| RFC 9226 | Bioctal: Hexadecimal 2.0 | April 2022 |
| Breen | Experimental | [Page] |
The prevailing hexadecimal system was chosen forcongruence with groups of four binary digits, but its designexhibits an indifference to cognitive factors.An alternative is introduced that is designed to reducebrain cycles in cases where a hexadecimal number should bereadily convertible to binary by a human being.¶
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Octal has long been used to represent groups of three binarydigits as single characters, and that system has theconsiderable merit of not requiring any digitsother than those already familiar from decimal numbers.Unfortunately, the increasing use of 16-bit machines andother machines that have word lengths that are evenly divisibleby four (but not by three) has led to the widespread adoption ofhexadecimal.Table 1 presents the digits of the hexadecimal alphabet.¶
| Value | Digit |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | A |
| 11 | B |
| 12 | C |
| 13 | D |
| 14 | E |
| 15 | F |
The choice of alphabet is clearly arbitrary: On the exhaustion ofthe decimal digits, the first letters of the Latin alphabet areused in sequence for the remaining hexadecimal digits.An arbitrary alphabet may be acceptable on an interim orexperimental basis.However, given the diminishing likelihood of a return to 18-bitcomputing,a review of this choice of alphabet is merited before its use,like that of the QWERTY keyboard, becomes too deeply establishedto permit the easy adoption of a more logical alternative.¶
One problem with the hexadecimal alphabet is well known: It containstwo vowels, and numbers expressed in hexadecimal have been foundto collide with words offensive to vegetarians and other groups.¶
Imposing a greater constraint on the solution space, however, isthe difficulty of mentally converting a numberexpressed in hexadecimal to (or from) binary.Consider the hexadecimal digit 'D', for example.First, one must remember that 'D' represents a value of 13 --and, while it may be easy to recall that 'F' is 15 with all bits set,for digits in the middle of the non-decimal range, such as'C' and 'D', one may resort to counting ("A is ten, B is eleven, ...").Next, one must subtract eight from that number to arrive at anumber that is in the octal range.Thus, the benefit of representing one additional bit incurs thecost of two additional mental operations before one arrives atthe position where the task that remains reduces to the difficultyof converting the remaining three digits to binary.¶
These mental steps are not difficult per se,since a child could do them, but if it is possible to avoidemploying children, then it should be avoided.An appeal to the authority of cognitive psychologyis perhaps also due here, in particular tothe "seven plus or minus two" principle[Miller]-- either because octal is within the upper end of that range (nine)and hexadecimal is not,or else because the difference in the size of the alphabetsis greater than the lower end of that range (five).Either way, it is almost certainly relevant.¶
Various alternatives have already been suggested.Some of these are equally arbitrary, e.g., in selecting thelast six letters of the Latin alphabet rather thanthe first six letters.¶
The scheme that comes closest to solving the main problemto date is described by Bruce A. Martin[Martin] who proposes new characters for the entire octal alphabet.While his principal motivation is to distinguish hexadecimalnumbers from decimals, the design of each character useshorizontal lines to directly represent the "ones" of thecorresponding binary number, making mental translationto binary a trivial task.¶
Unfortunately for this and other proposals involving new symbols,proposals to change the US-ASCII character set[USASCII] might no longer beaccepted.Also, it seems unrealistic to expect keyboards or printertype elements (whether of the golf ball or daisy wheel kind)to be replaced to accommodate new character designs.¶
Table 2 presents the hexadecimal alphabet once again, this timein a sequence of two octaves with values increasing left to rightand top to bottom.¶
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 8 | 9 | A | B | C | D | E | F |
Arranged thus, the binary representation of each digit in thesecond octave is the same as the digit above it, but with themost significant of the four bits set to '1' instead of '0'.¶
The incongruity of two decimal digits in the second octavealso suggests that, in blindly aligning with four bits,hexadecimal (six plus ten, neither of which are powers of two)misses an opportunity to align also with three bits.¶
Bioctal restores congruence by replacing the second row withcharacters mnemonically related to the corresponding characterin the first octave.¶
Table 3 shows the compelling result.¶
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| c | j | z | w | f | s | b | v |
The mnemonic basis is the shape of the lowercase character.This is seen directly for '2', '5', and '6'.For '3', '4', and '7', the corresponding letters arethe result of a quarter-turn clockwise (assuming an "open" '4').The choice of 'c' and 'j' for '0' and '1' avoids vowels andlowercase 'L', the latter being confusable with '1' in some fonts.¶
With this choice of letters, it is immediately evident thatboth problems with hexadecimal are solved.Mental conversion is now straightforward: if the digit is aletter, then the most significant of the four binary bits is '1',and the remaining three bits are the same as for the Arabic numeralwith the same shape in the first octave.¶
Several objections can be anticipated, the first of whichconcerns the name.The term "bioctal" is already used to refer to the combinationof two octal characters into a single field on, for example,paper tape (e.g.,[UNIVAC]).However, if the word "bioctal" must be disadvantaged relativeto words such as "biannual" in the number of meanings it isallowed to have, then it is the paper tapers who must give way:in that context, the "octal" part of "bioctal" refers to the numberof distinct values that three bits can have, while the "bi" refersto a doubling of the number of bits, not values.A meaning depending on such a discordant etymology does notdeserve to endure.¶
Second, it may be argued that the use of hexadecimal hasalready become too entrenched to be changed in the short term:Bioctal should be introduced only after those working in theindustry who have grown accustomed to hexadecimal have retired.Such a dilatory contention cannot be allowed to impedethe march of progress.Instead, any data entry technician who claims to have difficulty withbioctal may be reassigned to duties involving only binary numbers.¶
A third possible objection is that numbers in bioctal do notsort numerically.However, this assumes a sort based on the US-ASCII order of symbols;it is quite possible that bioctal numbers sort naturally in somelesser known variety of EBCDIC.Further, resistance to numeric sorting may be an indicator of virtue,being suggestive of an alphabet with a certain strength of character.¶
One difficulty remains: Not all computers support lowercase letters.While this is indeed true,it should be confirmed in any particular instance:the author has observed that in many cases a machine having akeyboard with buttons marked only with uppercase letters alsosupports lowercase letters.In any case, it is permissible to use uppercase lettersinstead of the lowercase ones ofTable 3;the morphology mnemonic continues to work for most bioctal digitsin uppercase, although an extra mental cycle is required for 'B'.¶
The letters 'b' and 'f' appear in both the bioctal and hexadecimalalphabets, which makes potential misinterpretation a concern.A case of particular hazard arises where two embedded systems engineerswork to develop a miniature lizard detectordesigned to be worn like a wristwatch.One engineer works on the lizard proximity sensor andthe other on a minimal two-character display.The interface between the circuits is 14 bits.To make things easier, the engineer working on the display arrangesfor these bits to be set in a pattern that allows them to be useddirectly as two seven-bit US-ASCII characters indicating themost significant lacertilian species detected in the vicinity of thedevice.Due to the use of an old US-ASCII table (i.e., one in hex, not bioctal)and human error, some of the values specified as outputs forthe detection subsystem are in hexadecimal, not the bioctal theengineer developing that subsystem expects --including, in the case of one type of lizard, "4b 4f".The result is that the detector displays "NL" (No Lizards)when it should display "KO" (Komodo dragon).This may be considered prejudicial to the security of the userof the device.¶
Extensive research has uncovered no other security-related scenariosto date.¶
This document has no IANA actions.¶
Bioctal is a significant advance over hexadecimal technologyand promises to reduce the small (but assuredly non-zero)contribution to anthropogenic global warming of mental hex-to-binaryconversions.Since the mnemonic basis of the alphabet is independent of Englishor any other particular natural language,there is no reason that it should not be adopted immediately aroundthe world, excepting perhaps certain islands of Indonesiato whichVaranus komodoensis is native.¶
The author is indebted toR. Goldberg for assistance withSection 4.¶