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Background on Symbolic Computation

  • Wikipedia considers symbolic computation to be simply computer algebra.
  • While computer algebra is a form of symbolic computation, there are plenty of other applications.
    • Programming languages
    • Compilers
    • Artificial intelligence

Lisp & Symbolic Computation

Lisp dialects have a homoiconic syntax: the code is data, and data is code. Lists being the structure of the language syntax, code can be manipulated just like lists.

  • This is what (historically) gave LISP its name (LISt Processor)
  • The concept of “quoting” is fairly unique to just Lisp.
  • It leads to a natural way to manipulate and work on code in the language.
  • Key point: we can manipulate code before it is evaluated!

John McCarthy (1958)

Recursive Functions of Symbolic Expressions and their Computation by Machine


  • Even in the best software designs, it’s hard to avoid repetitive patterns.
  • What if our language let us extend its syntax to account for these patterns?

Exercise for Home

Find a piece of code you wrote (in any language) which repeats a syntax pattern you couldn’t avoid by writing a function, class, etc.

What do I mean by “extend syntax”?

We can implement most all of the functionality we need in Python using functions. But can we implement something like Racket’s let in Python?

let (x = 10,
     y = 20) in:
   print(x, y)

(Python does not support above)

How about C Macros?

The C Preprocessor lets us do simple text substitutions:

#define FOREVER for (;;)

main () {
        printf("Hello, World!\n");

(they can get a little more complicated than that…)

But what happens when we want to do more complex things? Like manipulate the body of that “FOREVER loop”?

C Macros

At some point, textual source manipulation cannot serve the purpose we need anymore. Let this source from MicroPython serve as an example:

STATIC mp_obj_t machine_spi_init(...) {
STATIC MP_DEFINE_CONST_FUN_OBJ_KW(machine_spi_init_obj, 1, machine_spi_init);

STATIC mp_obj_t machine_spi_deinit(...) {
STATIC MP_DEFINE_CONST_FUN_OBJ_1(machine_spi_deinit_obj, machine_spi_deinit);

STATIC mp_obj_t mp_machine_spi_read(...) {
MP_DEFINE_CONST_FUN_OBJ_VAR_BETWEEN(mp_machine_spi_read_obj, 2, 3, mp_machine_spi_read);

STATIC mp_obj_t mp_machine_spi_readinto(...) {
MP_DEFINE_CONST_FUN_OBJ_VAR_BETWEEN(mp_machine_spi_readinto_obj, 2, 3, mp_machine_spi_readinto);

Lisp Macros

Hopefully it’s become apparent that symbolic computation is the right tool for the job when it comes to macros.

Lisp Macros:

  • Compile time
  • Syntax \(\to\) Syntax

Lisp Functions:

  • Run time
  • Data \(\to\) Data
  • Lisp dialects usually make the run time available during the compile time, so the normal language can be used to write macros.

Old-School Lisp Macros

Early Lisp macro systems operated on the simple contract of functions which take syntax, manipulate it, and returns a list containing the new syntax:

(defmacro repeat-forever (&rest body)
  `(prog ()
     a   ,@body
         (go a)))

;; we can then use the macro like this:
  (format t "HELLO WORLD~%"))

Old-School: More Examples

“let” as a macro:

(defmacro let (bindings &rest body)
  `((lambda ,(mapcar #'car bindings)
    ,@(mapcar #'cadr bindings)))

;; we can then use let like this:
(let ((a 10)
      (b 20))
  (format t "~A ~A~%" a b))

Old-School: Another Example

Suppose we wanted to define a syntax like this:

(numeric-case num

We could write a macro like this:

(defmacro numeric-case (num negative zero positive)
  `(let ((result ,num))
       ((< result 0) ,negative)
       ((= result 0) ,zero)
       (t ,positive))))

What could possibly go wrong?

Fixing numeric-case with gensym

gensym is here to save us when we need really obscure symbol names:

(defmacro numeric-case (num negative zero positive)
  (let ((sym (gensym)))
    `(let ((,sym ,num))
         ((< ,sym 0) ,negative)
         ((= ,sym 0) ,zero)
         (t ,positive)))))

More Macro Issues

  • What happens if the programmer redefined one of the functions we used (e.g., < or =) in the previous example?

Unhygienic Macros

Modern Lisp dialects typically provide what is called hygienic macros: macro systems which eliminate the issues we discovered with old-school Lisp macros (to varying degrees)

Racket’s Hygienic Macros

  • define-syntax defines compile-time syntax: a function that takes a “syntax” and returns a “syntax”.
  • Typical syntax operations provide a convenient way to manipulate the syntax in a hygenic manner.
  • You can also go unhygienic: syntax->datum converts syntax to lists, symbols, etc., and datum->syntax goes back.

What is a “syntax”?

Syntax literals can be written using #' [1]:

> #'(if (> 0 x) y z)
#<syntax:readline-input:1:2 (if (> 0 x) y z)>
> (define stx #'(if (> 0 x) y z))

We can convert this to a list if we wish:

> (syntax->datum stx)
'(if (> 0 x) y z)

And back:

> (datum->syntax stx (syntax->datum stx))
#<syntax (if (> 0 x) y z)>

If you didn’t have access to the original syntax object, you could pass #f as the first argument to datum->syntax.

[1]Note this is completely different from the function-namespace thing in old-school Lisps.

Going Unhygienic

We could write our let macro without considerations for hygiene:

(define-syntax (my-let stx)
    (let ([stx-list (syntax->datum stx)])
      `((lambda ,(map car (cadr stx-list))
          ,@(cddr stx-list))
        ,@(map cadr (cadr stx-list))))))

A little bit yucky, but it worked.

Doing Things Hygienic

syntax-case acts like match but for syntax objects:

(define-syntax (my-let stx)
  (syntax-case stx ()
    [(_ ([name expr] ...) body ...)
     #'((lambda (name ...)
          body ...)
        expr ...)]))

define-syntax-rule Shorthand

define-syntax-rule is a shorthand for a define-syntax with a syntax-case of a single rule inside.

(define-syntax-rule (my-let ([name expr] ...) body ...)
  ((lambda (name ...)
     body ...)
   expr ...))

Application of Macros: Anaphora

In natural language, anaphora is a reference to a previously defined noun:

Susan dropped \(\underbrace{\text{the plate}}_{\text{referent}}\). \(\underbrace{\text{It}}_{\text{anaphor}}\) shattered loudly!

Lisp programmers call a similar technique the same name:

(printf "~a~%"
        (aif (member 10 lst)
          "10 not in the list"))

Available in a Racket Package

The “anaphoric” package provides aif, awhen, acond, and aand.

Anaphoric If

Example from Fear of Macros:

(require racket/stxparam)

(define-syntax-parameter it
  (lambda (stx)
    (raise-syntax-error (syntax-e stx) "outside of anaphora")))

(define-syntax-rule (aif predicate consequent alternative)
  (let ([result predicate])
    (if result
        (syntax-parameterize ([it (make-rename-transformer #'result)])