Numeric­Tower

The numeric tower

This is an analysis of the R5RS provision that the full numeric tower may be subsetted. Five boolean feature settings characterize different kinds of numeric towers: exactness-preserving, ratios, inexact, and complex, and exact-complex. These refer respectively to the closure of exact numbers under rational operations (except /), exact non-integer values, inexact rationals, non-real numbers, and exact non-real numbers. Of course, exact-complex can only be true if complex is true.

So here are some possible numeric towers. I use + if a feature is present and - if it is absent, and give a general description of the resulting tower, with any examples I know of.

||exactness-preserving||ratios||inexact||complex||exact-complex||Description||Examples

-	-	-	-	-	A "toy" tower with fixnums (bounded exact integers) only. Possibly appropriate where no numerical work will be done.	SigScheme
-	-	+	-	-	A Scheme where all numbers are real and all arithmetic happens in essentially constant time.	Chicken (default), Oaklisp, Sixx, Sizzle
-	-	+	+	-	The complex-number equivalent of the preceding.	Stalin
-	+	+	+	-	Limited-range exact and inexact numbers.	Scheme 7
-	+	+	+	+	Limited-range exact and inexact numbers, including exact non-reals.	Wraith
+	-	-	-	-	Unlimited-range integers only. Possibly usable for crypto work. Fortunately, ratios are fairly cheap given bignums.	None known
+	-	+	-	-	A good all-round compromise, providing exact integers and inexact real numbers.	ISLisp, Bigloo, Scheme 9, Elk
+	-	+	+	-	The same as the preceding, but with inexact complex numbers as well.	SCM
+	+	-	-	-	Exact rational numbers only. Reasonable for some purposes, but will often run very slowly without inexact support, so unsuitable for scientific-type work.	Dream
+	-	+	+	+	The same as the preceding, but with exact non-real numbers. Even less suitable for scientific-type work, given that in practice non-real numbers are usually inexact.	None known
+	+	+	-	-	Full numeric tower with real numbers only.	Psyche
+	+	+	+	-	Full numeric tower with inexact non-reals only.	Gauche, Guile, SISC, KSi, UMB
+	+	+	+	+	Full numeric tower.	All R6RS systems, all CL systems, Racket, MIT, Gambit, Chicken with the numbers egg, Scheme48/scsh, Kawa, Chibi, Chez, STklos

Vincent Manis's comments

[These comments refer to an earlier version of this page. --cowan]

I'd prefer to couch my reply in terms of what data types are provided by an implementation. That maps pretty well onto John's four characteristics, but it seems easier for me to think about. I'll try to match my responses to John's ++-- strings.

Of the towers John lists, the following seem worth standardizing.

1. ----: this seems appropriate in implementations for limited embedded processors.
2. --+-, --++: there seem to be two kinds of possible implementation: a Chicken-like system that provides fixnums and flonums, and a system that provides flonums only (and therefore there is no such thing as an exact integer). Lua, with default build options, works this way, as does JavaScript. Both variations seem defensible to me. [The second variation will not work, because exact numbers must exist and be disjoint from inexact ones. --cowan]
3. +-+-, +-++: I believe that the implication here is that bignums are provided as well. [Yes, but they might be very small bignums. --cowan]
4. +++-, ++++: This adds ratios to the previous towers.

The others don't seem of much interest, even though some implementors have chosen among them.

Adding complex numbers to an implementation that doesn't support them can be done almost entirely as a library module, apart from issues such as read and write syntax. Similarly, adding ratios to an implementation that supports bignums again can be done almost entirely as a library module. (Proof: Chicken's numbers library, which John references.)

I therefore think that there are three defensible core towers, namely (a) ----, (b) --+-, and (c) +-+-, along with two library modules (ratios and complexes) that may or may not happen to be imported automatically. With careful design, it ought to be possible to provide reference implementations of these modules that would work on any implementation that supports bignums/ratios or inexact reals, respectively. This would allow the implementor to have his/her cake (smaller implementation, less work), and eat it (supporting a fuller tower).

I hope that the Report can be written in such a way that this smaller number of towers is either preferred or required. This maximizes code portability, while still making it possible to build small implementations. [That basically rules out bignums without flonums. --jcowan]

If the Report does allow multiple towers (which I think is a foregone conclusion), a built-in procedure with a name something like numeric-features should be required, this will allow portable code to verify that it is running on a sufficiently-capable system. numeric-features could return a list of tower criteria that the system supports. The criteria, and their symbol names, should be defined by the Report. [My proposal at CondExpandCowan provides this at compile time rather than run time. --jcowan]