Why does this matter? As Steve Yegge said, "If you don't know how compilers work, then you don't know how computers work." Yegge describes 8 problems that can be solved with compilers (or equally with interpreters, or with Yegge's typical heavy dosage of cynicism).
Syntax and Semantics of Scheme Programs
The syntax of a language is the arrangement of characters to form correct statements or expressions; thesemantics is the meaning of those statements or expressions. For example, in the language of mathematical expressions (and in many programming languages), the syntax for adding one plus two is "1 + 2" and the semantics is the application of the addition operator to the two numbers, yielding the value 3. We say we are evaluating an expression when we determine its value; we would say that "1 + 2" evaluates to 3, and write that as "1 + 2" ⇒ 3.Scheme syntax is different from most other languages you may be familiar with. Consider:
Java has a wide variety of syntactic conventions (keywords, infix operators, three kinds of brackets, operator precedence, dot notation, quotes, commas, semicolons), but Scheme syntax is much simpler:
Java Scheme if (x.val() > 0) {
fn(A[i] + 1,
return new String[] {"one", "two"});
}(if (> (val x) 0)
(fn (+ (aref A i) 1)
(quote (one two)))
- Scheme programs consist solely of expressions. There is no statement/expression distinction.
- Numbers (e.g. 1) and symbols (e.g. A) are called atomic expressions; they cannot be broken into pieces. These are similar to their Java counterparts, except that in Scheme, operators such as + and > are symbols too, and are treated the same way as A and fn.
- Everything else is a list expression: a "(", followed by zero or more expressions,
followed by a ")". The first element of the list determines what it means:
- A list starting with a keyword, e.g. (if ...), is a special form; the meaning depends on the keyword.
- A list starting with a non-keyword, e.g. (fn ...), is a function call.
In this page we will cover all the important points of Scheme (omitting some minor details), but we will take two steps to get there, defining a simplified language first, before defining the near-full Scheme language.
Language 1: Lispy Calculator
Lispy Calculator is a subset of Scheme using only five syntactic forms (two atomic, two special forms, and the procedure call). Lispy Calculator lets you do any computation you could do on a typical calculator—as long as you are comfortable with prefix notation. And you can do two things that are not offered in typical calculator languages: "if" expressions, and the definition of new variables. Here's an example program, that computes the area of a circle of radius 10, using the formula π r2:
(begin
(define r 10)
(* pi (* r r)))
Here is a table of all the allowable expressions:| Expression | Syntax | Semantics and Example |
|---|---|---|
| variable reference | var | A symbol is interpreted as a variable name;
its value is the variable's
value. Example: r⇒ 10 (assuming r was previously defined to be 10) |
| constant literal | number | A number
evaluates to itself. Examples: 12 ⇒ 12or-3.45e+6 ⇒ -3.45e+6 |
| conditional | (iftest conseq alt) | Evaluate test; if true,
evaluate and return conseq; otherwise alt. Example: (if (> 10 20) (+ 1 1) (+ 3 3)) ⇒ 6 |
| definition | (definevarexp) | Define a new variable and give it
the value of evaluating the expression exp. Examples: (define r 10) |
| procedure call | (proc arg...) | If proc is
anything other than one of the symbols if, define, or quote then it is treated as a procedure. Evaluate proc
and all the args, and then the procedure is applied to the list of arg values. Example: (sqrt (* 2 8)) ⇒ 4.0 |
In the Syntax column of this table, var must be a symbol,number must be an integer or floating point number, and the other italicized words can be any expression. The notation arg... means zero or more repetitions of arg.
What A Language Interpreter Does
A language interpreter has two parts:- Parsing: The parsing component takes an input program in the form of a sequence of characters, verifies it according to thesyntactic rules of the language, and translates the program into an internal representation. In a simple interpreter the internal representation is a tree structure (often called an abstract syntax tree) that closely mirrors the nested structure of statements or expressions in the program. In a language translator called a compiler there is often a series of internal representations, starting with an abstract syntax tree, and progressing to a sequence of instructions that can be directly executed by the computer. The Lispy parser is implemented with the function parse.
- Execution: The internal representation is then processed according to the semantic rules of the language, thereby carrying out the computation. Lispy's execution function is called eval (note this shadows Python's built-in function of the same name).
program (str) ➡ parse ➡ abstract syntax tree (list) ➡ eval ➡ result (object)
And here is a short example of what we want parse and eval to be able to do:
>> program = "(begin (define r 10) (* pi (* r r)))">>> parse(program) ['begin', ['define', 'r', 10], ['*', 'pi', ['*', 'r', 'r']]]>>> eval(parse(program)) 314.1592653589793
Parsing: parse, tokenize and read_from_tokens
Parsing is traditionally separated into two parts: lexical analysis, in which the input character string is broken up into a sequence of tokens, and syntactic analysis, in which the tokens are assembled into an abstract syntax tree. The Lispy tokens are parentheses, symbols, and numbers. There are many tools for lexical analysis (such as Mike Lesk and Eric Schmidt'slex), but we'll use a very simple tool: Python's str.split. The function tokenize takes as input a string of characters; it adds spaces around each paren, and then calls str.split to get a list of tokens:
def tokenize(chars):
"Convert a string of characters into a list of tokens."
return chars.replace('(', ' ( ').replace(')', ' ) ').split()>>> program = "(begin (define r 10) (* pi (* r r)))">>> tokenize(program)
['(', 'begin', '(', 'define', 'r', '10', ')', '(', '*', 'pi', '(', '*', 'r', 'r', ')', ')', ')']Our function parse will take a string representation of a program as input, call tokenize to get a list of tokens, and then call read_from_tokens to assemble an abstract syntax tree.read_from_tokens looks at the first token; if it is a ')' that's a syntax error. If it is a '(', then we start building up a list of sub-expressions until we hit a matching ')'. Any non-parenthesis token must be a symbol or number. We'll let Python make the distinction between them: for each non-paren token, first try to interpret it as an int, then as a float, and finally as a symbol. Here is the parser:
def parse(program):
"Read a Scheme expression from a string."
return read_from_tokens(tokenize(program))
def read_from_tokens(tokens):
"Read an expression from a sequence of tokens."
if len(tokens) == 0:
raise SyntaxError('unexpected EOF while reading')
token = tokens.pop(0)
if '(' == token:
L = []
while tokens[0] != ')':
L.append(read_from_tokens(tokens))
tokens.pop(0) # pop off ')'
return L
elif ')' == token:
raise SyntaxError('unexpected )')
else:
return atom(token)
def atom(token):
"Numbers become numbers; every other token is a symbol."
try: return int(token)
except ValueError:
try: return float(token)
except ValueError:
return Symbol(token)
parse works like this:>>> program = "(begin (define r 10) (* pi (* r r)))">>> parse(program) ['begin', ['define', 'r', 10], ['*', 'pi', ['*', 'r', 'r']]]We have made some choices about the representation of Scheme objects. Here we make the choices explicit:
Symbol = str # A Scheme Symbol is implemented as a Python str List = list # A Scheme List is implemented as a Python list Number = (int, float) # A Scheme Number is implemented as a Python int or floatWe're almost ready to define eval. But we need one more concept first.
Environments
The function eval takes two arguments: an expression, x, that we want to evaluate, and an environment, env, in which to evaluate it. An environment is a mapping from variable names to their values. By default, eval will use a global environent that includes the names for a bunch of standard functions (like sqrt and max, and also operators like *). This environment can be augmented with user-defined variables, using the expression (define variable value). For now, we can implement an environment as a Python dict of {variable: value} pairs.
import math
imoort operator as op
Env = dict # An environment is a mapping of {variable: value}
def standard_env():
"An environment with some Scheme standard procedures."
env = Env()
env.update(vars(math)) # sin, cos, sqrt, pi, ...
env.update({
'+':op.add, '-':op.sub, '*':op.mul, '/':op.div,
'>':op.gt, '=':op.ge, '
Note: it is customary in Scheme for begin to be a special form that takes a sequence of arguments,
evaluares each one, and returns the last one (discarding the other values, and using them only for their side effects, such as printing something). To make things easier for now, we implement begin as a function, not a special form.Evaluation: eval
We are now ready for the implementation of eval.
As a refresher, we repeat the table of Lispy Calculator forms:
| Expression | Syntax | Semantics and Example |
|---|---|---|
| variable reference | var | A symbol is interpreted as a variable name;
its value is the variable's
value. Example: r⇒ 10 (assuming r was previously defined to be 10) |
| constant literal | number | A number
evaluates to itself. Examples: 12 ⇒ 12or-3.45e+6 ⇒ -3.45e+6 |
| conditional | (iftest conseq alt) | Evaluate test; if true,
evaluate and return conseq; otherwise alt. Example: (if (> 10 20) (+ 1 1) (+ 3 3)) ⇒ 6 |
| definition | (definevarexp) | Define a new variable and give it
the value of evaluating the expression exp. Examples: (define r 10) |
| procedure call | (proc arg...) | If proc is
anything other than one of the symbols if, define, or quote then it is treated as a procedure. Evaluate proc
and all the args, and then the procedure is applied to the list of arg values. Example: (sqrt (* 2 8)) ⇒ 4.0 |
Notice how closely the code for eval follows the table:
def eval(x, env=global_env):
"Evaluate an expression in an environment."
if isinstance(x, Symbol): # variable reference
return env[x]
elif not isinstance(x, List): # constant literal
return x
elif x[0] == 'if': # conditional
(_, test, conseq, alt) = x
exp = (conseq if eval(test, env) else alt)
return eval(exp, env)
elif x[0] == 'define': # definition
(_, var, exp) = x
env[var] = eval(exp, env)
else: # procedure call
proc = eval(x[0], env)
args = [eval(arg, env) for arg in x[1:]]
return proc(*args)We're done! You can see it all in action:
>>> eval(parse("(define r 10)"))>>> eval(parse("(* pi (* r r))"))
314.1592653589793Interaction: A REPL
It is tedious to have to enter "eval(parse(...))" all the time. One of Lisp's great legacies is the notion of an interactive read-eval-print loop: a way for a programmer to enter an expression, and see it immediately read, evaluated, and printed, without having to go through a lengthy build/compile/run cycle. So let's define the function repl (which stands for read-eval-print-loop), and the function schemestr which returns a string representing a Scheme object.
def repl(prompt='lis.py> '):
"A prompt-read-eval-print loop."
while True:
val = eval(parse(raw_input(prompt)))
if val is not None:
print(schemestr(val))
def schemestr(exp):
"Convert a Python object back into a Scheme-readable string."
if isinstance(exp, List):
return '(' + ' '.join(map(schemestr, exp)) + ')'
else:
return str(exp)
Here is repl in action:>>> repl() lis.py> (define r 10) lis.py> (* pi (* r r)) 314.159265359 lis.py> (if (> (* 11 11) 120) (* 7 6) oops) 42 lis.py>
Language 2: Full Lispy
We will now extend our language with three new special forms, giving us a much more nearly-complete Scheme subset:| Expression | Syntax | Semantics and Example | ||||||
|---|---|---|---|---|---|---|---|---|
| quotation | (quote exp)|
Return the exp literally; do not evaluate it. | Example:(quote (+ 1 2)) ⇒ (+ 1 2) assignment | (set!var
exp) | Evaluate exp and assign that value tovar, which must have been previously defined (with adefine or as a parameter to an enclosing procedure). | Example: (set! r2 (* r r)) procedure | (lambda (var...)exp) | Create a procedure
with parameter(s) named var... and exp as the body. | Example: (lambda (r) (* pi (* r r))) |
The lambda special form (an obscure nomenclature choice that refers to Alonzo Church's lambda calculus) creates a procedure. We want procedures to work like this:
lis.py> (define circle-area (lambda (r) (* pi (* r r))) lis.py> (circle-area 10) 314.159265359The procedure call (circle-area 10) causes us to evaluate the body of the procedure, (* pi (* r r)), in an environment in which pi and * have the same global values they always did, but now r has the value 10. However, it wouldn't do to just set r to be 10 in the global environment. What if we were using r for some other purpose? We wouldn't want a call to circle-area to alter that value. Instead, we want to arrange for there to be a local variable named r that we can set to 10 without worrying about interfering with any other variable that happens to have the same name. We will create a new kind of environment, one which allows for both local and global variables.
The idea is that when we evaluate (circle-area 10), we will fetch the procedure body, (* pi (* r r)), and evaluate it in an environment that has r as the sole local variable, but also has access to the global environment. In other words, we want an environment that looks like this, with the local (blue) environment nested inside the outer (red) global environment:
| pi: 3.141592653589793 *: <built-in function mul> ... |
When we look up a variable in such a nested environment, we look first at the innermost level, but if we don't find the variable name there, we move to the next outer level.
It is clear that procedures and environments are intertwined, so let's define them together:
class Procedure(object):
"A user-defined Scheme procedure."
def __init__(self, parms, body, env):
self.parms, self.body, self.env = parms, body, env
def __call__(self, *args):
return eval(self.body, Env(self.parms, args, self.env))
class Env(dict):
"An environment: a dict of {'var':val} pairs, with an outer Env."
def __init__(self, parms=(), args=(), outer=None):
self.update(zip(parms, args))
self.outer = outer
def find(self, var):
"Find the innermost Env where var appears."
return self if (var in self) else self.outer.find(var)
global_env = standard_env()
We see that every procedure has three components: a list of parameter names, a body expression, and an environment
that tells us what non-local variables are accessible from the body.
An environment is a subclass of dict, so it has all the methods that dict has. In addition there are two methods: the constructor__init__ builds a new environment by taking a list of parameter names and a corresponding list of argument values, and creating a new environment that has those {variable: value} pairs as the inner part, and also refers to the given outer environment. The method find is used to find the right environment for a variable: either the inner one or an outer one.
To see how these all go together, here is the new definition of eval. Note that the clause for variable reference has changed: we now have to call env.find(x) to find at what level the variable x exists; then we can fetch the value of x from that level. (The clause fordefine has not changed, because a define always adds a new variable to the innermost environment.) There are two new clauses: for set!, we find the environment level where the variable exists and set it to a new value. With lambda, we create a new procedure object with the given parameter list, body, and environment.
def eval(x, env=global_env):
"Evaluate an expression in an environment."
if isinstance(x, Symbol): # variable reference
return env.find(x)[x]
elif not isinstance(x, List): # constant literal
return x
elif x[0] == 'quote': # quotation
(_, exp) = x
return exp
elif x[0] == 'if': # conditional
(_, test, conseq, alt) = x
exp = (conseq if eval(test, env) else alt)
return eval(exp, env)
elif x[0] == 'define': # definition
(_, var, exp) = x
env[var] = eval(exp, env)
elif x[0] == 'set!': # assignment
(_, var, exp) = x
env.find(var)[var] = eval(exp, env)
elif x[0] == 'lambda': # procedure
(_, parms, body) = x
return Procedure(parms, body, env)
else: # procedure call
proc = eval(x[0], env)
args = [eval(arg, env) for arg in x[1:]]
return proc(*args)To appreciate how procedures and environments work together, consider this program and the environment that gets formed when we evaluate (account1 -20.00):
|
|
Each rectangular box represents an environment, and the color of the box matches the color of the variables that are newly defined in the environment. In the last two lines of the program we define account1 and call(account1 -20.00); this represents the creation of a bank account with a 100 dollar opening balance, followed by a 20 dollar withdrawal. In the process of evaluating (account1 -20.00), we will eval the expression highlighted in yellow. There are three variables in that expression. amt can be found immediately in the innermost (green) environment. But balance is not defined there: we have to look at the green environment's outer env, the blue one. And finally, the variable + is not found in either of those; we need to do one more outer step, to the global (red) environment. This process of looking first in inner environments and then in outer ones is called lexical scoping. Env.find(var) finds the right environment according to lexical scoping rules.
Let's see what we can do now:
>>> repl() lis.py> (define circle-area (lambda (r) (* pi (* r r)))) lis.py> (circle-area 3) 28.274333877 lis.py> (define fact (lambda (n) (if ( (fact 10) 3628800 lis.py> (fact 100) 9332621544394415268169923885626670049071596826438162146859296389521759999322991 5608941463976156518286253697920827223758251185210916864000000000000000000000000 lis.py> (circle-area (fact 10)) 4.1369087198e+13 lis.py> (define first car) lis.py> (define rest cdr) lis.py> (define count (lambda (item L) (if L (+ (equal? item (first L)) (count item (rest L))) 0))) lis.py> (count 0 (list 0 1 2 3 0 0)) 3 lis.py> (count (quote the) (quote (the more the merrier the bigger the better))) 4 lis.py> (define twice (lambda (x) (* 2 x))) lis.py> (twice 5) 10 lis.py> (define repeat (lambda (f) (lambda (x) (f (f x))))) lis.py> ((repeat twice) 10) 40 lis.py> ((repeat (repeat twice)) 10) 160 lis.py> ((repeat (repeat (repeat twice))) 10) 2560 lis.py> ((repeat (repeat (repeat (repeat twice)))) 10) 655360 lis.py> (pow 2 16) 65536.0 lis.py> (define fib (lambda (n) (if ( (define range (lambda (a b) (if (= a b) (quote ()) (cons a (range (+ a 1) b))))) lis.py> (range 0 10) (0 1 2 3 4 5 6 7 8 9) lis.py> (map fib (range 0 10)) (1 1 2 3 5 8 13 21 34 55) lis.py> (map fib (range 0 20)) (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765)We now have a language with procedures, variables, conditionals (if), and sequential execution (the begin procedure). If you are familiar with other languages, you might think that a while or for loop would be needed, but Scheme manages to do without these just fine. The Scheme report says "Scheme demonstrates that a very small number of rules for forming expressions, with no restrictions on how they are composed, suffice to form a practical and efficient programming language." In Scheme you iterate by defining recursive functions.
How Small/Fast/Complete/Good is Lispy?
In which we judge Lispy on several criteria:True Story
To back up the idea that it can be very helpful to know how interpreters work, here's a story. Way back in 1984 I was writing a Ph.D. thesis. This was before LaTeX, before Microsoft Word for Windows—we used troff. Unfortunately, troff had no facility for forward references to symbolic labels: I wanted to be able to write "As we will see on page @theorem-x" and then write something like "@(set theorem-x \n%)" in the appropriate place (the troff register \n% holds the page number). My fellow grad student Tony DeRose felt the same need, and together we sketched out a simple Lisp program that would handle this as a preprocessor. However, it turned out that the Lisp we had at the time was good at reading Lisp expressions, but so slow at reading character-at-a-time non-Lisp expressions that our program was annoying to use.From there Tony and I split paths. He reasoned that the hard part was the interpreter for expressions; he needed Lisp for that, but he knew how to write a tiny C routine for reading and echoing the non-Lisp characters and link it in to the Lisp program. I didn't know how to do that linking, but I reasoned that writing an interpreter for this trivial language (all it had was set variable, fetch variable, and string concatenate) was easy, so I wrote an interpreter in C. So, ironically, Tony wrote a Lisp program (with one small routine in C) because he was a C programmer, and I wrote a C program because I was a Lisp programmer.
In the end, we both got our theses done (Tony, Peter).
The Whole Thing
The whole program is here: lis.py.Further Reading
To learn more about Scheme consult some of the fine books (byFriedman and Fellesein,Dybvig,Queinnec,Harvey and Wright orSussman and Abelson), videos (by Abelson and Sussman), tutorials (byDorai,PLT, orNeller), or thereference manual.
I also have another page describing a more advanced version of Lispy.
Peter Norvig