PS8: Open to Interpretation

• Due: Thu., May 04
• Notes:
  • This is the last required pset, which covers material through the lecture on Thu. Apr. 27. It will be followed by a completely optional, extra credit PS09 that covers additional material from the following week.
  • Although formally “due” on Thu., May 04, this (and all your other as-yet-incomplete CS251 work) must actually be completed by the end of finals period.
  • This pset has 150 points total: 25 for one solo problem (now complete) and 125 for five regular problems.
  • The problems needn’t be done in order. Feel free to jump around.
• Submission:
  • In your yourFullName CS251 Spring 2017 Folder, create a Google Doc named yourFullName CS251 PS8.
  • At the top of your yourFullName CS251 PS8 doc, include your name, problem set number, date of submission, and an approximation of how long each problem part took.
  • For all parts of all problems, include all answers (including Racket and SML code) in your PS8 google doc. Format code using a fixed-width font, like Consolas or Courier New. You can use a small font size if that helps.
  • For Problems 2, include your interpretation/compilation derivations. It is easiest to write these via nested bullets, as shown in both the pset description and the lecture notes.
  • For Problem 3:
    • Include the modified parts of yourAccountName-ps8-postfix.rkt in your Google Doc. (You only need to include the modified parts, not the entire contents of the PostFix interpreter!)
    • Drop a copy of your yourAccountName-ps8-postfix.rkt in your ~/cs251/drop/ps08 drop folder on cs.wellesley.edu.

  • In Problem 1 and Problems 4–6, you will modify the following code files from starter files populating your ~wx/cs251/sml/ps8 directory on your wx Virtual Machine:

    • Problem 1: LazySequence.sml
    • Problem 4: IntexArgChecker.sml.sml
    • Problem 5: BindexToPostFix.sml
    • Problem 6: Simprex.sml and SimprexEnvInterp.sml

    Drop a copy of your entire ps8 directory into your ~/cs251/drop/ps08 drop folder on cs.wellesley.edu.

  • For Problem 1 and Problems 4–6 you should include all modified functions from the four files listed above in your Google Doc (so that I can comment on them). In the Google Doc, you need only include the functions you modified, not all functions.

Starting this Problem Set

Problem 1 and Problems 4–6 involve starter files in the ~wx/cs251/sml/ps8 directory in your wx Virtual Machine.

To create this directory, execute the following two commands in a wx VM shell:

cd ~/cs251/sml
git pull origin master

If you performed the above commands before the LazySequence.sml starter file for Problem 1 was posted, you may simply exexecute the commands again to get the single extra file.

1. Solo Problem (now complete): Lazy Sequences (25 points)

This is a solo problem. This means you must complete it entirely on your own without help from any other person and without consulting resources other than course materials or online documentation. You may ask Lyn for clarification, but not for help.

Background

This is a module/ADT problem that shares aspects of the FunSet and OperationTreeSet problems from PS7. It involves a data structure implementation involving multiple tree nodes, where some of the tree nodes have functional components.

The problem involves a list-like data structure called a sequence that has the following signature:

signature SEQUENCE = sig
    (* The type of a sequence *)
    type ''a t 

    (* An empty sequence *)
    val empty : ''a t

    (* Create a sequence of (hi - lo) values fcn(lo), fcn(lo+1), ..., fcn(hi-1) *)
    val segment: int -> int -> (int -> ''a) -> ''a t

    (* Convert a lengnth-n list into a sequence of n values *)                          
    val fromList: ''a list -> ''a t

    (* Concatenate two sequences: given a length-m sequence s and length-n sequence t, 
       returns a single length-m+n sequence that has all values of s followed by all 
       values of t *)
    val concat : ''a t -> ''a t -> ''a t

    (* Return the length of a sequence = number of values in it *)                              
    val length : ''a t -> int

    (* Return the nth value of a sequence (0-indexed). 
       Raise an IndexOutOfBounds exception for an out-of-bounds index. *)                                
    val get : int -> ''a t -> ''a

    (* Return a sequence that results from applying f to each elt *)
    val map : (''a -> ''b) -> ''a t -> ''b t

    (* Return a list with all elements in the sequence. The ith element of the resulting
       list should be the ith element of the given sequence. *)
    val toList : ''a t -> ''a list

end

It would be straightforward to implement a sequence as a list. But this implementation can have issues.

As a concrete example, consider the following inefficient ways to increment and double a number:

(* A slow incrementing function *)
fun linearIncrement n =
  let fun loop num ans =
  if num = 0 then ans else loop (num-1) (ans+1)
  in loop n 1
 end

(* A very slow doubling function *)
fun quadraticDouble n =
  let fun loop num ans =
  if num = 0 then ans else loop (num-1) (linearIncrement ans)
  in loop n n
 end

For example, quadraticDouble 10000 returns fairly quickly, but quadraticDouble 100000 takes many seconds.

Now suppose we want to create the sequence

val reallyBigSeq = segment ~200000000 200000000 quadraticDouble

There are two big problems if we try to represent this as a list. First, the list would need to contain 400 million elements. Second, calculating each entry in the list (especially for large inputs) would take a very long time. So the list implementation is bad in terms of both space and time.

An alternative implementation is not to caclulate and store each element when the sequence is created, but to calculate each element on the fly only when the get operation is called to return it. This is the basis of the LazySequence structure that you will implement in this problem. It is based on the following datatype:

datatype ''a t = Segment of int * int * (int -> ''a) (* lo, hi, fcn *)
       | ThunkList of (unit -> ''a) list
       | Concat of ''a t * ''a t

In programming languages, the term “lazy” means “not computed until necessary”. In this context, an element of a sequence will not be calculated until it needs to be known.

The Segment constructor

In the case of reallyBigSeq, the sequence can be represented via the sum-of-product value

Segment(~200000000,200000000,quadraticDouble)

This simply keeps track of the information for constructing the sequence. Now if we wish to get the value of this sequence at index 20000017, we can apply quadraticDouble to (200000017 + ~200000000) = 17 and return the value quadraticDouble(17). This way, we only “pay” for calculations done at the indices on which we actually perform get. This leads to the following implementation of the segment function:

fun segment lo hi fcn = Segment(lo,hi,fcn)

Thunks

Lazy data structures often employ something called a thunk, which is simply a function of zero arguments. Wrapping a calculation in a function of zero arguments is a way to delay it until later ratger than performing it now. For example, consider testThunk:

val testThunk = fn () => quadraticDouble 100000

Even though calculating quadraticDouble 100000 takes many seconds, defining testThunk is immediate, because no calculation is performed until testThunk is called, as in testThunk(). This is called “dethunking”, which can be defined as follows:

fun dethunk thunk = thunk()

So testThunk() can equivalently be written dethunk(testThunk)

Note that every SML function necessarily takes exactly one argument, so the notion that thunks take “zero” arguments is loose terminology. In fact a thunk takes the “unit value”, written (), which is the only value of type unit and can be viewed as a tuple with zero components.

The upside of a thunk is that the delayed computation is never performed if the thunk is never dethunked. The downside is that the calculation is performed every time the thunk is dethunked, so computations can be repeated unnecessarily.

Note that the type of testThunk is unit -> int, where unit specifies that the () value must be supplied to the thunk in order to extract its value. In general, the type of a thunk for a value of type ''a is unit -> ''a, so we use this as the definition of a type abbreviation named thunkTy:

type ''a thunkTy = unit -> ''a

The ThunkList constructor

Whereas the Segment constructor supports “regular” lazy sequences whose values are determined by some function over a range of integers, the ThunkList constructor is intended to support more “irregular” sequences in which there is not a simple correspondence between sequence values and a range of integers. For example, consider these sequences:

val numSeq = fromList [300005, 20074, 4000017, 1002]
val quadSeq = map quadraticDouble numSeq

Suppose we want all the applications of quadraticDouble to be delayed until the sequence elements are extracted. E.g., get 1 quadSeq should calculate quadraticDouble 20074 when the get is performed, but not before that. If the elements at indices 0 and 2 are never accessed via get, then the time-intensive calculations quadraticDouble 300005 and quadraticDouble 4000017 are never performed.

The ThunkList constructor takes a list of thunks:

ThunkList of ''a thunkTy list

Using ThunkList, the quadSeq sequence can be represented as something like:

ThunkList[ fn () => quadraticDouble 300005,      
           fn () => quadraticDouble 20074,      
           fn () => quadraticDouble 4000017, 
           fn () => quadraticDouble 1002 ]

Based on this idea, we can use ThunkList to represent all “irregular” sequences constructed from lists via fromList:

fun fromList xs = ThunkList (List.map (fn x => (fn () => x)) xs)

This means that in numSeq, all the numbers will be thunked, and so will be equivalent to:

ThunkList[ fn () => 300005,      
           fn () => 20074,      
           fn () => 4000017, 
           fn () => 1002 ]]

The Concat constructor

The final constructor in the ''a t sum-of-product sequence datatype is the Concat constructor:

Concat of ''a t * ''a t

This is simply used to glue together two sequences:

fun concat seq1 seq2 = Concat(seq1,seq2)

So in the end, any instance of the ''a t sum-of-product sequence datatype is a binary tree of Concat nodes whose leaves are either Segment nodes or ThunkList nodes.

Your task

The starter file for this problem is LazySequence.sml in the ~wx/cs251/sml/ps8 directory in your wx Virtual Machine. (You may need to perform git pull origin master to get the file if you previously pulled the ps8 directory.)

LazySequence.sml contains the SEQUENCE signature from above and the following partial implementation of the LazySequence structure:

  structure LazySequence :> SEQUENCE = struct

     type ''a thunkTy = unit -> ''a

     datatype ''a t = Segment of int * int * (int -> ''a) (* lo, hi, fcn *)
                 | ThunkList of ''a thunkTy list
                      | Concat of ''a t * ''a t

     fun segment lo hi fcn = Segment(lo,hi,fcn)
     fun fromList xs = ThunkList (List.map (fn x => (fn () => x)) xs)
     fun concat seq1 seq2 = Concat(seq1,seq2)
     fun dethunk thunk = thunk()                            

     val empty = ThunkList []

     fun length seq = 0 (* replace this stub *)

     fun get n seq = raise Unimplemented (* replace this stub *)

     fun map f seq = empty (* replace this stub *)

     fun toList seq = [] (* replace this stub *)

  end

The segment, fromList, concat functions and empty value have already been defined for you. Your task is to implement the remaining four functions: length, get, map, and toList. Pay attention to the following notes in your implementation.

Notes:

• [2017/05/04] Lyn made a change to utils/Utils.sml on this date that is important for avoiding type errors in LazySequence.sml. Be sure to run git pull origin master to grab this change.

• Be careful to use the explicitly qualified List.map and List.length functions when operating on a list, but unqualified map and length operators when opreating on a sequence. Any attempt to use the unqualified map and length on a list will result in a type error.

• The length function should always return an integer and should never raise an exception.

• The map function should always return a sequence and should never raise an exception, even if the mapped function might raise an error on one of the values. For example, in

  val mapSeq2 = map (fn n => 20 div n) (fromList [3, 0, 5, 2, 4])

  no exception should be raised when defining mapSeq2 or evaluating get 0 mapSeq2, get 2 mapSeq2, get 3 mapSeq2, or get 4 mapSeq2. But get 1 mapSeq2 should raise a divide-by-zero exception.

• The map function should preserve the “shape” of its input sum-of-product constructor tree. So it should return a Concat output for a Concat input, a Segment output for a Segment input, and a ThunkList output for a ThunkList input.

• [2017/05/04] In the implementation of the map function, it will be necessary to combine two function values into one. What is the most natural way to do this? (Hint: it’s something we’ve studied before!)

• The get function should raise an IndexOutOfBounds exception for a sequence s when called on a index outside the range 0 and length(s) - 1.

• If calculating one or more values of a sequence s raises an exception, toList s should raise the exception associated with the value having the lowest index.

• The get function should not be used in the implementation of toList.

• The LazySequence structure is followed by numerous testing functions and test cases. (The expected resuls of all the test cases appear in a comment at the end of the file LazySequence.sml.) Here we illustrate the four functions that are used for testing get in the context of these two example sequences:

  val mapSeq1 = map (fn n => 20 div n) (fromList [3, 5, 2, 4])
  val mapSeq2 = map (fn n => 20 div n) (fromList [3, 0, 5, 2, 4])

  • (testGet seq) returns a list of (index, value-at-index) pairs for each valid index in seq. If calculating at least one of the values raises an exception, testGet raises the exception encountered for the lowest index at which an exception is raised by get.

    - testGet mapSeq1;
    val it = [(0,6),(1,4),(2,10),(3,5)] : (int * int) list
    
    - testGet mapSeq2;
    
    uncaught exception Div [divide by zero]
      raised at: /tmp/emacs-region28242nLJ:129.31-129.34
                 /tmp/emacs-region28242nLJ:104.29

  • (testGetRange seq lo hi) returns a list of (index, value-at-index) pairs for each index in seq in the range lo (inclusive) to hi (exclusive). If calculating at least one of the values raises an exception, testGet raises the exception encountered for the lowest index at which an exception is raised by get.

    - testGetRange mapSeq1 1 3;
    val it = [(1,4),(2,10)] : (int * int) list
    
    - testGetRange mapSeq1 3 6;
    
    uncaught exception IndexOutOfBounds
      raised at: /tmp/emacs-region28242nLJ:68.16-68.34
                 /tmp/emacs-region28242nLJ:108.29
    
    - testGetRange mapSeq2 2 5;
    val it = [(2,4),(3,10),(4,5)] : (int * int) list
    
    - testGetRange mapSeq2 0 3;
    
    uncaught exception Div [divide by zero]
      raised at: /tmp/emacs-region28242nLJ:129.31-129.34
                /tmp/emacs-region28242nLJ:108.29

  • (testGetHandleExceptions eltToString seq) returns a list of (index, string-for-value-or-exception-at-index) pairs for each valid index in seq. The eltToString function must be supplied to turn each sequence value into a string. If an exception is encountered, the exception string is used in place of a value at that index.

    - testGetHandleException Int.toString mapSeq1;
    val it = [(0,"6"),(1,"4"),(2,"10"),(3,"5")] : (int * string) list
    
    - testGetHandleException Int.toString mapSeq2;
    val it = [(0,"6"),(1,"Error: Div -- divide by zero"),(2,"4"),(3,"10"),(4,"5")] : (int * string) list

  • (testGetRangeHandleExceptions eltToString seq lo hi) returns a list of (index, string-for-value-or-exception-at-index) pairs for each index in seq in the range lo (inclusive) to hi (exclusive). The eltToString function must be supplied to turn each sequence value into a string. If an exception is encountered, the exception string is used in place of a value at that index.

    - testGetRangeHandleException Int.toString mapSeq1 ~1 6;
    val it =
      [(~1,"Error: IndexOutOfBounds -- ~1"),(0,"6"),(1,"4"),(2,"10"),(3,"5"),
       (4,"Error: IndexOutOfBounds -- 4"),(5,"Error: IndexOutOfBounds -- 5")]
      : (int * string) list
    
    - testGetRangeHandleException Int.toString mapSeq2 ~1 6;
    val it =
      [(~1,"Error: IndexOutOfBounds -- ~1"),(0,"6"),
       (1,"Error: Div -- divide by zero"),(2,"4"),(3,"10"),(4,"5"),
       (5,"Error: IndexOutOfBounds -- 5")] : (int * string) list

2. Metaprogramming Derivations (30 points)

2a. Deriving Implementations (10 points)

As we discussed in the Metaprogramming lecture, there are two basic reasoning rules involving interpreters, translators (a.k.a. compilers), and program implementation:

The Interpreter Rule (I)

The Translator Rule (T)

In practice, we often elide the word “machine” for interpreter machines and translator machines. For example, we will refer to an “L interpreter machine” as an “L interpreter”, and an “S-to-T translator machine” as an “S-to-T translator” or “S-to-T compiler”. We will also often elide the word “program”; e.g., we will refer to a “P-in-L program” as “P-in-L”. So an “L interpreter” means an “L interpreter machine” while a “S-in-L interpreter” means an “S-in-L interpreter program”. Similar remarks hold for translators (also called compilers): an “S-to-T translator” means an “S-to-T translator machine”, while a “S-to-T-in-L translator” means an “S-to-T-in-L translator program”.

Example

For example, we considered the following elements in class:

• a 251-web-page-in-HTML program (i.e., a 251 web page written in HTML)
• an HTML-interpreter-in-C program (i.e., a web browser written in C)
• a C-to-x86-compiler-in-x86 program (i.e., a C-to-x86 compiler written in x86)
• a x86 computer (i.e., an x86 interpreter machine)

They can be used to build a 251 web page display machine as follows:

$\cfrac{ \text{251-web-page-in-HTML program} \qquad \cfrac{ \cfrac{ \text{HTML-interpreter-in-C program} \qquad \cfrac{ \text{C-to-x86-compiler-in-x86 program} \qquad \text{x86 computer} }{ \text{C-to-x86 compiler machine} } (I) }{ \text{HTML-interpreter-in-x86 program} } (T) \qquad \text{x86 computer} }{ \text{HTML interpreter machine} } (I) }{ \text{251 web page machine} } (I)$

Feel free to abbreviate by dropping the words “program” and “machine”. “… in …” denotes a program, while “… compiler” and “… translator” denote translator machines and “… interpreter” and “… computer” denote interpreter machines.

$\cfrac{ \text{251 web page in HTML} \qquad \cfrac{ \cfrac{ \text{HTML interpreter in C} \qquad \cfrac{ \text{C-to-x86 compiler in x86} \qquad \text{x86 computer} }{ \text{C-to-x86 compiler} } (I) }{ \text{HTML interpreter in x86} } (T) \qquad \text{x86 computer} }{ \text{HTML interpreter} } (I) }{ \text{251 web page machine} } (I)$

Writing Derivations

Rather that displaying derivations using horizontal lines, it is recommended that you use nested bullets to indicate the same structure as the inference rules (though drawn “upside down”). For example, the derivation above could also be written:

• 251 web page machine (I)
  • 251-web-page-in-HTML program
  • HTML interpreter machine (I)
    • HTML-interpreter-in-x86 program (T)
      • HTML-interpreter-in-C program
      • C-to-x86 compiler machine (I)
        • C-to-x86-compiler-in-x86 program
        • x86 computer
    • x86 computer

Questions

1. (5 points) Suppose you are given the following:

   • A P-in-Java program
   • a Java-to-C-compiler-in-Racket program (i.e., a Java-to-C compiler written in Racket);
   • a Racket-interpreter-in-x86 program (i.e., a Racket interpreter written in x86 machine code);
   • a C-to-x86-compiler-in-x86 program (i.e., a C-to-x86 compiler written in x86 code);
   • an x86 computer (i.e., an x86 interpreter machine).

   Using the two reasoning rules above, construct a “proof” that demonstrates how to execute the P-in-Java program on the computer. That is, show how to construct a P machine.

2. (5 points) Suppose you are given:

   • a GraphViz-in-C program
   • a C-to-LLVM-compiler-in-x86 program
   • an LLVM-to-JavaScript-compiler-in-x86 program
   • a JavaScript-interpreter-in-ARM program
   • an x86 computer (i.e., an x86 interpreter machine, such as your laptop)
   • an ARM computer (i.e., an ARM interpreter machine, such as a smartphone)

   Show how to construct a GraphViz machine.

2b. Bootstrapping (20 points)

In his Turing Award lecture-turned-short-paper Reflections on Trusting Trust, Ken Thompson describes how a compiler can harbor a so-called “Trojan horse” that can make compiled programs insecure. Read this paper carefully and do the following tasks to demonstrate your understanding of the paper:

1. (8 points) Stage II of the paper describes how a C compiler can be “taught” to recognize '\v' as the vertical tab character. Using the same kinds of components and processes used in Problem 1, we can summarize the content of Stage II by carefully listing the components involved and describing (by constructing a proof) how a C compiler that “understands” the code in Figure 2.2 can be generated. (Note that the labels for Figures 2.1 and 2.2 are accidentally swapped in the paper.) In Stage II, two languages are considered:

   • the usual C programming language
   • C+\v = an extension to C in which '\v' is treated as the vertical tab character (which has ASCII value 11).

   Suppose we are given the following:

   • a C-to-binary compiler (Here, “binary” is the machine code for some machine. This compiler is just a “black box”; we don’t care where it comes from);
   • a C+\v-to-binary-compiler-in-C (Figure 2.3 in Thompson’s paper);
   • a C+\v-to-binary-compiler-in-C+\v (what should be Figure 2.2 in Thompson’s paper);
   • a machine that executes the given type of binary code.

   Construct a proof showing how to use the C+\v-to-binary-compiler-in-C+\v source code to create a C+\v-to-binary-compiler-in-binary program.

2. (12 points) Stage III of the paper describes how to generate a compiler binary harboring a “Trojan horse”. Using the same kinds of components and processes used in Problem 1, we can summarize the content of Stage III by carefully listing the components involved and describing (by constructing a proof) how the Trojaned compiler can be generated. In particular, assume we have the parts for this stage:

   • a C-to-binary-compiler (again, just a “black box” we are given);
   • a C-to-binary-compiler-in-C without Trojan Horses (Figure 3.1 in Thompson’s paper);
   • a C-to-binary-compiler_TH-in-C with two Trojan Horses (Figure 3.3 in Thompson’s paper);
   • a login-in-C program with no Trojan Horse;
   • a binary-based computer;

   The subscript _TH indicates that a program contains a Trojan Horse. A C-to-binary compiler_TH has the “feature” that it can insert Trojan Horses when compiling source code that is an untrojaned login program or an untrojaned compiler program. That is, if P is a login or compiler program, it is as if there is a new rule:

   The Trojan Horse Rule (TH)

   $$\cfrac{ P\text{-in-C program} \qquad \text{C-to-binary-compiler}_{TH} }{ P_{TH}\text{-in-binary} } (TH)$$

   Using these parts along with the two usual rules (I) and (T) and the new rule (TH), show how to derive a Trojaned login program login_TH-in-binary that is the result of compiling login-in-C with a C compiler that is the result of compiling C-to-binary-compiler-in-C. The interesting thing about this derivation is that login_TH-in-binary contains a Trojan horse even though it is compiled using a C compiler whose source code (C-to-binary-compiler-in-C) contains no code for inserting Trojan horses!

3. Extending PostFix (30 points)

fIn lecture we studied several different versions of an interpreter for the PostFix language implemented in Racket. This pset involves starting with the following version of the interpreter:

• ps8-postfix-starter.rkt

This is a slightly modified version of the file postfix-transform-fancy.rkt studied in lecture.

Begin by making a copy of ps8-postfix-starter.rkt named yourAccountName-ps8-postfix.rkt and load this into Dr. Racket. Near the bottom of this file is the following PostFix program named sos (for “sum of squares”). Racket’s semi-colon comments have been used to explain the commands in the program:

;; Sum-of-squares program 
(define sos
  '(postfix 2 ; let's call the arguments a and b, from top down
            1 nget ; duplicate a at top of stack
            mul ; square a
            swap ; stack now has b and a^2 from top down
            1 nget mul ; square b
            add ; add b^2 + a^2 and return
            ))

Let’s run the program on the arguments 5 and 12:

> (postfix-run sos '(5 12))
About to execute commands (1 nget mul swap 1 nget mul add) on stack (5 12)
  after executing 1, stack is (1 5 12)
  after executing nget, stack is (5 5 12)
  after executing mul, stack is (25 12)
  after executing swap, stack is (12 25)
  after executing 1, stack is (1 12 25)
  after executing nget, stack is (12 12 25)
  after executing mul, stack is (144 25)
  after executing add, stack is (169)
169

Note that the stack that results from executing each command on the previous stack is displayed, line by line. This behavior is controlled by the print-stacks? flag at the top of the program:

(define print-stacks? #t)

If we set the flag to #f, the intermediate stack display will be turned off:

(define print-stacks? #f)

> (postfix-run sos '(5 12))
169

Turn the print-stacks? flag on when it’s helpful to understand or debug a PostFix program.

1. (10 points)

   Consider the following Racket function g that is defined near the bottom of the PostFix intepreter file:

   (define (g a b c)
     (- c
        (if (= 0 (remainder a 2))
            (quotient b (- a c))
            (* (+ b c) a))))

   In this part, your goal is to flesh out the definition of a three-argument PostFix program pfg that performs the same calculation as g on three arguments:

   (define pfg
      '(postfix 3 
          ;; Flesh out and comment the commands in this PostFix program 
        ))

   Here are some examples with a correctly defined pfg:

   > (apply g '(10 2 8))
   7
   
   > (postfix-run pfg '(10 2 8))
   7
   
   > (apply g '(-7 2 3))
   38
   
   > (postfix-run pfg '(-7 2 3))
   38
   
   > (apply g '(5 4 5))
   -40
   
   > (postfix-run pfg '(5 4 5))
   -40

   Notes:

   • Please comment the commands in pfg like those in sos.

   • You have been provided with a testing function (test-pfg) that will test your pfg function on 5 sets of arguments:

     ;; Tests on an incorrect definition of pfg: 
     > (test-pfg)
     Testing pfg on (10 2 8): ***different results***
       g: 3
       pfg: -3
     Testing pfg on (11 2 8): ***different results***
       g: -102
       pfg: 102
     Testing pfg on (-6 3 8): ***different results***
       g: 10
       pfg: -10
     Testing pfg on (-7 2 3): ***different results***
       g: 38
       pfg: -38
     Testing pfg on (5 4 5): ***different results***
       g: -40
       pfg: -40
     
     ;; Tests on a correct definition of pfg: 
     > (test-pfg)
     Testing pfg on (10 2 8):  same result for g and pfg = 3
     Testing pfg on (11 2 8):  same result for g and pfg = -102
     Testing pfg on (-6 3 8):  same result for g and pfg = 10
     Testing pfg on (-7 2 3):  same result for g and pfg = 38
     Testing pfg on (5 4 5):  same result for g and pfg = -40

2. (5 points) Extend PostFix by adding the following three commands:

   • le is like lt, but checks “less than or equal to” rather than “less than”
   • ge is like gt, but checks “greater than or equal to” rather than “greater than”
   • and expects two integers at the top of the stack. It replaces them by 0 if either is 0; otherwise it replaces them by 1.

   For example:

   > (postfix-run '(postfix 0 4 5 le) '())
   1
   
   > (postfix-run '(postfix 0 5 5 le) '())
   1
   
   > (postfix-run '(postfix 0 5 4 le) '())
   0
   
   > (postfix-run '(postfix 0 4 5 ge) '())
   0
   
   > (postfix-run '(postfix 0 4 4 ge) '())
   1
   
   > (postfix-run '(postfix 0 5 4 ge) '())
   1
   
   > (postfix-run '(postfix 0 0 0 and) '())
   0
   
   > (postfix-run '(postfix 0 0 1 and) '())
   0
   
   > (postfix-run '(postfix 0 1 0 and) '())
   0
   
   > (postfix-run '(postfix 0 0 17 and) '())
   0
    
   > (postfix-run '(postfix 0 17 0 and) '())
   0
   
   > (postfix-run '(postfix 0 1 1 and) '())
   1
   
   > (postfix-run '(postfix 0 1 17 and) '())
   1
   
   > (postfix-run '(postfix 0 17 17 and) '())
   1
    
   > (postfix-run '(postfix 0 17 23 and) '())
   1

   Notes:

   • Do not modify postfix-exec-command for this part. Instead, just add three bindings to the list postfix-relops.

   • The testing function (test-3b) will test all of le, ge, and and in the context of the single PostFix program test-sorted:

     (define test-sorted '(postfix 3 ; let's call the arguments a, b, and c, from top down
                              2 nget le ; is a <= b?
                              3 nget 3 nget ge ; is c >= b?
                              and ; is a <= b and c >= b?
                              ))
     
     > (test-3b) ; Uses the test-sorted program in its definition 
     Trying test-sorted on (4 5 6): works as expected; result = 1
     Trying test-sorted on (4 5 5): works as expected; result = 1
     Trying test-sorted on (4 4 5): works as expected; result = 1
     Trying test-sorted on (4 4 4): works as expected; result = 1
     Trying test-sorted on (4 6 5): works as expected; result = 0
     Trying test-sorted on (5 6 4): works as expected; result = 0
     Trying test-sorted on (5 4 6): works as expected; result = 0
     Trying test-sorted on (6 5 4): works as expected; result = 0
     Trying test-sorted on (6 4 5): works as expected; result = 0
     Trying test-sorted on (5 5 4): works as expected; result = 0
     Trying test-sorted on (5 4 4): works as expected; result = 0

3. (5 points) Extend PostFix with a dup command that duplicates the top element of the stack (which can be either an integer or executable sequence). For example:

   (define sos-dup '(postfix 2 dup mul swap dup mul add))
   
   > (postfix-run sos-dup '(3 4))
   About to execute commands (dup mul swap dup mul add) on stack (3 4)
     after executing dup, stack is (3 3 4)
     after executing mul, stack is (9 4)
     after executing swap, stack is (4 9)
     after executing dup, stack is (4 4 9)
     after executing mul, stack is (16 9)
     after executing add, stack is (25)
   25
   
   (define cmd-dup '(postfix 1 (dup dup mul add swap) dup 3 nget swap exec exec pop))
   
   > (postfix-run cmd-dup '(4))
   About to execute commands ((dup dup mul add swap) dup 3 nget swap exec exec pop) on stack (4)
     after executing (dup dup mul add swap), stack is ((dup dup mul add swap) 4)
     after executing dup, stack is ((dup dup mul add swap) (dup dup mul add swap) 4)
     after executing 3, stack is (3 (dup dup mul add swap) (dup dup mul add swap) 4)
     after executing nget, stack is (4 (dup dup mul add swap) (dup dup mul add swap) 4)
     after executing swap, stack is ((dup dup mul add swap) 4 (dup dup mul add swap) 4)
   About to execute commands (dup dup mul add swap) on stack (4 (dup dup mul add swap) 4)
     after executing dup, stack is (4 4 (dup dup mul add swap) 4)
     after executing dup, stack is (4 4 4 (dup dup mul add swap) 4)
     after executing mul, stack is (16 4 (dup dup mul add swap) 4)
     after executing add, stack is (20 (dup dup mul add swap) 4)
     after executing swap, stack is ((dup dup mul add swap) 20 4)
     after executing exec, stack is ((dup dup mul add swap) 20 4)
   About to execute commands (dup dup mul add swap) on stack (20 4)
     after executing dup, stack is (20 20 4)
     after executing dup, stack is (20 20 20 4)
     after executing mul, stack is (400 20 4)
     after executing add, stack is (420 4)
     after executing swap, stack is (4 420)
     after executing exec, stack is (4 420)
     after executing pop, stack is (420)
   420
   
   > (postfix-run '(postfix 0 dup) '())
   ERROR: dup requires a nonempty stack ()

   Notes:

   • Implement dup by adding a cond clause to postfix-exec-command.

   • Test your dup implementation using the above test cases. Your dup implementation should ensure there is at least one value on the stack, and give an appropriate error message when there isn’t.

4. (10 points) In this part you will extend PostFix with a rot command that behaves as follows. The top stack value should be a positive integer rotlen that we’ll call the rotation length. Assume there are n values v₁, …, v_n below rotlen on the stack, where n is greater than or equal to rotlen. Then the result of performing the rot command is to rotate the top rotlen elements of the stack by moving v₁ after v_rotlen. I.e., the order of values on the stack after rot should be v₂, …, v_rotlen, v₁, v_rotlen+1, …, v_n. So the first rotlen elements of the stack have been rotated by one unit, while the order of the remaining elements on the stack is unchanged.

   Here are examples involving rot:

   (define rot-test '(postfix 6 4 rot 3 rot 2 rot))
   
   > (postfix-run rot-test '(8 7 6 5 9 10))
   About to execute commands (4 rot 3 rot 2 rot) on stack (8 7 6 5 9 10)
     after executing 4, stack is (4 8 7 6 5 9 10)
     after executing rot, stack is (7 6 5 8 9 10)
     after executing 3, stack is (3 7 6 5 8 9 10)
     after executing rot, stack is (6 5 7 8 9 10)
     after executing 2, stack is (2 6 5 7 8 9 10)
     after executing rot, stack is (5 6 7 8 9 10)
   5
   
   > (postfix-run '(postfix 3 (mul add) rot) '(5 6 7))
   About to execute commands ((mul add) rot) on stack (5 6 7)
     after executing (mul add), stack is ((mul add) 5 6 7)
   ERROR: rot length must be a positive integer but is (mul add)
   
   > (postfix-run '(postfix 3 -1 rot) '(5 6 7))
   About to execute commands (-1 rot) on stack (5 6 7)
     after executing -1, stack is (-1 5 6 7)
   ERROR: rot length must be a positive integer but is -1
   
   > (postfix-run '(postfix 3 4 rot) '(5 6 7))
   About to execute commands (4 rot) on stack (5 6 7)
     after executing 4, stack is (4 5 6 7)
   ERROR: not enough stack values for rot (4 5 6 7)
   
   > (postfix-run '(postfix 0 rot) '())
   About to execute commands (rot) on stack ()
   ERROR: rot requires a nonempty stack but is ()

   Notes:

   • Implement rot by adding a cond clause to postfix-exec-command.

   • Racket supplies a list-tail function that is very helpful for implementing rot:

     > (list-tail '(10 20 30 40 50) 0)
     '(10 20 30 40 50)
     
     > (list-tail '(10 20 30 40 50) 1)
     '(20 30 40 50)
     
     > (list-tail '(10 20 30 40 50) 2)
     '(30 40 50)
     
     > (list-tail '(10 20 30 40 50) 3)
     '(40 50)
     
     > (list-tail '(10 20 30 40 50) 4)
     '(50)
     
     > (list-tail '(10 20 30 40 50) 5)
     '()

     Racket does not provide a similar list-head function, but I have provided it in the ps8-postfix-starter.rkt file. It works this way:

     > (list-head '(10 20 30 40 50) 0)
     '()
     
     > (list-head '(10 20 30 40 50) 1)
     '(10)
     
     > (list-head '(10 20 30 40 50) 2)
     '(10 20)
     
     > (list-head '(10 20 30 40 50) 3)
     '(10 20 30)
     
     >  (list-head '(10 20 30 40 50) 4)
     '(10 20 30 40)
     
     >  (list-head '(10 20 30 40 50) 5)
     '(10 20 30 40 50)

   • Test your rot implementation using the above test cases. Your rot implementation should give appropriate error messages in various situations, like those in the test cases.

4. Static Argument Checking in Intex (15 points)

As described in the lecture slides on Intex, it is possible to statically determine (i.e., without running the program) whether an Intex program contains a argument reference with an out-of-bound index. In this problem you will flesh out skeleton implementations of the two approaches for static argument checking sketched in the slides. The skeleton implementations are in the file ps8/IntexArgChecker.sml.

1. Top-down checker (7 points): In the top-down approach, the checkTopDown function on a program passes the number of arguments to the checkExpTopDown function on expressions, which recursively passes it down the subexpressions of the abstract syntax tree. When an Arg expression is reached, the index is examined to determine a boolean that indicates whether it is in bounds. The boolean on all subexpressions are combined in the upward phase of recursion to determine a boolean for the whole tree. Complete the following skeleton to flesh out this approach.

   (* val checkTopDown: pgm -> bool *)
   fun checkTopDown (Intex(numargs,body)) =
     checkExpTopDown numargs body
   
   (* val checkExpTopDown : int -> Intex.exp -> bool *)
   and checkExpTopDown numargs (Int i) = raise Unimplemented
     | checkExpTopDown numargs (Arg index) = raise Unimplemented
     | checkExpTopDown numargs (BinApp(_,exp1,exp2)) = raise Unimplemented

   Uncomment the definition of topDownChecks to test your implementation.

2. Bottom-up checker (6 points): In the bottom-up approach, the checkExpBottomUp function returns a pair (min, max) of the minimum and maximum argument indices that appear in an expression. If an expression contains no argument indices, it should return the (valOf Int.maxInt, valOf Int.minInt), which is a pair of SML’s (1) maximum integer and (2) minumum integer. The funtions Int.min and Int.max are useful for combining such pairs. (Note that Int.maxInt is the identity for Int.min, and Int.minInt is the identity for Int.max.)

   The checkBottomUp function on programs returns true if all argument references are positive and less than or equal to the programs number of arguments, and false otherwise. Complete the following skeleton to flesh out this approach.

   (* checkBottomUp: pgm -> bool *)
   fun checkBottomUp (Intex(numargs,body)) =
     let val (min,max) = checkExpBottomUp body
      in 0 < min andalso max <= numargs
     end
      
   (* val checkExpBottomUp : Intex.exp -> int * int *)
   and checkExpBottomUp (Int i) = (valOf Int.maxInt, valOf Int.minInt)
     | checkExpBottomUp (Arg index) = raise Unimplemented
     | checkExpBottomUp (BinApp(_,exp1,exp2)) = raise Unimplemented

   Uncomment the definition of bottomUpChecks to test your implementation.

3. Static vs dynamic checking (2 points): A static argument index checker flags programs that might dynamically raise an index-out-of-bounds error, but does not guarantee they will dynamically raise such an error. Give an example of an Intex program for which checkTopDown and checkBottomUp returns false but does not raise an argument-out-of-bounds error when run on particular arguments. Hint: such a program can raise a different error.

5. Bindex To PostFix (15 points)

Background

In lecture, we studied the following intexToPostFix function that automatically translates Intex programs to equivalent PostFix programs:

fun intexToPostFix (Intex.Intex(numargs, exp)) =
  PostFix.PostFix(numargs, expToCmds exp 0)
    (* 0 is a depth argument that statically tracks 
       how many values are on stack above the arguments *)

and expToCmds (Intex.Int i) depth = [PostFix.Int i]
  | expToCmds (Intex.Arg index) depth = [PostFix.Int (index + depth), PostFix.Nget]
   (* specified argument is on stack at index + depth *)
  | expToCmds (Intex.BinApp(binop,exp1,exp2)) depth =
    (expToCmds exp1 depth) (* 1st operand is at same depth as whole binapp *)
    @ (expToCmds exp2 (depth + 1)) (* for 2nd operand, add 1 to depth 
                                      to account for 1st operand *)
    @ [PostFix.Arithop (binopToArithop binop)]

and binopToArithop Intex.Add = PostFix.Add
  | binopToArithop Intex.Sub = PostFix.Sub
  | binopToArithop Intex.Mul = PostFix.Mul
  | binopToArithop Intex.Div = PostFix.Div
  | binopToArithop Intex.Rem = PostFix.Rem

For example, given the Intex program intexP2 expressed as an Intex.pgm datatype instance

val intexP2 = Intex(4, BinApp(Mul,
                              BinApp(Sub, Arg 1, Arg 2),
                              BinApp(Div, Arg 3, Arg 4)))

the intexToPostFix translation is as follows:

- intexToPostFix(intexP2);
val it =
  PostFix
    (4,
     [Int 1,Nget,Int 3,Nget,Arithop Sub,Int 4,Nget,Int 6,Nget,Arithop Div,
      Arithop Mul]) : PostFix.pgm

With the helper function translateString shown below, the input and output of translation can be expressed as strings in s-expression format:

fun translateString intexPgmString =
  PostFix.pgmToString (intexToPostFix (Intex.stringToPgm intexPgmString))

- IntexToPostFix.translateString("(intex 4 (* (- ($ 1) ($ 2)) (/ ($ 3) ($ 4))))");
val it = "(postfix 4 1 nget 3 nget sub 4 nget 6 nget div mul)" : string

There are two key aspects to the translation from Intex to PostFix:

1. The expToCmds function translates every Intex expression to a sequence of PostFix commands that, when executed, will push the integer value of that expression onto the stack. In the case of translating a binary application expression, these sequence is composed by appending the sequences for the two operands followed by the appropriate arithmetic operator command.

2. The trickiest part of expToCommands is knowing how many integers are on the stack above the program arguments, so that references to Intex arguments by index can be appropriately translated. This is handled by a depth argument to expToCommands that keeps track of this number. Note that the first operand of a binary application has the same depth as the whole binary application expression, but the second operand has a depth one greater than the first to account for the integer value of the first operand pushed onto the stack.

Your Task

In this problem, you will implement a similar translator from Bindex programs to PostFix programs.

The file ps8/BindexToPostFix.sml contains a skeleton of the Bindex to PostFix translator in which all Bindex expressions translate to a command sequence consisting of the single integer command 42:

fun makeArgEnv args = Env.make args (Utils.range 1 ((length args) + 1))
(* returned env associates each arg name with its 1-based index *)

fun envPushAll env = Env.map (fn index => index + 1) env
(* add one to the index for each name in env *)

fun envPush name env = Env.bind name 1 (envPushAll env)

fun bindexToPostFix (Bindex.Bindex(args, body)) =
  PostFix.PostFix(length args, expToCmds body (makeArgEnv args))

(* In expToCmds, env statically tracks the depth of each named variable
   value on the stack *)
and expToCmds (Bindex.Int i) env = [PostFix.Int 42] (* replace this stub *)
  | expToCmds (Bindex.Var name) env = [PostFix.Int 42] (* replace this stub *)
  | expToCmds (Bindex.BinApp(binop, rand1, rand2)) env = [PostFix.Int 42] (* replace this stub *)
  | expToCmds (Bindex.Bind(name, defn, body)) env = [PostFix.Int 42] (* replace this stub *)

and binopToArithop Bindex.Add = PostFix.Add
  | binopToArithop Bindex.Sub = PostFix.Sub
  | binopToArithop Bindex.Mul = PostFix.Mul
  | binopToArithop Bindex.Div = PostFix.Div
  | binopToArithop Bindex.Rem = PostFix.Rem

When loaded, the BindexToPostFix.sml file performs numerous tests of the translator. Initially, all of these fail. Your task is to redefine the clauses of expToCmds so that all the test cases succeed.

Notes:

• The expToCmds function in the Bindex to PostFix translator uses an environment to track the depth of each variable name in the program. Recall that an environment (defined by the Env structure in utils/Env.sml) maps names to values. In this case, it maps names in a Bindex program to the current stack depth of the value associated with that name. Carefully study the helper functions makeArgEnv, envPushAll, and envPush to understand what they do; all are helpful in the contex of this problem.

• As in the Intex to PostFix translator, in the Bindex to PostFix translator, each Bindex expression should translate to a sequence of PostFix commands that, when executed, pushes exactly one value — the integer value of that expression — onto the stack.

• For simplicity, you may assume there are no unbound variables in the Bindex program you are translating. With this assumption, you may use the Option.valOf function to extract the value v from a SOME(v) value.

• The BindexToPostFix structure contains a testTranslator function that takes the name of a file containing a Bindex program and a list of argument lists and (1) displays the Bindex program as an s-expression string, (2) displays the PostFix program resulting from the translation as an s-expression string and (3) tests the behavior of both programs on each of the argument lists in the list of argument lists. If the behaviors match, it prints an approprate message; if they don’t match, it reports an an error with the two different result values. For instance, in the initial skelton, the behavioral tests fail on the program in ../bindex/avg.bdx:

  Testing Bindex program file ../bindex/avg.bdx
  
  Bindex program input to translation:
  (bindex (a b) (/ (+ a b) 2))
  
  PostFix program output of translation:
  (postfix 2 42)
  
  Testing args [3,7]: *** ERROR IN TRANSLATION ***
    Bindex result: 5
    PostFix result: 42
  
  Testing args [15,5]: *** ERROR IN TRANSLATION ***
    Bindex result: 10
    PostFix result: 42

  However, when the translator is working, the following will be displayed:

  Testing Bindex program file ../bindex/avg.bdx
  
  Bindex program input to translation:
  (bindex (a b) (/ (+ a b) 2))
  
  PostFix program output of translation:
  (postfix 2 1 nget 3 nget add 2 div)
  
  Testing args [3,7]: both programs return 5
  
  Testing args [15,5]: both programs return 10

• You have succeeded when all of the test cases succeed when you load the .sml file.

6. Simprex (35 points)

Background

Rae Q. Cerf of Toy-Languages-я-Us likes the sigma construct from the Bindex lecture, but she wants something more general. In addition to expressing sums, she would also like to express numeric functions like factorial and exponentiation that are easily definable via simple recursion. The functions that Rae wants to define all have the following form:

f(n) = z, if n ≤ 0
f(n) = c(n, f(n-1)), if n > 0

Here, z is an integer that defines the value of f for any non-positive integer value and and c is a binary combining function that combines n and the value of f(n−1) for any positive n. Expanding the definition yields:

f(n) = c(n, c(n-1, c(n−2, ... c(2, c(1, z)))))

For example, to define the factorial function, Rae uses:

z_fact = 1
c_fact(i, a) = i*a

To define the exponentation function bⁿ, Rae uses:

z_expt = 1
c_expt(i, a) = b*a

In this case, c_expt ignores its first argument, but the fact that c_expt is called n times is important.

As another example, Rae defines the sum of the squares of the integers between 1 and n using

z_sos = 0
c_sos(i, a) = (i*i) + a

The examples considered so far don’t distinguish right-to-left vs. left-to-right evaluation, A simple example of that is

z_sub = 0
c_sub(i, a) = i-a

For example, when n is 4, the result is the value of 4 - (3 - (2 - (1 - 0))), which is 2, and not the value of 1 - (2 - (3 - (4 - 0))), which is -2.

Another example that distinguishes right-to-left vs. left-to-right evaluation is using Horner’s method to calculate the value of the polynomial x^(n-1) + 2*(x^(n-2)) + 3*(x^(n-3)) + ... + (n-1)*x + n:

z_horner = 0
c_horner(i, a) = i + x*a

For example, when n is 4 and x is 5, the result is (4 + 5*(3 + 5*(2 + 5*(1 + 5*0)))) = 194, which is the value of 5^3 + 2*5^2 + 3*5^1 + 4 = (125 + 50 + 15 + 4) = 194.

Rae designs an extension to Bindex named Simprex that adds a new simprec construct for expressing her simple recursions:

(simprec E_zero (Id_num Id_ans E_combine ) E_arg)

This simprec expression evaluates E_arg to the integer value n_arg and E_zero to the integer value n_zero. If n_arg ≤ 0, it returns n_zero. Otherwise, it returns the value

c(n_arg, c(n_arg−1, c(n_arg−2, ... c(2, c(1, nzero)))))

where c is the binary combining function specified by (Id_num Id_ans E_combine). This denotes a two-argument function whose two formal parameters are named Id_num and Id_ans and whose body is E_combine. The Id_num parameter ranges over the numbers from n_arg down to 1, while the Id_ans parameter ranges over the answers built up by c starting at n_zero. The scope of Id_num and Id_ans includes only E_combine; it does not include E_zero or E_arg.

Rae’s simprec expression is closely related to the notion of primitive recursive functions defined in the theory of computation.

Here are some sample Simprex programs:

;; Program that calculuates the factorial of n
(simprex (n) (simprec 1 (i a (* i a)) n))

;; Exponentiation program raising base b to power p 
(simprex (b p) (simprec 1 (i ans (* b ans)) p))

;; Program summing the squares of the numbers from 1 to hi
(simprex (hi) (simprec 0 (i sumSoFar (+ (* i i) sumSoFar)) hi))

;; Program distingishing right-to-left and left-to-right evaluation via subtraction
(simprex (n) (simprec 0 (i ans (- i ans)) n))

;; Calculating x^(n-1) + 2*(x^(n-2)) + 3*(x^(n-3)) + ... + (n-1)*x + n via Horner's method
(simprex (n x) (simprec 0 (i a (+ i (* x a)) n)))

Your Tasks

After completing her design, Rae is called away to work on another language design problem. Toy-Languages-я-Us is impressed with your CS251 background, and has hired you to implement the Simprex language, starting with a version of the Sigmex implementation. (Sigmex is the name of the language that results from extending Bindex with the sigma construct from lecture.) Your first week on the job, you are asked to complete the following tasks that Rae has specified in a memo she has written about finishing her project.

1. (15 points) Rae’s memo contains the following Simprex test programs. Give the results of running each of the programs on the argument 3. Show your work so that you may get partial credit if your answer is incorrect.

   ;; program 1 (2 points)
   (simprex (a) (simprec 0 (b c (+ 2 c)) a))
   
   ;; program 2 (3 points)
   (simprex (x) (simprec 0 (n sum (+ n (* x sum))) 4))
   
   ;; program 3 (4 points)
   (simprex (y) (simprec 0 (a b (+ b (sigma c 1 a (* a c)))) y))
   
   ;; program 4 (6 points)
   (simprex (n) 
     (simprec (simprec (* n (- n 3)) (q r r) (* n n))
              (c d (+ d (simprec 0 (x sum (+ sum (- (* 2 x) 1))) c))) 
              (simprec -5 (a b (+ 1 b)) (* n n))))

2. (20 points)

   Rae has created a skeleton implementation of Simprex by modifying the files for the Sigmex (= Bindex + sigma) implementation to contain stubs for the simprec construct. Her modified files, which are named Simprex.sml and SimprexEnvInterp.sml, can be found in the ps8 folder.

   Finish Rae’s implementation of the Simprex language by completing the following four tasks, which Rae has listed in her memo:

   1. (2 points) Extend the definition of sexpToExp in Simprex.sml to correctly parse simprec expressions.

   2. (2 points) Extend the definition of expToSexp in Simprex.sml to correctly unparse simprec expressions.

   3. (5 points) Extend the definition of freeVarsExp in Simprex.sml to correctly determine the free variables of a simprec expression.

   4. (11 points) Extend the definition of eval in SimprexEnvInterp.sml to correctly evaluate simprec expressions using the environment model.

   Notes:

   • This problem is similar the the Sigmex (= Bindex + sigma) language extension we implemented in lecture. You may wish to study the solution files bindex/SigmexSolns.sml and bindex/SigmexEnvInterp.sml as part of doing this problem.

   • In Simprec.sml, the exp type is defined to be:

     and exp = Int of int (* integer literal with value *)
             | Var of var (* variable reference *)
             | BinApp of binop * exp * exp (* binary operator application with rator, rands *)
             | Bind of var * exp * exp (* bind name to value of defn in body *)
             | Sigma of var * exp * exp * exp (* E_lo, E_hi, E_body *)
             | Simprec of exp * var * var * exp * exp 
                          (* zeroExp * numVar * ansVar * combExp * argExp *)

     The s-expression notation (simprec E_zero (I_num I_ans E_combine) E_arg) is represented in SML as

     Simprec (<exp for E-zero>, <string for I_num>, <string for I_ans>, 
              <exp for E_combine>, <exp for E_arg>)

     For example, the expression (simprec 1 (x a (* x a)) n) is represented in SML as Simprec(Int 1, "x", "a", BinApp(Mul, Var "x", Var "a"), Var "n")

   • You can test your modifications of sexpToExp, expToExp, and freeVarsExp via use "Simprex.sml" after you have first uncommented the test cases at the end of Simprex.sml. This will display the results of various test cases for freeVarsExp before displaying the Simprex structure. It will also test sexpToExp and expToSexp. If you uncomment the test cases before you have correctly modified sexpToExp and expToSexp, any attempt to use "Simprec.sml" or use "SimprecEnvInterp.sml" will fail with a SyntaxError.

   • You can test your modifications of eval via use "SimprexEnvInterp.sml" after you have first uncommented the test cases at the end of SimprexEnvInterp.sml. This will display the results of various test cases for eval before displaying the SimprexEnvInterp structure.

Extra Credit: Self-printing Program (15 points)

In his Turing-lecture-turned-short-paper Reflections on Trusting Trust, Ken Thompson notes that it is an interesting puzzle to write a program in a language whose effect is to print out a copy of the program. We will call this a self-printing program. Write a self-printing program in any general-purpose programming language you choose. Your program should be self-contained — it should not read anything from the file system.

Important Note: There are lots of answers on the Internet, but you will get zero points if you just copy one of those. (I can tell!) Do this completely one your own, without looking anything up online.

Extra Credit: PostFix Iterations (20 points)

1. (5 points) Study and test the following mystery PostFix program of one argument, which is provided near the end of ps8-postfix-starter.rkt. Describe the function that it calculates on its one argument, and give a brief, high-level description of how it calculates that function.

   ;; What function does this compute? 
   (define mystery
     '(postfix 1
               1
               (swap dup 0 eq 
                 (swap)
                 (swap 2 nget mul swap 1 sub swap 3 vget exec)
                 sel exec)
               3 rot 3 vget exec))

   Note: vget is a command that is like nget, except that it can return any value (including an executable sequence), not just an integer. It has been included in your ps8-postfix-starter.rkt file for use in this extra credit problem.

2. (15 points) Write a PostFix program that takes a single argument (assumed to be a nonnegative integer n) and iteratively calculates the nth Fibonacci number.