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

- 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.

The discussions of the withdrawn SRFI-33: "Integer Bitwise-operation Library" seemed to founder on consistency of procedure names and arity; and on perceived competition with the boolean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

__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

__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

__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`index`th 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"

__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`start`th 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`start`th 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"

__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`count`h bit of the integer`k`.For any non-negative integers

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

__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.

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`.

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Editor: David Van Horn Last modified: Mon Jan 3 16:24:00 EST 2005