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2012-11-27 12:48:55
94Fixed typo: "vai" for "via."history

Notes about Results

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WG1 - Core

#460 Semantics of eqv?

The semantics of eqv? are highly contended, and as a result we are bringing this up for a third and final vote.

The earlier votes on #125 (eqv? may not return true for procedures) and #229 (eqv? is unspecified for NaNs) were voted on and treated as orthogonal. There have been no objections to these, and so the results still stand. We're only focusing on the core eqv? semantics for inexact numbers.

This is fundamentally a tale of three equivalences: mathematical, operational, and representational.

Ultra-brief history: R2RS was purely mathematical, defining `eqv?` on numbers in terms of `=`. R3RS defined a simple operational equivalence by distinguishing exact and inexact. This was not a complete operational equivalence because R3RS already had multiple precisions and implicit NaNs and signed zeros. R[45]RS dropped the discussion of operational equivalence but kept the exactness separate. R6RS re-introduced the notion of operational equivalence, this time with a complete definition. The R7RS 7th draft introduced an incomplete notion of representational equivalence - two numbers are `eqv?` iff they have the same IEEE-style representation.

For this final vote there are three proposals under consideration, corresponding to the complete forms of each equivalence. Variations may be proposed and added if a suitable rationale is given.

The r5rs proposal follows R5RS exactly, in the spirit of mathematical equivalence (assuming exacts and inexacts are different types to begin with). The advantage of this is it's very simple and it appeals to idealists - people who want to pretend that they are computing real mathematical values without such thing as NaNs or negative zeros. The disadvantage of this is, as with most things that appeal to idealists, it does not match reality. Our computers use crude hacks for efficiency, and even if someone manages to build an ideal Scheme, it will likely be impractical and most implementations will continue to use those hacks. Moreover, mathematical equivalence is already available via the = procedure. This is not a practical equivalence relation for a standard. The text for the true case of eqv? for inexact numbers under r5rs is:

(3) obj1 and obj2 are both numbers, have the same exactness, and are numerically equal (see `=`).

The r6rs proposal follows R6RS exactly in the spirit of operational equivalence (with a small correction to avoid making everything unspecified via NaN transitivity). The advantages of this is that it's exactly what you want to distinguish if two values will always behave the same, for example for compiler optimizations or memoization. The disadvantage is that the definition is complicated and difficult to nail down - it doesn't account for non-standard extensions an implementation may provide which could distinguish certain new values. The r6rs text is:

(3.1) obj1 and obj2 are both exact numbers and are numerically equal (see `=`) (3.2) obj1 and obj2 are both inexact numbers, are numerically equal (see `=`), and yield the same results (in the sense of `eqv?` and excluding `+nan.0`) when passed as arguments to any other procedure that can be defined as a finite composition of Scheme’s standard arithmetic procedures.

Finally, the representational proposal is based on the previous same-bits in the spirit of representational equivalence. Two numbers are eqv? if they are represented the same way. This is potentially finer grained than operational equivalence - it may in fact make useless distinctions, but it is generally safer to over-distinguish than to under-distinguish. The representational text is:

(3.1) obj1 and obj2 are both exact numbers and are numerically equal (see `=`) (3.2) obj1 and obj2 are both numbers of the same composite numeric type, whose corresponding fields are all `eqv?` * numbers in the style of the IEEE 754-2008 floating point standard are considered composites of their radix, precision, maximum exponent, sign, exponent, and significand * non-real complex numbers are composites of their real and imaginary parts