gistfile1.js

// http://blog.livedoor.jp/dankogai/archives/51859373.html

var x = { a: 1, b: 2, c: 3 };
var y = { a: 1, b: 2, c: 3, d: 4 };
var z = {a: 1, b: 2, c: 3 };
var w = {a: 1, b: 2, c: 3, d: 4 };
x.b = w;
y.b = x;
z.b = y;
w.b = x;

Object.equals(x, z); // false
Object.equals(z, x); // true
Object.equals(y, w); // true
Object.equals(w, y); // false

/* How to check deep equality precisely?
cf. "Deep equality revisited"
    S. Abiteboul and J. Van den Bussche
    Deductive and Object-Oriented Databases, Lecture Notes in Computer Science, vol 1013, pages 213-228. Springer, 1995.
*/

// I think this extreme case is forgivable since even uneval() of firefox was fooled.
// (though it was souped down since Firefox 12)
// add the snippet below and here is what I got in firefox:
try {
console.log(uneval(x))
console.log(uneval(z));
}catch(e){
console.log('uneval() is missing');
}
// ({a:1, b:{a:1, b:{}, c:3, d:4}, c:3})
// ({a:1, b:{a:1, b:{a:1, b:{a:1, b:{}, c:3, d:4}, c:3}, c:3, d:4}, c:3})
// as the result suggests, x and z differ in "when", not "where" it bumps into self-reference.
// it should be documented, however.

I thought that Object.equals is a structural equivalence for
values and a reference is not a value. In this case, it should
examine the structures of x and z as values of infinite trees
and should not be sensitive to self-reference since the "values"
do not contain references by definition. (I'm talking about the
formal definition of values for a particular structural
equivalence relation, not the implementation of values in
JavaScript.)

I'm not disagreeing with your opinion that the current
implementation of Object.equals somehow makes sense, as long as
its formal definition is well-defined. I understand that the
formal definition is different from what I thought.

Unfortunately, I found that the current implementation of
Object.equals is not well-defined, at least as an equivalence
relation. The (updated) example shows that Object.equals is not
symmetric. I think the name equals implies that it is an
equivalence relation (not necessarily a structural equivalence), i.e.,
a relation which is reflexive, symmetric and transitive. So I have to
conclude that Object.equals is not well-defined.