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 Any useful equality predicate must satisfy [1]. Unfortunately, satisfying [2] also may be too difficult; the equivalence of behavior for procedural objects is an unsolvable problem. We are thus forced to settle for an equality predicate which may make more distinctions than are strictly necessary.

LISP has two standard equality predicates: EQUAL and EQ. We exhibited a definition of EQUAL in Part Zero. In Part Zero we also gave a description of EQ, but defined it only on atoms; LISP usually extends EQ to all S-expressions in such a way as to distinguish the results of different calls to CONS (regardless of the arguments given to CONS). Variable references and variable binding "preserve EQness".

In the absence of RPLACA ("pure LISP"), EQ and EQUAL both satisfy desideratum [1]. EQUAL, however, makes fewer unnecessary distinctions than EQ. By desideratum [2], EQUAL is therefore preferred to EQ. (The technique of "hash-consing" [Goto] can be used in this situation to make EQ and EQUAL effectively the same.)

In the presence of side effects such as RPLACA, EQUAL fails to make sufficiently many distinctions. Each call to CONS produces distinct objects, which EQUAL may fail to distinguish. In this case, EQUAL fails desideratum [1]. Thus, in the presence of RPLACA, EQ is the preferred equality predicate.

In summary, indeed "the more things change, the more they remain the same". Two distinct objects may look the same because one masquerades as the other; they can be operationally distinguished only by purposely altering the behavior of just one of them. Thus the ability to decide whether two objects are the same is directly correlated with the ability to perform side effects on them.

As we saw in the preceding section, side effects can become rather complicated. To help keep this complexity under control, we ought to abstract and package common patterns of their use.

Suppose we have a procedure PRINT-NUMBER which prints numbers:

Now people find this program very useful and use it in all their programs.

Normally we want to print numbers in radix 10 (decimal), but occasionally (for example, in a debugging aid) we want to print numbers in other radices, such as 8 or 16. One might generalize the PRINT-NUMBER program to take the radix as an extra argument: