Title

Author

Status

• Received: 2005/01/03
• Draft: 2005/01/03 - 2005/03/03

Abstract

• hashing;
• Galois-field[2] calculations of error-detecting and error-correcting codes;
• cryptography and ciphers;
• pseudo-random number generation;
• register-transfer-level modeling of digital logic designs;
• Fast-Fourier transforms;
• packing and unpacking numbers in persistant data structures;
• space-filling curves with applications to dimension reduction and sparse multi-dimensional database indexes; and
• generating approximate seed values for root-finders and transcendental function algorithms.

Rationale

The discussions of the withdrawn SRFI-33: "Integer Bitwise-operation Library" seemed to founder on consistency of procedure names and arity; and on perceived competition with the boolean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

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 one or more arguments. 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.

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

Development of the Bit order and Lamination procedures were driven by digital logic modeling and the implementation of the Hilbert-Peano space-filling function.

Gray codes are useful for communication between asynchronous processes in both hardware and software; and are integral to the implementation of the Hilbert-Peano space-filling function.

Specification

Bitwise Operations

Function: logand n1 n2 ...

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

Example:

(number->string (logand #b1100 #b1010) 2)
   => "1000"

Function: logior n1 n2 ...

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

Example:

(number->string (logior #b1100 #b1010) 2)
   => "1110"

Function: logxor n1 n2 ...

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

Example:

(number->string (logxor #b1100 #b1010) 2)
   => "110"

Function: lognot n

Returns the integer which is the 2s-complement of the integer argument.

Example:

(number->string (lognot #b10000000) 2)
   => "-10000001"
(number->string (lognot #b0) 2)
   => "-1"

Function: bitwise-if 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.

Function: logtest j k

(logtest j k) == (not (zero? (logand j k)))

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

Integer Properties

Function: logcount 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

Function: 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

Bit Within Word

Function: logbit? index j

(logbit? index j) == (logtest (expt 2 index) j)

(logbit? 0 #b1101) => #t
(logbit? 1 #b1101) => #f
(logbit? 2 #b1101) => #t
(logbit? 3 #b1101) => #t
(logbit? 4 #b1101) => #f

Function: 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

Function: logical:ones n

Returns the smallest non-negative integer having n binary ones.

Function: 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"

Function: copy-bit-field to start end from

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 4 0) 2)
        => "1101100000"
(number->string (copy-bit-field #b1101101010 0 4 -1) 2)
        => "1101101111"

Function: ash 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"

Bit order and Lamination

Function: logical:rotate k count len

Returns the low-order len bits of k cyclically permuted count bits towards high-order.

Example:

(number->string (logical:rotate #b0100 3 4) 2)
=> "10"
(number->string (logical:rotate #b0100 -1 4) 2)
=> "10"

Function: bit-reverse k n

Returns the low-order k bits of n with the bit order reversed. The low-order bit of n is the high order bit of the returned value.

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

Function: bitwise:laminate k1 ...

Returns an integer composed of the bits of k1 ... interlaced in argument order. Given k1, ... kn, the n low-order bits of the returned value will be the lowest-order bit of each argument.

Function: bitwise:delaminate count k

Returns a list of count integers comprised of every counth bit of the integer k.

For any non-negative integers k and count:

(eqv? k (bitwise:laminate (bitwise:delaminate count k)))

Bits as Booleans

Function: integer->list k len

Function: integer->list k

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

Function: 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.

Function: booleans->integer bool1 ...

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

Gray code

A Gray code is an ordering of non-negative integers in which exactly one bit differs between each pair of successive elements. There are multiple Gray codings. An n-bit Gray code corresponds to a Hamiltonian cycle on an n-dimensional hypercube.

Gray codes find use communicating incrementally changing values between asynchronous agents. De-laminated Gray codes comprise the coordinates of Peano-Hilbert space-filling curves.

Function: integer->gray-code k

Converts k to a Gray code of the same integer-length as k.

Function: gray-code->integer k

Converts the Gray code k to an integer of the same integer-length as k.

For any non-negative integer k,

(eqv? k (gray-code->integer (integer->gray-code k)))

Function: = k1 k2

Function: gray-code<? k1 k2

Function: gray-code>? k1 k2

Function: gray-code<=? k1 k2

Function: gray-code>=? k1 k2

These procedures return #t if their Gray code arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically nondecreasing, or monotonically nonincreasing.

For any non-negative integers k1 and k2, the Gray code predicate of (integer->gray-code k1) and (integer->gray-code k2) will return the same value as the corresponding predicate of k1 and k2.

Implementation

Copyright

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