Integers as Bits

Abstract

Treating integers as two's-complement strings of bits is an arcane but important domain of computer science.

It is used for:

For more information see:

SRFI 60: Integers as Bits

Rationale

This proposal describes the SLIB module logical, which has been used for those purposes listed above.

The discussions of the withdrawn SRFI-33: "Integer Bitwise-operation Library" seemed to founder on consistency of procedure names and arity; and on perceived competition with the boolean arrays of SRFI-47.

I have implemented both logical number operations and boolean arrays; and have not been conflicted as to their application. I used boolean arrays to construct very fast indexes for database tables having millions of records. To avoid running out of RAM, creation of megabit arrays should be explicit; so the boolean array procedures put their results into a passed array. In contrast, these procedures are purely functional.

Bits and Complements

A bit-index in these descriptions is nonnegative with the least significant bit at index 0. A positive integer has a finite number of "1" bits. A negative integer has a finite number of "0" bits.

The reference implementation is written using only Scheme integer operations. Thus the only exposure of the underlying representation is the ranges of fixnums.

The complement describes the representation of negative integers. With one's-complement fixnums, the range of integers is -(2^n) to 2^n, and there are two possible representations of 0. With two's-complement fixnums, the range of integers is -(2^n+1) to 2^n.

Since we treat integers as having two's-complement negations, the two's-complement of an integer is simply its negation. The one's-complement of an integer is computed by lognot:

(define (lognot n) (- -1 n))

Bitwise Operations and Integer Properties

The logior, logxor, logand, lognot, logtest, logbit? (logbitp), ash, logcount, and integer-length procedures are from Common-Lisp. Logior, logxor, and logand have been extended to accept any arity. Opportunities to use an n-ary version of logtest have not been frequent enough to justify its extension.

In the ```Bitwise Operations```, rather than striving for orthogonal completeness, I have concentrated on a nearly minimal set of bitwise logical functions sufficient to support the uses listed above.

Although any two of logior, logxor, and logand (in combination with lognot) are sufficient to generate all the two-input logic functions, having these three means that any nontrivial two-input logical function can be synthesized using just one of these two-input primaries with zero or one calls to lognot.

bitwise-if is what SRFI-33 calls bitwise-merge.

The SRFI-33 aliases: bitwise-ior, bitwise-xor, bitwise-and, bitwise-not, bitwise-merge, any-bits-set?, and bit-count are also provided.

log2-binary-factors (alias first-set-bit) is a useful function which is simple but non-obvious:

(define (log2-binary-factors n)
  (+ -1 (integer-length (logand n (- n)))))

Bit Within Word and Field of Bits

The Bit Within Word and Field of Bits procedures are used for modeling digital logic and accessing binary data structures in software.

I have changed to copy-bit-field argument order to be consistent with the other Field of Bits procedures: the start and end index arguments are last. This makes them analogous to the argument order to substring and SRFI-47 arrays, which took their cue from substring.

These start and end index arguments are not compatible with SRFI-33's size and position arguments (occurring first) in its bit-field procedures. Both define copy-bit-field; the arguments and purposes being incompatible.

A procedure in slib/logical.scm, logical:rotate, rotated a given number of low-order bits by a given number of bits. This function was quite servicable, but I could not name it adequately. I have replaced it with rotate-bit-field with the addition of a start argument. This new function rotates a given field (from positions start to end) within an integer; leaving the rest unchanged.

Another problematic name was logical:ones, which generated an integer with the least significant k bits set. Calls to bit-field could have replaced its uses . But the definition was so short that I just replaced its uses with:

(lognot (ash -1 k))

The bit-reverse procedure was then the only one which took a width argument. So I replaced it with reverse-bit-field.

The Lamination and Gray-code functions were moved to slib/phil-spc.scm

Bits as Booleans

Bits as Booleans provides the procedures to convert between integers and lists of booleans. There is no comparable facility in SRFI-33.

Specification

Bitwise Operations

[procedure] logand n1 ...
[procedure] bitwise-and n1 ...

Returns the integer which is the bit-wise ```AND``` of the integer arguments.

Example:
(number->string (logand #b1100 #b1010) 2)
    => "1000"
[procedure] logior n1 ...
[procedure] bitwise-ior n1 ...

Returns the integer which is the bit-wise ```OR``` of the integer arguments.

Example:
(number->string (logior #b1100 #b1010) 2)
    => "1110"
[procedure] logxor n1 ...
[procedure] bitwise-xor n1 ...

Returns the integer which is the bit-wise ```XOR``` of the integer arguments.

Example:
(number->string (logxor #b1100 #b1010) 2)
    => "110"
[procedure] lognot n
[procedure] bitwise-not n

Returns the integer which is the ```one's-complement``` of the integer argument.

Example:
(number->string (lognot #b10000000) 2)
    => "-10000001"
(number->string (lognot #b0) 2)
    => "-1"
[procedure] bitwise-if mask n0 n1
[procedure] bitwise-merge mask n0 n1

Returns an integer composed of some bits from integer n0 and some from integer n1. A bit of the result is taken from n0 if the corresponding bit of integer mask is 1 and from n1 if that bit of mask is 0.

[procedure] logtest j k
[procedure] any-bits-set? j k
(logtest j k) == (not (zero? (logand j k)))

(logtest #b0100 #b1011) => #f
(logtest #b0100 #b0111) => #t

Integer Properties

[procedure] logcount n
[procedure] bit-count n

Returns the number of bits in integer n. If integer is positive, the 1-bits in its binary representation are counted. If negative, the 0-bits in its two's-complement binary representation are counted. If 0, 0 is returned.

Example:
(logcount #b10101010)
    => 4
(logcount 0)
    => 0
(logcount -2)
    => 1
[procedure] integer-length n

Returns the number of bits neccessary to represent n.

Example:
(integer-length #b10101010)
    => 8
(integer-length 0)
    => 0
(integer-length #b1111)
    => 4
[procedure] log2-binary-factors n
[procedure] first-set-bit n

Returns the number of factors of two of integer n. This value is also the bit-index of the least-significant `1' bit in n.

Example
(require 'printf)
(do ((idx 0 (+ 1 idx)))
      ((> idx 16))
    (printf "%s(%3d) ==> %-5d %s(%2d) ==> %-5d\n"
            'log2-binary-factors
            (- idx) (log2-binary-factors (- idx))
            'log2-binary-factors
            idx (log2-binary-factors idx)))
-|
log2-binary-factors(  0) ==> -1    log2-binary-factors( 0) ==> -1
log2-binary-factors( -1) ==> 0     log2-binary-factors( 1) ==> 0
log2-binary-factors( -2) ==> 1     log2-binary-factors( 2) ==> 1
log2-binary-factors( -3) ==> 0     log2-binary-factors( 3) ==> 0
log2-binary-factors( -4) ==> 2     log2-binary-factors( 4) ==> 2
log2-binary-factors( -5) ==> 0     log2-binary-factors( 5) ==> 0
log2-binary-factors( -6) ==> 1     log2-binary-factors( 6) ==> 1
log2-binary-factors( -7) ==> 0     log2-binary-factors( 7) ==> 0
log2-binary-factors( -8) ==> 3     log2-binary-factors( 8) ==> 3
log2-binary-factors( -9) ==> 0     log2-binary-factors( 9) ==> 0
log2-binary-factors(-10) ==> 1     log2-binary-factors(10) ==> 1
log2-binary-factors(-11) ==> 0     log2-binary-factors(11) ==> 0
log2-binary-factors(-12) ==> 2     log2-binary-factors(12) ==> 2
log2-binary-factors(-13) ==> 0     log2-binary-factors(13) ==> 0
log2-binary-factors(-14) ==> 1     log2-binary-factors(14) ==> 1
log2-binary-factors(-15) ==> 0     log2-binary-factors(15) ==> 0
log2-binary-factors(-16) ==> 4     log2-binary-factors(16) ==> 4

Bit Within Word

[procedure] logbit? index n
[procedure] bit-set? index n
Example
(logbit? index n) == (logtest (expt 2 index) n)

(logbit? 0 #b1101) => #t
(logbit? 1 #b1101) => #f
(logbit? 2 #b1101) => #t
(logbit? 3 #b1101) => #t
(logbit? 4 #b1101) => #f
[procedure] copy-bit index from bit

Returns an integer the same as from except in the indexth bit, which is 1 if bit is #t and 0 if bit is #f.

Example:
(number->string (copy-bit 0 0 #t) 2)       => "1"
(number->string (copy-bit 2 0 #t) 2)       => "100"
(number->string (copy-bit 2 #b1111 #f) 2)  => "1011"

Field of Bits

[procedure] bit-field n start end

Returns the integer composed of the start (inclusive) through end (exclusive) bits of n. The startth bit becomes the 0-th bit in the result.

Example:
(number->string (bit-field #b1101101010 0 4) 2)
    => "1010"
(number->string (bit-field #b1101101010 4 9) 2)
    => "10110"
[procedure] copy-bit-field to from start end

Returns an integer the same as to except possibly in the start (inclusive) through end (exclusive) bits, which are the same as those of from. The 0-th bit of from becomes the startth bit of the result.

Example:
(number->string (copy-bit-field #b1101101010 0 0 4) 2)
    => "1101100000"
(number->string (copy-bit-field #b1101101010 -1 0 4) 2)
    => "1101101111"
(number->string (copy-bit-field #b110100100010000 -1 5 9) 2)
    => "110100111110000"
[procedure] ash n count
[procedure] arithmetic-shift n count

Returns an integer equivalent to (inexact->exact (floor (* n (expt 2 count)))).

Example:
(number->string (ash #b1 3) 2)
    => "1000"
(number->string (ash #b1010 -1) 2)
    => "101"
[procedure] rotate-bit-field n count start end

Returns n with the bit-field from start to end cyclically permuted by count bits towards high-order.

Example:
(number->string (rotate-bit-field #b0100 3 0 4) 2)
    => "10"
(number->string (rotate-bit-field #b0100 -1 0 4) 2)
    => "10"
(number->string (rotate-bit-field #b110100100010000 -1 5 9) 2)
    => "110100010010000"
(number->string (rotate-bit-field #b110100100010000 1 5 9) 2)
    => "110100000110000"
[procedure] reverse-bit-field n start end

Returns n with the order of bits start to end reversed.

Example
(number->string (reverse-bit-field #xa7 0 8) 16)
    => "e5"

Bits as Booleans

[procedure] integer->list k len
[procedure] integer->list k

integer->list returns a list of len booleans corresponding to each bit of the non-negative integer k. #t is coded for each 1; #f for 0. The len argument defaults to (integer-length k).

[procedure] list->integer list

list->integer returns an integer formed from the booleans in the list list, which must be a list of booleans. A 1 bit is coded for each #t; a 0 bit for #f.

integer->list and list->integer are inverses so far as equal? is concerned.

[procedure] booleans->integer bool1 ...

Returns the integer coded by the bool1 ... arguments.

Implementation

slib/logical.scm implements the integers-as-bits procedures for R4RS or R5RS compliant Scheme implementations.

Author

Copyright (C) Aubrey Jaffer (2004, 2005). All Rights Reserved.

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The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

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