CS 3520 Homework 6

Due: Wednesday, October 16th, 2019 11:59pm

Difficulty:  ★★☆☆

Start with more-lazy.rkt.

Part 1 — Conditional and Lazy Pairs

Implement an interpreter with lazy evaluation and the following grammar:

  <Exp> = <Number>
        | <Symbol>
        | {+ <Exp> <Exp>}
        | {* <Exp> <Exp>}
        | {lambda {<Symbol>} <Exp>}
        | {<Exp> <Exp>}
        | {let {[<Symbol> <Exp>]} <Exp>}
        | {if0 <Exp> <Exp> <Exp>}
        | {pair <Exp> <Exp>}
        | {fst <Exp>}
        | {snd <Exp>}

That is, a language with single-argument functions and application, an if-zero conditional, and pair, fst, and snd operations. (The language does not include recursive bindings or records.) Unlike cons, the pair operation does not require its second argument to be a list (and we do not have an empty-list value, anyway).

Implement your interpreter with the eager Plait language, not a lazy language.

Evaluation of the interpreted langauge must be lazy. In particular, if a function never uses the value of an argument, then the argument expression should not be evaluated. Similarly, if the first or second part of a pair is never needed, then the first or second expression should not be evaluated. Even if the first or second part of a particular pair is used multiple times, the expression to compute the first or second part should be evaluated only once.

From the starting code, expand the parse function to support the new forms: if0, pair, fst, and snd. Also, as in HW 4, provide an interp-expr function; the interp-expr wrapper for interp should take an expression and return either a number S-expression, `function for a function result, or `pair for a pair result. (Meanwhile, the interp function should never return the symbol `pair, just like the starting interp function never returns the symbol `function.) Note that pair results must distinct from function results, so you will need to modify interp and not just use encodings via parse.

  (test (interp-expr (parse `10))
        `10)
  (test (interp-expr (parse `{+ 10 17}))
        `27)
  (test (interp-expr (parse `{* 10 7}))
        `70)
  (test (interp-expr (parse `{{lambda {x} {+ x 12}}
                              {+ 1 17}}))
        `30)
  
  (test (interp-expr (parse `{let {[x 0]}
                               {let {[f {lambda {y} {+ x y}}]}
                                 {+ {f 1}
                                    {let {[x 3]}
                                      {f 2}}}}}))
        `3)
  
  (test (interp-expr (parse `{if0 0 1 2}))
        `1)
  (test (interp-expr (parse `{if0 1 1 2}))
        `2)
  
  (test (interp-expr (parse `{pair 1 2}))
        `pair)
  (test (interp-expr (parse `{fst {pair 1 2}}))
        `1)
  (test (interp-expr (parse `{snd {pair 1 2}}))
        `2)
  (test (interp-expr (parse `{let {[p {pair 1 2}]}
                               {+ {fst p} {snd p}}}))
        `3)
  
  ;; Lazy evaluation:
  (test (interp-expr (parse `{{lambda {x} 0}
                              {+ 1 {lambda {y} y}}}))
        `0)
  (test (interp-expr (parse `{let {[x {+ 1 {lambda {y} y}}]}
                               0}))
        `0)
  (test (interp-expr (parse `{fst {pair 3
                                        {+ 1 {lambda {y} y}}}}))
        `3)
  (test (interp-expr (parse `{snd {pair {+ 1 {lambda {y} y}}
                                        4}}))
        `4)
  (test (interp-expr (parse `{fst {pair 5
                                        ;; Infinite loop:
                                        {{lambda {x} {x x}}
                                         {lambda {x} {x x}}}}}))
        `5)
  
  (test (interp-expr 
         (parse 
          `{let {[mkrec
                  ;; This is call-by-name mkrec
                  ;;  (simpler than call-by-value):
                  {lambda {body-proc}
                    {let {[fX {lambda {fX}
                                {body-proc {fX fX}}}]}
                      {fX fX}}}]}
              {let {[fib
                     {mkrec
                      {lambda {fib}
                        ;; Fib:
                        {lambda {n}
                          {if0 n
                               1
                               {if0 {+ n -1}
                                    1
                                    {+ {fib {+ n -1}}
                                       {fib {+ n -2}}}}}}}}]}
                ;; Call fib on 4:
                {fib 4}}}))
        `5)

  (test (interp-expr 
         (parse 
          `{let {[mkrec
                  ;; This is call-by-name mkrec
                  ;;  (simpler than call-by-value):
                  {lambda {body-proc}
                    {let {[fX {lambda {fX}
                                {body-proc {fX fX}}}]}
                      {fX fX}}}]}
             {let {[nats-from
                    {mkrec
                     {lambda {nats-from}
                       ;; nats-from:
                       {lambda {n}
                         {pair n {nats-from {+ n 1}}}}}}]}
               {let {[list-ref
                      {mkrec
                       {lambda {list-ref}
                         ;; list-ref:
                         {lambda {n}
                           {lambda {l}
                             {if0 n
                                  {fst l}
                                  {{list-ref {+ n -1}} {snd l}}}}}}}]}
                 ;; Call list-ref on infinite list:
                 {{list-ref 4} {nats-from 2}}}}}))
        `6)

Part 2 — Stress Test

There’s no new implementation for this part. Just make sure that your interpreter is able to run the following two tests within a few seconds. If it takes tens of seconds or even more than a minute, then your interpreter is not lazy enough—probably because it re-interps the expression for a pair’s first or second part every time the first or second part is accessed, which turns the linear-time program into a quadratic-time program.

The handin server will not run this test, but we will check your implementation against this test when grading. It must produce the right answer quickly.


  (test (interp-expr 
         (parse 
          `{let {[mkrec
                  {lambda {body-proc}
                    {let {[fX {lambda {fX}
                                {body-proc {fX fX}}}]}
                      {fX fX}}}]}
             {let {[nats-to
                    {mkrec
                     {lambda {nats-to}
                       {lambda {n}
                         {if0 n
                              {pair 0 0}
                              {let {[l {nats-to {+ n -1}}]}
                                {pair {+ {fst l} 1}
                                      l}}}}}}]}
               {let {[sum
                      {mkrec
                       {lambda {sum}
                         {lambda {n}
                           {lambda {l}
                             {if0 n
                                  0
                                  {+ {fst l}
                                     {{sum {+ n -1}} {snd l}}}}}}}}]}
                 {{sum 10000} {nats-to 10000}}}}}))
        `50005000)

  (test (interp-expr 
         (parse 
          `{let {[mkrec
                  {lambda {body-proc}
                    {let {[fX {lambda {fX}
                                {body-proc {fX fX}}}]}
                      {fX fX}}}]}
             {let {[nats-to
                    {mkrec
                     {lambda {nats-to}
                       {lambda {n}
                         {if0 n
                              {pair 0 0}
                              {let {[l {nats-to {+ n -1}}]}
                                {pair l
                                      {+ {snd l} 1}}}}}}}]}
               {let {[sum
                      {mkrec
                       {lambda {sum}
                         {lambda {n}
                           {lambda {l}
                             {if0 n
                                  0
                                  {+ {snd l}
                                     {{sum {+ n -1}} {fst l}}}}}}}}]}
                 {{sum 10000} {nats-to 10000}}}}}))
        `50005000)

Last update: Monday, August 3rd, 2020
mflatt@cs.utah.edu