SICP/README.md

# SICP

**This is currently (2021/06/11) work in progress.**

These are my solutions to the CS classic [Structure and Interpretation of
Computer Programs](https://mitpress.mit.edu/sites/default/files/sicp/index.html).
I have looked up the answer for some exercises on the
[Scheme Community Wiki](http://community.schemewiki.org/?SICP-Solutions).
I have marked such exercises in their respective script.

You can use the Scheme implementation by the MIT to run these scripts. In Arch,
execute `pacman -S mit-scheme` to install it. Then run the scripts via
`mit-scheme --quiet < script.scm`. You can also use the shell script `./run
script.scm`.

I haven't completely solved the following exercises.

- Exercise 1.13. I wasn't able to do the proof.
- Exercise 4.78. I managed to use the amb-evaluator for the query system, but it
  does not work for joined queries.
- Exercise 4.79. I did not attempt to solve this exercise because of not
  finishing the previous one, and it is the last one in the chapter.
- Exercise 5.52. I have implemented the basic structure of a Scheme to C
  translator, but I only finished a basic proof-of-concept. Most of the
  additional work would be similar to 5.51.

I had a great time working through this book. I feel like my mental capabilities
improved throughout the process, and finishing all the exercises gives me an
incredible feeling of accomplishment. I have written short summaries for each of
the chapters below.


# Chapter 1

The first chapter of SICP starts by explaining the Scheme syntax. The first
couple of exercises are simple enough. However, already at 1.5, the book
foreshadows some of the difficulty that is about to come.

```scheme
(define (p) (p))
(define (test x y)
  (if (= x 0) 0 y))
```

The goal is to decide whether Scheme uses applicative-order-evaluation or
normal-order-evaluation based on the above code. I have initially found the
exercise confusing, but the code triggering an infinite loop is a clear
indication of Scheme (or at least my version of Scheme, MIT Scheme) using
applicative-order-evaluation.

After this exercise, things get more comfortable again. The book proceeds to
introduce if-else clauses, conditionals, as well as recursion. The book uses
these primitives to compare iterative and recursive procedures based on a couple
of typical CS example functions such as computing Fibonacci numbers, greatest
common divisor, and fast exponentiation.

Two new insights I had how using modulo instead of subtracting the divisor
speeds up the GCD algorithm I learned in middle school and how exponentiation
can run in O(log n) by halving even exponents.

I wasn't able to prove the Golden Ration exercise at the time of working through
this chapter. My knowledge of induction and proofs was too limited. I found that
depressing at the time, and I wish they hadn't included that exercise.

Nevertheless, the book moves on to further essential CS concepts such as Prime
numbers and the Fermat primality test. Funnily enough, I used that probabilistic
Prime test for a Project Euler exercise, wondering why I wasn't able to get the
correct results. It turns out that this test detects probable primes (the book
mentions that a little later and introduces the Miller-Rabin test that
pseudoprimes cannot fool). On the one hand, it was cool to use an algorithm from
a book directly. On the other hand, I was undoubtedly a bit annoyed by that
story.

The book moves on to discuss the runtime of some of the algorithms discussed to
this point. It introduces some other mathematical concepts, such as calculating
roots via the fixed-point method, Euler expansions, and the Newton method for
finding minima/maxima. It was cool to see how the fixed-point method can be used
to implement the Newton method if you plug the derivate of a function into it. I
did my project presentation for math in high school about the Newton method. So
this brought up cool memories. I wish I still had that presentation.

Finally, SICP introduces the evaluation model for stateless functions and
concludes with some exercises that require second-order procedures: procedures
that take other procedures as arguments.


# Chapter 2

Chapter 2 starts by introducing compound data structures to represent pairs and
rational numbers.  Abstraction barriers allow implementing procedures on data
types independent of the underlying representation. For example, we could reduce
a rational-number to its lowest denominator at creation or display time. The
book introduces interval arithmetic to deepen the understanding of data
abstractions.

Next, the book shows how to create more complex data structures such as lists
and trees from cons. Higher-order procedures such as map and fold operate on
these structures, for example, to update each element or to aggregate data.

The book then expands on the idea of higher-order procedures by introducing a
picture language as shown in the following image. We can manipulate a painter
with different transformations to create more complex images. The book does not
present a way to paint to the screen, so I have implemented the painter to
create a Python script that can then draw the images via the PIL library.

![Corner Split](shared/draw-corner-split-3.png)

The next section introduces symbolic data that we utilize to implement a system
for symbolic differentiation. One of my favorite things about the book is that
it references concepts from other disciplines, such as calculus. I am happy that
my high school knowledge of these topics is still present enough for me to work
through the exercises.

Next, we explore sets and different ways to present them. The section finishes
with the implementation of Huffman Encoding trees.

The rest of this chapter shows how to implement an algebra system utilizing a
data-directed programming style. We create packages for different types of
numbers, such as rational, complex, and imaginary numbers. We install methods
for all basic algebraic operations, and the functions dispatch the correct
procedure depending on the data type.

Over the next sections and many exercises, we expand the system to automatically
simplify the numbers by creating a hierarchy of data types. Eventually, we
extend the system to support polynomials and even rational polynomials by
extending our previous rational numbers implementation.

I found these exercises challenging but incredibly rewarding. The algebra system
was the point where I gave up when I worked through the book initially, so I
felt a sense of accomplishment when I finished it on my second attempt.


# Chapter 3

Chapter 3 introduces statefulness into the computation model. I want to point
out how far we have come without explaining variables. It is one of the reasons
why I enjoyed the book so much. Even though I was already familiar with
functional programming, the book taught me how to think purely, leading to more
solid code.

The initial section shows how we can use message dispatching to maintain the
balance of a bank account. The general idea is to define variables within the
scope of a procedure. Any procedure defined in the same context has access to
these variables. By returning a procedure, we can thus manage the variables,
such as the bank account balances, after leaving the original context.

For this approach, the interpreter needs to know how to resolve variables in a
specific context. The book introduces the environment model of computation to
handle variables within different contexts. As we would expect from an
imperative programming language, there are nested frames, and the interpreter
looks up variables starting from the current frame going outwards.

Based on our new understanding of statefulness, we learn about mutable data
structures such as queues and tables. By implementing some of these data
structures in Scheme, I understood and appreciated them more deeply.

The chapter about mutable data structures finishes with a simulator for digital
circuits and a constraint solver. That is probably the only part in the book
where I had wished that there were more exercises. There are some exercises for
both tools, but they don't go too deep.

Of course, once we have introduced statefulness, that opens the possibility for
race conditions when multiple parts of the program access variables
concurrently. The book explains nicely how transfers from different bank
accounts yield different results depending on the execution order. We can use
resources to manage concurrent accesses, but that can lead to additional
problems like deadlocks. The book explains all of that beautifully within a
single section.

Lastly, we learn about the stream model for computation. Streams are delayed
lists which means that the interpreter computes the cdr-arguments on demand.
This paradigm allows us to reimplement a couple of procedures arguably more
elegantly. Just take a look at the beautiful implementation of the Fibonacci
sequence.

```scheme
(define fibs
  (cons-stream 0
    (cons-stream 1
      (add-streams (stream-cdr fibs)
                   fibs))))
```

The chapter ends by explaining how the stream paradigm can resolve the
concurrency problem, at least partially. We can use streams to represent events
that happen over time. Nevertheless, if we get streams from multiple sources,
it's still unclear how to merge them deterministically. The final sentiment has
changed much in the last thirty years. Concurrency is still a challenge, for
example, in embedded development.

# Chapter 4