let x := 5 in [add x 3]¹
⇒ [add 5 3]
⇒ 8.
And of course, shadowing may occur between non-adjacent scopes: let x := 3 in let y := 4 in let x := 5 in ….

In technical terms: when substituting, only substitute “free occurrences” of an Id in E₁, not any “bound occurrences”².

As an example, for the expression let x := 3 in <let y := 4 in [add x <let x := 5 in [add x y]>]>, we have the syntax tree drawn at the right albeit with add instead of 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 "let x := 3 in <let y := 4 in [add x <let x := 5 in [add x y]>]>"))
              (+ 3 (+ 5 4)))
(check-expect (subst "x" 3        (string->expr "<let y := 4 in [add x <let x := 5 in [add x y]>]>"))
              (string->expr                     "<let y := 4 in [add 3 <let x := 5 in [add x y]>]>"))

hint/spoiler: in let zed := Expr_initialize in Expr_body, 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):
   ⇒ let x := 3 in <let y := 4 in [add x <let x := 5 in [add x y]>]>
   ⇒ <let y := 4 in [add 3 <let x := 5 in [add x y]>]>
   ⇒ <[add 3 <let x := 5 in [add x 4]>]>
   ⇒ <[add 3 <[add 5 4]>]>
   ⇒ 12
   Complete the blanks in a similar way, below:
   1. ⇒ let y := 3 in let x := 5 in [add x y]
      ⇒                                             
      ⇒                           
      ⇒ 8
   2. ⇒ let y := 3 in let x := y in [add x y]
      ⇒                                                                     
      ⇒                           
      ⇒ 6
   3. ⇒ let x := 5 in let y := 3 in [add let x := y in [add x y] x]
      ⇒                                                                                                                                       
      ⇒                                                                                             
      ⇒                                                   
      ⇒                           
      ⇒ 11
   4. ⇒ let x := 5 in let x := [add x 1] in [add x 2]
      ⇒ let      := [add      1] in [add      2]
      ⇒ let      :=      in [add      2]
      ⇒ [add      2]
      ⇒     

2. Update Z2 to Z3, by the necessary changes to enable shadowing.
   You are encouraged to build on your own previous solution, but you may also use the Z2-soln (.rkt)

   • The change should be quite small, but is surgically precise.
   • Recall that Z2's subst is essentially the same code as change-name in AncTrees.
     Hint: What is needed for Z3's (and Z4's) subst is like a anc-tree's change-name, but it stops when it reaches the name it’s changing.
   • Please label each section of lines you change with a comment “;>>>Z3”.

Expr ::= … | FuncExpr | FuncApplyExpr  ;>>>Z4

FuncExpr ::= fun Id => Expr            ;>>>Z4   interpretation: function-value; the parameter follows thonk and the body =>.
FuncApplyExpr ::= pass Expr |> Expr    ;>>>Z4   interpretation: call the function (secondfirst Expr), passing it an argument (1st2nd Expr).  TO BE CLEAR: The | of |> is literal punctuation, not a BNF/grammar or.

Here is the function (λ (x) (+ (* 3 x) 1)) written in Z4: fun x => [add [mul x 3] 1].
And, here is the (uniterated) collatz function in Z4:

fun n => zero [mod n 2] ? [mul n 0.5] : [add [mul 3 n] 1]

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:

let tripleAndInc                       bind the name tripleAndInc…
:=  fun x => [add [mul x 3] 1]          …to a function-value
in  pass 5 |> tripleAndInc             …and then apply it to 5

pass 5 |> fun x => [add [mul x 3] 1]   Equivalently: apply a function-literal

3. First, write the following four functions as Z4 programs not as racket programs -- you'll write words like fun and mul. (You can later modify them to make test-cases for your Z4 interpreter.)
   1. A constant function that always returns (say) 17.
   2. the function sqr, which squares its input; give it a name with a Z4 LetExpr whose body we’ll just leave as “…”. I.e. write the Z4 equivalent of racket (let {[sqr (lambda (x) (* x x))]} …).
   3. the factorial function, written in Z4 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 Z5), 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 Z4 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 Z3 so that it allows functions to be represented; label each section of lines you change with a comment “;>>>Z4”.
   1. Add a struct/class for representing FuncExprs internally.
   2. parse! (and tests).
   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 pass Expr₀ |> Expr₁:

      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.

Prolog

As mentioned: Your prolog queries can be inside a comment of a (racket) file, at the top of your submitted hardcopy, thanks! Only include your added rules/facts, not the rest of the provided .pl file.

5. Add another super-person (hero, villain, or neither) and at least three more color-or-weapon preferences to the Prolog superhero knowledge base from lecture.
6. Two super-people are compatible if they share a fighting style, or a color-preference (and they aren’t the same person). Moreover, team(A,B,C) is true if A and B are compatible, and B and C are compatible, but A and C aren’t the same.
   1. define a rule compatible(A,B).
   2. What query will solve for every super-person compatible with bubbles?
   3. define team(A,B,C).
   4. What query will solve for every team starting with bubbles and ending with rafael?

7. Write friendly: We say two super-people are friendly if they are connected through some chain of compatible people. For example, bubbles and rafael would be friendly if compatible(bubbles,magikarp), compatible(magikarp,if0bob), and compatible(if0bob,rafael). (That is: friendly is the reflexive, transitive closure of the compatible relation.) Everybody, of course, is trivially friendly with themselves (since they’re connected by a chain of 0 compatible others).

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("let x := 5 in [add x 3]"))
= eval(string->expr("[add 5 3]"))
= eval(string->expr("8"))

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