MASSACHVSETTS INSTITVTE OF TECHNOLOGY
      Department of Electrical Engineering and Computer Science

                          6.945 Spring 2009
                            Problem Set 4

  Issued: Wed. 25 Feb. 2009                    Due: Wed. 4 Mar. 2009

Reading:
    SICP, From Chapter 4: section 4.1.7--4.2 (from PS03)
                          section 4.3; (pp. 412--437)

Code: utils.scm, ghelper.scm, syntax.scm, rtdata.scm,
      load-analyze.scm, analyze.scm, repl.scm
      load-amb.scm, analyze-amb.scm repl-amb.scm
      multiple-dwelling.scm


		       Heavy Evaluator Hacking

In this problem set we build interpreters in a different direction.
We start with the essential EVAL/APPLY interpreter, written as an
analyzer of the syntax into a compiler of compositions of execution
procedures -- a small combinator language.  We will warm up by making
modifications to this evaluator.

Next, we will change the evaluator to include AMB expressions.  To add
AMB, the execution procedures will all have a different shape: in
addition to the environment, each will take two "continuation
procedures" SUCCEED and FAIL.  In general, when a computation comes up
with a value it will invoke SUCCEED with the proposed value and a
complaint department which, if invoked, will try to produce an
alternate value.  If a computation cannot come up with a value, it
will invoke the complaint department passed to it in the FAIL
continuation.

An important lesson to be learned here is how to use continuation
procedures to partially escape the expression structure of the
language.  By construction, a functional expression has a unique
value.  However, in the AMB system an expression may be ambiguous as
to its value...  Think about how we arrange that to make sense!

---------------------------------------------------------------------

	     Separating Syntactic Analysis from Execution
		      (Compiling to Combinators)

It is important to read SICP section 4.1.7 carefully here.  When you
load "load-analyze.scm" you will get an evaluator similar to the one
described in this section.  


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Problem 4.1: Warmup

It is often valuable to have procedures that can take an indefinite
number of arguments.  The addition and multiplication procedures in
Scheme are examples of such procedures.  Traditionally, a user may
specify such a procedure in a definition by making the bound-variable
specification of a lambda expression a symbol rather than a list of
formal parameters.  That symbol is expected to be bound to the list of
arguments supplied.  For example, to make a procedure that takes
several arguments and returns a list of the squares of the arguments
supplied, one may write:

(lambda x (map square x))

or

(define (ss . x) (map square x))

and then

(ss 1 2 3 4) ==> (1 4 9 16)

Modify the analyzing interpreter to allow this construction.  

Hint: you do not need to change the code involving DEFINE or LAMBDA
in syntax.scm!  This is entirely a change in analyze.scm

Demonstrate that your modification allows this kind of procedure, and
that it does not cause other troubles.
-------------

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Problem 4.2: Infix notation

Many people like infix notation for small arithmetic expressions.  It
is not hard to write a special form, (INFIX <infix-string>), that
takes a character string, parses it as an infix expression with the
usual precedence rules, and reduces it to Lisp.  Note that to do this
you really don't have to delve into the combinator target mechanism of
the evaluator, since this can be accomplished as a "macro" in the same
way that COND and LET are implemented (see syntax.scm).

So, for example, we should be able to write the program:

(define (quadratic a b c)
  (let ((discriminant (infix "b**2-4*a*c")))
    (infix "(-b+sqrt(discriminant))/(2*a)")))

Hint: Do not try to parse numbers!  That is hard -- let Scheme do it
for you: use string->number (see MIT Scheme documentation).  Just
pass the substring that specifies the number to string->number to get
the numerical value.

Write the INFIX special form, install it in the evaluator, and
demonstrate that it works.

Please! Unless you have lots of time to burn, do not write a complete
infix parser for some entire language, like Python (easy) or Java
(hard)!  We just want parsing of simple arithmetic expressions.
-------------


---------------------------------------------------------------------

		 AMB and Nondeterministic Programming

Now comes the real fun part of this problem set!  Please read section
4.3 of SICP carefully before starting this part.  This interpreter
requires a change in the interface structure of the combinators that
code compiles into, so it is quite different.  Of course, our system
differs from the one in SICP in that it is implemented with generic
extension capability.  The loader for the interpreter extended for AMB
is "load-analyze.scm".

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Problem 4.3: Warmup: Programming with AMB

Run the multiple-dwelling program (to get a feeling for how to use the
system).

Do exercises 4.38, 4.39, and 4.40 (p. 419) from SICP.  

Note: we supply the multiple-dwelling.scm program so you need not type
it in. 
-------------

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Problem 4.4:

Modify the AMB interpreter to record and report the number of
alternatives examined in exploring a search space.  What is this
number for the simple multiple-dwelling program?  For your best
improvement of it from your work in exercise 4.40, above.
-------------