# Understanding Lisp: Part 1

Note: this post is still in progress. When I started writing these notes several months ago, I quickly realized that it is very difficult to describe the Lisp language and McCarthy’s original paper on Lisp without going deep into details (during my initial attempt to write a cursory overview of Lisp, I also realized that there were many aspects of the language and its history that I never truely understood). Posting these notes in an unfinished version is meant to serve as a forcing function to get me to finally finish this article, since it has remained dormant in the past month. I hope that, even in its current unfinished form, it still provides some interesting insight.

I’ve been slowly making my way through John McCarthy’s seminal paper on Lisp, Recursive Functions of Symbolic Expressions: Their Computation by Machine, Part 1. It was only after a week or so of digesting the content that I started to understand the significance of his ideas and the broader context within which the Lisp programming language was developed. In this article, I will attempt to summarize what I’ve learned so far about the history and development of this fascinating programming language.1

In trying to understand the Lisp language and its origins, I think it helps to ask two separate questions. First, why is a language or a formalism such as Lisp (as we discuss, Lisp is much more than simply a programming language) needed or desirable? Answering this question first requires considering the shortcomings of the programming languages available at the time when McCarthy started his work, as well as the theoretical motivations that prompted McCarthy to work on Lisp in the first place. These topics are addressed in the first Section, and center around the topic of recursive functions and programming, which are fundamental to the design and spirit of Lisp and many subsequent programming languages in the functional programming tradition. The second question is: what is the nature of the actual Lisp language that McCarthy and his colleagues first proposed and implemented, and what, if anything, is special about it? The topic is discussed in the second section, with a focus on the Lisp eval function and the idea of self interpreting code (both of which were initially theoretical ideas that even McCarthy, at the time of writing his article, didn’t realize had many practical applications in programming language design).

# Lisp the Idea

## The Historical Context and Recursive Programs

First, a rather obvious point about the historical context: Lisp was developed during a time when only a few modern programming languages existed (specifically, during the late 1950s). The conventional wisdom is that Lisp is the second oldest programming language in continuous use behind Fortran. While it’s hard to find a readable and concise code example involving earlier versions of Fortran, a good starting point is to consider the following program in a modern version of Fortran: 2


PROGRAM MAIN
INTEGER N, X
EXTERNAL SUB1
COMMON /GLOBALS/ N
X = 0
PRINT *, 'Enter number of repeats'
CALL SUB1(X,SUB1)
END

SUBROUTINE SUB1(X,DUMSUB)
INTEGER N, X
EXTERNAL DUMSUB
COMMON /GLOBALS/ N
IF(X .LT. N)THEN
PRINT *, 'x = ', X
X = X + 1
CALL DUMSUB(X,DUMSUB)
END IF
END


This program, which takes a given number $\texttt{N}$ from a user prompt and prints all intermediate numbers using a variable $\texttt{X}$ a subroutine $\texttt{SUB1}$, is implemented in Fortran 77, which became the Fortran standard in 1978 (nearly twenty years after Lisp). The details of this program are not important. What is important is the part that involves the second argument to $\texttt{SUB1}$ called $\texttt{DUMSUB}$. When this subroutine is called, as shown below: $$\small \texttt{CALL SUB1(X,SUB1)}$$ he second argument is the actual function $\texttt{SUB1}$. As discussed in the article cited above, Fortran 77 (in contrast to later versions of Fortran such as Fortran 90 and 95) does not fully support recursion (i.e., the ability for a function to call itself); a function in Fortran 77 can only make a recursive call if it calls itself as an argument.

To illustrate recursion a bit more, the code below shows two versions of this same function in the Python programming language, a recursive version ($\texttt{recursive_version}$, where the function calls itself on line 6) and a non-recursive version ($\texttt{non_recursive_version}$, which instead uses a while loop). In the former case, we get rid of the need to change the state or value of the variable $\texttt{x}$, and instead call the function directly with $\texttt{x + 1}$.

N = 10

def recursive_version(x):
if x < N:
print("x = %d" % x)
recursive_version(x+1)

def non_recursive_version(x):
while x < N:
print("x = %d" % x)
x += 1

recursive_version(0)
non_recursive_version(0)


Casual users of Python may only rarely encounter recursion and instead choose to always write their functions in the second iterative or imperative style (in this case, we mutate the value of $\texttt{x}$ during each pass in the $\texttt{while}$ loop). In fact, there are good reasons for avoiding recursion in languages such as Python; if you change the value $\texttt{N}=2000$, for example, Python will raise a $\texttt{RuntimeError}$ and complain that you have exceeded a $\texttt{maximum recursion depth}$ (this is related to certain internal properties of how Python is implemented that we won’t delve into here; see here for a more general discussion of recursion in Python).

## Recursive Functions: Some Theoretical Perspective

The point of the examples above is that recursion is not an essential part of classical imperative programming; this is reflected in the shortcomings of the languages discussed above, languages which came into existence long after McCarthy’s initial work on Lisp (case in point: in Fortran 77, which again came out nearly 20 years after Lisp, function recursion is not directly supported). In Lisp, however, recursion is not only well supported, but it is fundamental to the design and spirit of the language. This is perhaps one main reason why the language is hard to grasp. What’s more, the motivation for using recursion (especially in McCarthy’s original work) is related to theoretical considerations that are not likely to be understood by many programmers. This is precisely the topic I intend to explore a bit more in this section.

Before getting into the theoretical motivations, let’s ensure that we have the correct intuitions about recursion by looking again at an example in Python. This example is a naive implementation of the factorial function $n!$, which takes a positive integer $n$ and returns the product of $1 \times 2 \times … \times n$. For example, $6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720$. The factorial function is a classic recursive function, which can be expressed as the following recurrence (or recursive definition): $$\small n!=n \times (n-1)!$$ This can be implemented directly in the Python program shown below (again, we have to be careful not to make $N$ too large in order to avoid exceeding the maximum recursion depth):

N = 10

def factorial(n):
if n == 0:
return 1
return n * factorial(n - 1)

factorial(N)


I find it helpful to visualize the program above (and recursion more generally) using a tree representation, such as the one shown below (where $N=3$):

graph TD
A["mult(3,4)"] --> B(3)
A["factorial(3)"] --> |multplication| G(*)
A["factorial(3)"] --> |subtract 3-1| H[factorial 3]
H["factorial(2)"] --> I(2)
H["factorial(2)"] --> |multiplication| J(*)
H["factorial(2)"] --> |subtract 2-1| K[factorial 1]
K["factorial(1)"] --> l(1)
K["factorial(1)"] --> |multiplication| m(*)
K["factorial(1)"] --> |subtract 1-1| n[factorial 0]
n["factorial(0)"] --> |constant| 1


Here we can see that $\texttt{factorial}$ continues to be called until we reach a zero point, at which point $1$ is returned and multiplication is incrementally applied over the remaining numbers (or leaf nodes) in the tree (i.e., $1,2,3$). One important thing to observe is that we have explained away the factorial function in terms of three much simpler operations, namely multiplication, subtraction and returning 1 (or what is sometimes referred to as the $\texttt{constant}$ function). In other words, if we are concerned about whether the factorial function will reliably give us a solution for any number $N$, we can be reassured by seeing through its recursive definition that it relies on these much simpler operations. We can even decompose the multiplication function for positive numbers into a recursive function, as shown below:

def mult(x,y):
if x == 1:
return y
return y + mult(x-1, y)


which in turn relies on the constant and subtraction functions again and the addition function. If we apply this function to the values $3,4$, the recursive process can be visualized again as a tree, as shown below:


graph TD
A["mult(3,4)"] --> B(4)
A["mult(3,4)"] --> |subtract 3-1| H[factorial 3]
H["mult(2,4)"] --> I(4)
H --> |subtract 2-1| K[factorial 1]
K["mult(1,4)"] --> |constant| l(4)



As with the factorial function, the meaning of the multiplication function ceases to be anything other than a label. We can even proceed further down this path by now defining basic addition in terms of simpler functions. For completeness, we define addition recursively below using a more conventional mathematical notation for recursive definitions (or, more technically, primitive recursive definitions):

\small \begin{align} \texttt{add}(x,0) &= x \\ \texttt{add}(x,\texttt{succ}(y)) &= \texttt{succ}(\texttt{add}(x,y)) \end{align}

where the first equation shows the point at which the recursion reaches the end (e.g., the zero point in the first tree we considered above), and the second equation shows each subsequent step from this end point via the successor function $\texttt{succ}$, which for a given number $n$ simply returns $n+1$. This notation is more in line with how mathematicians, specifically those working in the field of recursive function theory, describe recursive functions. Notice that it is slightly backwards from how we have been implementing recursive in our Python program. An equivalent, though more programmatic, notation might looks as follows: $$\small \texttt{add}(x,y) = \begin{cases} x, & \text{when}\ y=0 \\ \texttt{succ}(\texttt{add}(x,y-1)), & \text{when}\ y > 0 \end{cases}$$ which shows more precisely how we would implement the function in Python or any comparable programming language (i.e., all we need to do is translate the when conditions to $\texttt{if}$ statements and $\texttt{for}$ loops). Note that this definition involves the composition of two functions (i.e., the $\texttt{succ}$ function is applied over the output of the $\texttt{add}$ function, or $\texttt{succ}(\texttt{add}(\cdot))$), which is another important aspect of building recursive functions that we will gloss over here but that nonetheless merits closer examination.

## Lisp as a Formalism (and a Programming Language)

Now, why is this discussion about recursion important? If we continue to decompose the basic arithmetic functions in the manner above, we will eventually converge on a set of functions called the basic primitive recursive functions, of which the successor function is a member. These functions are important in the mathematical theory of computation since they are the basic building blocks of a large portion of the set of computable functions. Loosely speaking, the computable functions are the set of functions that are theoretically guaranteed to be calculable by a computer. It is within this context that McCarthy first conceived of Lisp; he wanted to use Lisp as a tool for investigating recursive function theory. He was specifically interested in a new formalism for describing computable functions in neater way than Turing machines [i.e., the de-facto model of a computer used by theoretical computer scientists to study computation] or the general recursive definitions [e.g., the somewhat counter intuitive mathematical notation shown above] used in recursive function theory (as he writes in his History of Lisp). He adds that:

The fact that Turing machines constitute an awkward programming language doesn’t much bother recursive function theorists, because they almost never have any reason to write particular recursive definitions, since the theory concerns recursive functions in general. They often have reason to prove that recursive functions with specific properties exist, but this can be done by an informal argument without having to write them down explicitly…. Anyway, I decided to write a paper describing LISP both as a programming language and as a formalism for doing recursive function theory…. The paper had no influence on recursive function theorists, because it didn’t address the questions that interested them.

As McCarthy acknowledges, he largely failed at getting pure mathematicians interested in the Lisp formalism. Nonetheless, the style of programming that Lisp facilitates is closely aligned to the mathematical ideas we outlined above. Again, in McCarthy’s own words:

One mathematical consideration that influenced LISP was to express programs as applicative expressions built up from variables and constants using functions. I considered it important to make these expressions obey the usual mathematical laws allowing replacement of expressions by expressions giving the same value. The motive was to allow proofs of properties of programs using ordinary mathematical methods.

One mathematician who has been greatly influenced by Lisp is Gregory Chaitin (we already touched on some of his work in my post on Kolmogorov complexity), who once referred to Lisp as the only computer programming language that is mathematically respectable. He concurs with McCarthy on the lack of interest in Lisp among theoreticians and adds the following in the preface of his seminal work Algorithmic Information Theory:

But by a quirk of fate LISP has largely been ignored by theoreticians and has instead become the standard programming language for work on artificial intelligence. I believe that pure LISP is in precisely the same role in computational mathematics that set theory is in theoretical mathematics, in that it provides a beautifully elegant and extremely powerful formalism which enables concepts such as that of numbers and functions to be defined from a handful of more primitive notions.

Again, a key point to focus on here is describing complex concepts with only a handful of more primitive notions. Now, let’s get into the details of the actual Lisp language and see how this is done.

## The Main Ingredients of the Lisp Formalism

So now let’s look at how McCarthy defines Lisp in his paper (note that his presentation deviates from more modern presentations of the language e.g., Anatomy of Lisp, my personal favorite; we will try to explain the reason for this in the next section). One of the main contributions in this paper is the introduction of a new notation called a s-expression, or symbolic expression, which is the basic building block of Lisp programs. An s-expression is defined recursively as follows: \small \begin{align} 1.& \text{ Atomic symbols are s-expressions} \\ 2.& \text{ If } e_{1} \text{ and } e_{2} \text{ are s-expressions, so is } (e_{1} \cdot e_{2}) \end{align} where the set of atomic symbols includes strings (including the blank symbol), numeric values (in reality, McCarthy’s original formulation didn’t have a complete implementation of numbers, but we will ignore this in this article), booleans, and other primitives that one would expect to see in most modern programming languages. These atomic symbols can then be joined together to create more complex s-expressions, or ordered lists (i.e., part 2 of the definition) that use the special symbols $(,)$ (brackets) and $\cdot$ (the dot symbol). For example, the three strings $\texttt{first}$, $\texttt{second}$ and $\texttt{third}$ are types of atomic symbols (and hence are s-expressions), and can be joined together to create the following more complex s-expression:

\small \begin{align} \texttt{(first \cdot (second \cdot (third \cdot Nil)))} \end{align}

While it might not be entirely obvious (even to those already familiar with Lisp and s-expressions), this last representation is a lower-level representation of the ordered list $\texttt{(first second third)}$, where $\texttt{Nil}$ is a special end of list symbol. Shortly after introducing s-expressions in this manner, McCarthy acknowledges that it is more convenient to use the latter notation without the dots, which is the notation that it is more commonly encountered in modern presentations and implementations of Lisp. We will henceforth use the notation on the left as shorthand for the more precise representation with dots to the right of $\equiv$:

\small \begin{align} \texttt{(m_{1 } m_{2} … m_{n})} \equiv \texttt{(m_{1} \cdot (m_{2} \cdot ( … \cdot (m_{n} \cdot Nil ))))} \end{align}

It is important to note, however, that this lower-level notation (and s-expressions more generally) can be visualized as trees, which are similar to the tree representations that we considered in the beginning. In the tree below, we will use a prefix notation here by putting each $\cdot$ symbol in front of the items in the list (this prefix notation will become important when we discuss Lisp functions):


graph TD
A[.] --> B(first)
A --> H[.]
H --> I(second)
H -->  J(.)
J --> K(third)
J --> L(Nil)


S-expressions therefore make it possible to express the types of nested structures that we encountered in our recursive definitions and programs above. Beyond Lisp, s-expressions have proven to be a popular notation for representing hierarchical linguistic information in research on computational linguistics. For example, the structure below is a representation of the grammatical structure for the sentence This is an example of a sentence represented as an s-expression, where the grammatical categories (i.e., $\texttt{S, NP, VP, VBZ,..}$) are the prefixes in each constituent s-expression:

(ROOT
(S
(NP (DT This))
(VP (VBZ is)
(NP
(NP (DT an) (NN example))
(PP (IN of)
(NP
(NP (DT a) (NN sentence))
(VP (VBN represented)
(PP (IN as)
(NP (DT an) (NN s-expression))))))))))


Given our definition of s-expressions so far, however, this notation only permits us to construct more complex structure from atomic symbols, or what we might refer to as the data in our language. McCarthy therefore adds the following three additional ingredients:

1. Conditional expressions, which will allow us to define recursive functions and predicates.
2. Lambda abstraction, which will allow us to formally define functions and function application.
3. Primitive symbolic functions over s-expressions, which play the role of the basic primitive recursive functions that we described in the beginning.

### Conditional expressions

A conditional expression takes the following form: \small \begin{align} (p_{1} \to e_{1}, … , p_{n} \to e_{n}) \end{align} where each $p_{j}$ is a proposition (i.e., a true or false statement) and the corresponding $e_{j}$ is an arbitrary s-expression that follows from $p_{j}$ being true. For example, the following conditional expression defines the factorial function considered before (where $T$ in the second condition means $\texttt{true}$ and is special type of atomic symbol):

\small \begin{align} n! = ( n = 0 \to 1, T \to n * (n-1)!) \end{align}

Expanding this expressions for $n=2$ then gives the following result:

\small \begin{align} 2! &= (2 = 0 \to 1, T \to 2 * (2 - 1)! ) \\ &= 2 \cdot (2 - 1)! \\ &= 2 \cdot (1 = 0 \to 1, T \to 1 * (1 - 1)) \\ &= 2 \cdot 1 \cdot 0! \\ &= 2 \cdot 1 \cdot (0 = 0 \to 1, T \to 0 * (0 - 1)!) \\ &= 2 \cdot 1 \cdot 1 \end{align}

It is important to note that while conditional expressions are meant for creating recursive definitions, they can also be used to define non-recursive functions (in the same way that primitive recursive definitions can be used for defining non-recursive functions). The conditions can also be of arbitrary depth, as shown in the following example involving the $\texttt{sign}$ or signum function: \small \begin{align} \texttt{sign}(x) = (x < 0 \to -1, x = 0 \to 0, T \to 1) \end{align} McCarthy shows how to use conditional expressions to express connectives in classical propositional logic; this gets us a bit closer to McCarthy’s other motivation for developing Lisp, which was to support symbolic and logic-based artificial intelligence. Notice in the first expression that the right hand side of the first condition is itself a conditional expression. In addition to having arbitrary atomic conditions, each condition can contain an arbitrary number of conditions on the right hand side (which was tricky to do in imperative languages, especially the language available before Lisp, using only $\texttt{if}$ statements as he discusses in this paper again). \small \begin{align} p \land q &= (p \to (q \to T; T \to F), T \to F) \\ p \lor q &= (p \to T, q \to T, T \to F) \\ \neg p &= (p \to F, T \to T) \\ p \to q &= (p \to T, T \to T) \end{align} Also note that these definitions already involve some basic functions such as multiplication (*), equivalence (=) and less than ($<$). As before, we can now decompose these simpler function and try to narrow them down to a small set of primitives. Before we do this, however, we will briefly describe the second ingredient that McCarthy introduces, which is called lambda abstraction.

### Lambda Abstraction

In its simplest form, lambda abstraction is a notation for associating values to function components. For example, in our factorial example, the $\lambda(x)$ in the following (we use large parenthesis here to show the conditional expression, or what McCarthy calls the form of the function): \small \begin{align} \lambda(x)\bigg(x = 0 \to 1, T \to x * (x-1)! \bigg) \end{align} serves as a kind of placeholder for $x$ in the main equation and abstracts over all the possible numeric values that the equation might take. More technically, the lambda notation is a way of creating functions. When supplied with a set of arguments, the variables in the function form are substituted with the arguments tied to the lambda variables through a process called $\beta$-reduction. In the following example: \small \begin{align} \lambda(x)\bigg(x=0 \to 1, T \to x * (x-1)!\bigg)(2) &\equiv \bigg(2 = 0 \to 1, T \to 2 * (2 - 1)! \bigg) \\ &= 2 \end{align} the value 2 is substituted for all occurrences of $x$. There is a lot more to be said about this lambda notation, which is based on a much broader theory of computation called the lambda calculus developed by the mathematician Alonzo Church); we will unfortunately save this discussion for another time. In the parlance of everyday programming, these functions are often referred to as anonymous functions, since they lack any kind of name or variable identifier. In the example above, the lack of a name is problematic since we have no way of identifying the function when we make a recursive call in the second condition. To handle this, McCarthy introduces a function called $\texttt{label}$, which allows these functions to take names: \small \begin{align} \texttt{label}\big(\texttt{factorial}, \lambda(x) \bigg(x = 0 \to 1; T \to x * \texttt{factorial}(x - 1)\bigg)\big) \end{align} Interestingly, while anonymous lambda functions were popularized in programming by McCarthy and Lisp, they have found there way into modern programming languages, as shown below in Python (where the $\texttt{label}$ function can be achieved by doing ordinary variable assignment):

factorial = lambda x : (1 if x == 0 else x * factorial(x-1))


### Primitive symbolic functions (some examples)

Now with these two ingredients (i.e., conditional expressions and lambda notation), we can get to the set of primitive functions that McCarthy defines, or what he calls s-functions that operate over s-expressions. To represent these functions, McCarthy uses a bracket notation $[ \thinspace\thinspace]$ called m-expressions to distinguish function application over s-expressions from actual s-expressions (so many notations, I know, but this will all blend into a single notation in the next section). The 5 functions in the Table below are the basic functions that McCarthy defines, each of which operates over s-expressions (for readability, we will write these functions without the $\texttt{label}$ function and without lambdas).

FunctionDefinitionDescription
1. atom\begin{align} &\texttt{atom}[X] = \texttt{T} \\\ &\texttt{atom}[(X . A)] = \texttt{F} \end{align}checks if an s-expression is atomic
2. eq.\begin{align} &\texttt{eq}[X,X] = T \\\ &\texttt{eq}[X,A] = \texttt{F} \\\ &\texttt{eq}[X,(X . A)] = \texttt{undefined} \end{align}checks if two atomic s-expressions are the same
3. car\begin{align} &\texttt{car}[(X . A)] = X \\\ &\texttt{car}[((X . A) . Y)] = (X . A) \end{align}Returns the first element in s-expression
4. cdr\begin{align}&\texttt{cdr}[(X . A)] = A \\\ &\texttt{cdr}[((X . A) . Y)] = Y \end{align}Returns the last element in s-expression after first
5. cons\begin{align} &\texttt{cons}[X,A] = (X . A) \\\ &\texttt{cons}[(X . A),Y] = ((X . A) . Y) \end{align}joins two s-expressions.

The first two function operate over atomic expressions, whereas next the two functions operate over non-atomic expressions and the last one operates over both types of expressions. These are henceforth our basic ingredients; we can now build on these primitives and additionally use conditional expressions to build much more complex functions (we can in fact define all computable functions). McCarthy defines several such recursive functions, for example the function $\texttt{ff}$ below: $$\small \texttt{ff}[x] = ( \texttt{atomic}[x] \to x; T \to \texttt{ff}[\texttt{car}[x]])$$ which takes a (potentially complex) s-expression and returns the first atomic item in that expression. For example: \small \begin{align} \texttt{ff}[\texttt{first}] = \texttt{first} \texttt{ff}[\texttt{(first second third)}] = \texttt{first} \end{align} With the help of the boolean connectives we defined as conditional expressions, we can now define the \texttt{equal} function, which determines equality between arbitrary (i.e., potentially non-atomic) s-expressions $x$ and $y$: \small \begin{align} \texttt{equal}[x,y] &= \bigg(\big( \texttt{atom}[x] \land \texttt{atom}[y] \land \texttt{eq}(x,y) \big) \lor \\\ & \big( \neg \texttt{atom}[x] \land \neg \texttt{atom}[y] \land \texttt{equal}[\texttt{car}[x], \texttt{car}[y]] \land \texttt{equal}[ \texttt{cdr}[x], \texttt{cdr}[y]] \big) \to T; T \to F\bigg) \end{align} In other words, two s-expressions expressions are equal if they are atomic and equal according to our primitive function $\texttt{eq}$, or are non-atomic and satisfy the recursive constraint that each atomic expression starting from beginning and end of each complex expression via $\texttt{car}$ and $\texttt{cdr}$ will evaluate to true.

# Lisp the Language and Implementation

## Functions as S-expressions and Lisp as an Interpreter

We could continue on and define increasingly complex functions, but as you can see the notation is already getting a bit out of hand. The m-expression syntax takes inspiration from another programming language of the 1950s called Algol. While it was McCarthy’s initial intention to write functions in this style, such a notation was never widely adopted in the Lisp community (for this reason, it is not easy to follow the code in his original paper). Nonetheless, McCarthy does the following two rather remarkable things in the remainder of his paper.

### 1. M-expression as S-expressions: What Modern Lisp Looks Like

First, related to this issue of notation, he describes how to translation m-expressions into s-expressions, which brings up closer to modern day Lisp. In his own words again, he writes that

The project of defining M-expressions [i.e., the function notation given in the last section] precisely and compiling them or at least translating them into S-expressions was neither finalized nor explicitly abandoned. It just receded into the indefinite future, and a new generation of programmers appeared who preferred internal notation [i.e,. the s-expression notation we introduce below; we will explain more below about what he means what he says internal notation] to any FORTRAN-like or ALGOL-like notation that could be devised.

The rough translation that he initially proposes in his paper works as follows (as warned above, this is when the prefix notation enters the scene, which many people love to hate about Lisp): functions of the form $f[e_{1}, …,e_{n}]$ are translated as: \small \begin{align} \texttt{(f e_{1 } e_{2 } e_{n })} \end{align} n which the function name $f$ in placed in the prefix position (McCarthy adds a further rule that function and variables names should be capitalized, which we ignore here). Conditional expressions $(p_{1} \to e_{1}; p_{2} \to e_{2})$ are then translated in the following way using the special symbol $\texttt{cond}$: \small \begin{align} \texttt{(cond (p_{1} e_{1})(p_{2} e_{2})…)} \end{align} And finally, lambda abstraction takes the following form for a given expression or function form $\varepsilon*$ (i.e., the types of expressions inside of the large parentheses above): \small \begin{align} \texttt{(lambda (x_{1}, …, x_{n}) \varepsilon*)} \end{align} As an example, applying this translation to our factorial function yields the following s-expression (with the $\texttt{label}$ function having the same meaning as above):

(label factorial
(lambda (x) ;; argument
(cond ((= x 0) 1) ;; conditions
(T (mult x (factorial (- x 1)))))))


Below shows more complicated examples involving the $\texttt{ff}$ and $\texttt{equals}$ functions from before.

(label ff
(lambda (x)
(cond ((atomic x) x) ;; atomic item is found
(T (ff (car x)))))) ;;recursive call

(label equal
(lambda (x y) ;; arguments to compare
(cond
((or  ;; first cond. disjunction
(and (atom x) ;; first disjunct
(atom y)
(eq x y))
(and (not (atom x));;second disjunct
(not (atom y))
(equal (car x)
(car y))
(equal (cdr x)
(cdr y))))
T)
(T F)))) ;; second condition


Note that I am trying to stay faithful to McCarthy’s notation here. In Scheme, which is one popular implementation of Lisp, the $\texttt{equal}$ function above is valid if you replace the $\texttt{label}$ function with $\texttt{define}$, which we will do in the remainder of this article (see here). In the spirit of the decomposing complex operations to simpler functions, below we show how to implement in Scheme some of the functions (e.g., $\texttt{and}$, $\texttt{or}$, $\texttt{neg}$) that we glossed over in the code examples above and in the last section:

(define neg
(lambda (x)
(cond (x #f)  ;; #f is equal to F used above
(#t #t) ;; #t is equal to T used above
)))

#### A More Complete Eval

still under construction!

1. Part way through writing this article, I discovered Paul Graham’s paper The Roots of Lisp,which has the same goal of understanding what McCarthy discovered in his original paper; I have borrowed some of his explanations throughout this paper. I urge readers to look at this paper, which gets much deeper into the details of McCarthy’s original code, and specifically the eval function and its broader significance in programming (whereas here we focus more on the theoretical ideas that motivated Lisp and the broader historical context). ^
2. This example is taken from here ^
3. As an aside, I once heard a fantastic quip, perhaps originally from here, that In Ruby, everything is an object. In Clojure [a particular dialect of Lisp implemented on the JVM], everything is a list [or s-expression]. In Javascript, everything is a terrible mistake. ^