bc

bc(1)                        General Commands Manual                       bc(1)



이름
       bc - 무한 정확도 계산기 언어

문법
       bc [ -lws ] [  화일 ... ]

버전
       이 맨페이지는 GNU bc 버전 1.03에 대한 것이다.

설명
       bc 는 대화형으로 문장을 실행하는 무한 정확도의 숫자를 지원하는 언어이며 약간 C 언어와 비슷한 문법을 가지고 있다.  명령행
       옵션을 주면 표준 수학 라이브러리를 사용할 수 있다.  옵션을 주면 화일들을 처리하기에 앞서 수학 라이브러리가 정의된다.  bc 는
       우선 명령에서 주어진 화일 순서대로 처리한다.  화일을 모두 처리한 후 bc 는 표준 입력을 읽는다.  모든 코드는 읽는 즉시
       실행된다. ( 만약 화일 내의 코드에 처리를 중지하라는 명령이 있다면 bc는 표준입력에서 읽지 않을 것이다. )

       bc 현재 버전에서는 전통적인 bc 기능과 POSIX 표준 이외의 확장 기능을 포함하고 있다. 명령행 옵션을 주면 확장 기능에 대한
       경고 메세지를 보여주고 처리를 무시하게 할 수 있다. 이 문서에서는 GNU 버전의 처리기에서 사용하는 언어를 설명한다. 확장 기능도
       같이 설명한다.

   옵션
       -l     표준 수학 라이브러리를 정의한다.

       -w     POSIX bc에 대한 확장기능에 대하여 경고 메세지를 보여준다.

       -s     POSIX bc 언어로만 작동하게 한다.

   숫자
       bc에서 가장 기본적인 요소는 바로 숫자이다. 숫자는 임의 자리의 정확도를 갖는 숫자이다. 정확도란 정수 부분과 분수 부분을 모두
       포함한다.  모든 숫자는 내부적으로 10 진수 처리하며 10 진수로 계산이 이뤄진다.  (이번 버전에서는 나누기, 곱하기의 결과에
       대하여 버림을 한다 ) 숫자에는 두 가지 구성요소가 있는데 길이(length)와 스케일(scale)이 바로 그것이 다. 길이는
       숫자의 유효 숫자 갯수이며 스케일은 소수점 이하의 자리수를 말한다.  예를 들자면:
               .000001  은 길이 6, 스케일 6 이다.
               1935.000 은 길이 7, 스케일 3 이다.

   변수
       숫자는 두 가지 형태의 변수, 일반 변수 또는 배열에 저장된다. 둘 다 모두 이름을 가진다. 이름은 문자 다음에 문자, 숫자,
       언더스코어(_)를 쓰며 길이에는 제한이 없다.  arrays.  Both simple variables and array
       variables are named.  Names begin with a letter followed by any number of
       letters, digits and underscores.  All letters must be lower case.  (Full
       alpha-numeric names are an extension. In POSIX bc all names are a single
       lower case letter.)  The type of variable is clear by the context because
       all array variable names will be followed by brackets ([]).

       There are four special variables, scale, ibase, obase, and last.  scale
       defines how some operations use digits after the decimal point.  The
       default value of scale is 0. ibase and obase define the conversion base
       for input and output numbers.  The default for both input and output is
       base 10.  last (an extension) is a variable that has the value of the
       last printed number.  These will be discussed in further detail where
       appropriate.  All of these variables may have values assigned to them as
       well as used in expressions.

   COMMENTS
       Comments in bc start with the characters /* and end with the characters
       */.  Comments may start anywhere and appear as a single space in the
       input.  (This causes comments to delimit other input items.  For example,
       a comment can not be found in the middle of a variable name.)  Comments
       include any newlines (end of line) between the start and the end of the
       comment.

   EXPRESSIONS
       The numbers are manipulated by expressions and statements.  Since the
       language was designed to be interactive, statements and expressions are
       executed as soon as possible.  There is no "main" program.  Instead, code
       is executed as it is encountered.  (Functions, discussed in detail later,
       are defined when encountered.)

       A simple expression is just a constant. bc converts constants into
       internal decimal numbers using the current input base, specified by the
       variable ibase. (There is an exception in functions.)  The legal values
       of ibase are 2 through 16 (F).  Assigning a value outside this range to
       ibase will result in a value of 2 or 16.  Input numbers may contain the
       characters 0-9 and A-F. (Note: They must be capitals.  Lower case letters
       are variable names.)  Single digit numbers always have the value of the
       digit regardless of the value of ibase. (i.e. A = 10.)  For multi-digit
       numbers, bc changes all input digits greater or equal to ibase to the
       value of ibase-1.  This makes the number FFF always be the largest 3
       digit number of the input base.

       Full expressions are similar to many other high level languages.  Since
       there is only one kind of number, there are no rules for mixing types.
       Instead, there are rules on the scale of expressions.  Every expression
       has a scale.  This is derived from the scale of original numbers, the
       operation performed and in many cases, the value of the variable scale.
       Legal values of the variable scale are 0 to the maximum number
       representable by a C integer.

       In the following descriptions of legal expressions, "expr" refers to a
       complete expression and "var" refers to a simple or an array variable.  A
       simple variable is just a
              name
       and an array variable is specified as
              name[expr]
       Unless specifically mentioned the scale of the result is the maximum
       scale of the expressions involved.

       - expr The result is the negation of the expression.

       ++ var The variable is incremented by one and the new value is the result
              of the expression.

       -- var The variable is decremented by one and the new value is the result
              of the expression.

       var ++
               The result of the expression is the value of the variable and
              then the variable is incremented by one.

       var -- The result of the expression is the value of the variable and then
              the variable is decremented by one.

       expr + expr
              The result of the expression is the sum of the two expressions.

       expr - expr
              The result of the expression is the difference of the two
              expressions.

       expr * expr
              The result of the expression is the product of the two
              expressions.

       expr / expr
              The result of the expression is the quotient of the two
              expressions.  The scale of the result is the value of the variable
              scale.

       expr % expr
              The result of the expression is the "remainder" and it is computed
              in the following way.  To compute a%b, first a/b is computed to
              scale digits.  That result is used to compute a-(a/b)*b to the
              scale of the maximum of scale+scale(b) and scale(a).  If scale is
              set to zero and both expressions are integers this expression is
              the integer remainder function.

       expr ^ expr
              The result of the expression is the value of the first raised to
              the second. The second expression must be an integer.  (If the
              second expression is not an integer, a warning is generated and
              the expression is truncated to get an integer value.)  The scale
              of the result is scale if the exponent is negative.  If the
              exponent is positive the scale of the result is the minimum of the
              scale of the first expression times the value of the exponent and
              the maximum of scale and the scale of the first expression.  (e.g.
              scale(a^b) = min(scale(a)*b, max( scale, scale(a))).)  It should
              be noted that expr^0 will always return the value of 1.

       ( expr )
              This alters the standard precedence to force the evaluation of the
              expression.

       var = expr
              The variable is assigned the value of the expression.

       var <op>= expr
              This is equivalent to "var = var <op> expr" with the exception
              that the "var" part is evaluated only once.  This can make a
              difference if "var" is an array.

        Relational expressions are a special kind of expression that always
       evaluate to 0 or 1, 0 if the relation is false and 1 if the relation is
       true.  These may appear in any legal expression.  (POSIX bc requires that
       relational expressions are used only in if, while, and for statements and
       that only one relational test may be done in them.)  The relational
       operators are

       expr1 < expr2
              The result is 1 if expr1 is strictly less than expr2.

       expr1 <= expr2
              The result is 1 if expr1 is less than or equal to expr2.

       expr1 > expr2
              The result is 1 if expr1 is strictly greater than expr2.

       expr1 >= expr2
              The result is 1 if expr1 is greater than or equal to expr2.

       expr1 == expr2
              The result is 1 if expr1 is equal to expr2.

       expr1 != expr2
              The result is 1 if expr1 is not equal to expr2.

       Boolean operations are also legal.  (POSIX bc does NOT have boolean
       operations). The result of all boolean operations are 0 and 1 (for false
       and true) as in relational expressions.  The boolean operators are:

       !expr  The result is 1 if expr is 0.

       expr && expr
              The result is 1 if both expressions are non-zero.

       expr || expr
              The result is 1 if either expression is non-zero.

       The expression precedence is as follows: (lowest to highest)
              || operator, left associative
              && operator, left associative
              ! operator, nonassociative
              Relational operators, left associative
              Assignment operator, right associative
              + and - operators, left associative
              *, / and % operators, left associative
              ^ operator, right associative
              unary - operator, nonassociative
              ++ and -- operators, nonassociative

       This precedence was chosen so that POSIX compliant bc programs will run
       correctly. This will cause the use of the relational and logical
       operators to have some unusual behavior when used with assignment
       expressions.  Consider the expression:
              a = 3 < 5

       Most C programmers would assume this would assign the result of "3 < 5"
       (the value 1) to the variable "a".  What this does in bc is assign the
       value 3 to the variable "a" and then compare 3 to 5.  It is best to use
       parenthesis when using relational and logical operators with the
       assignment operators.

       There are a few more special expressions that are provided in bc.  These
       have to do with user defined functions and standard functions.  They all
       appear as "name(parameters)".  See the section on functions for user
       defined functions.  The standard functions are:

       length ( expression )
              The value of the length function is the number of significant
              digits in the expression.

       read ( )
              The read function (an extension) will read a number from the
              standard input, regardless of where the function occurs.   Beware,
              this can cause problems with the mixing of data and program in the
              standard input.  The best use for this function is in a previously
              written program that needs input from the user, but never allows
              program code to be input from the user.  The value of the read
              function is the number read from the standard input using the
              current value of the variable ibase for the conversion base.

       scale ( expression )
              The value of the scale function is the number of digits after the
              decimal point in the expression.

       sqrt ( expression )
              The value of the sqrt function is the square root of the
              expression.  If the expression is negative, a run time error is
              generated.

   STATEMENTS
       Statements (as in most algebraic languages) provide the sequencing of
       expression evaluation.  In bc statements are executed "as soon as
       possible."  Execution happens when a newline in encountered and there is
       one or more complete statements.  Due to this immediate execution,
       newlines are very important in bc. In fact, both a semicolon and a
       newline are used as statement separators.  An improperly placed newline
       will cause a syntax error.  Because newlines are statement separators, it
       is possible to hide a newline by using the backslash character.  The
       sequence "\<nl>", where <nl> is the newline appears to bc as whitespace
       instead of a newline.  A statement list is a series of statements
       separated by semicolons and newlines.  The following is a list of bc
       statements and what they do: (Things enclosed in brackets ([]) are
       optional parts of the statement.)

       expression
              This statement does one of two things.  If the expression starts
              with "<variable> <assignment> ...", it is considered to be an
              assignment statement.  If the expression is not an assignment
              statement, the expression is evaluated and printed to the output.
              After the number is printed, a newline is printed.  For example,
              "a=1" is an assignment statement and "(a=1)" is an expression that
              has an embedded assignment.  All numbers that are printed are
              printed in the base specified by the variable obase. The legal
              values for obase are 2 through BC_BASE_MAX.  (See the section
              LIMITS.)  For bases 2 through 16, the usual method of writing
              numbers is used.  For bases greater than 16, bc uses a multi-
              character digit method of printing the numbers where each higher
              base digit is printed as a base 10 number.  The multi-character
              digits are separated by spaces.  Each digit contains the number of
              characters required to represent the base ten value of "obase-1".
              Since numbers are of arbitrary precision, some numbers may not be
              printable on a single output line.  These long numbers will be
              split across lines using the "\" as the last character on a line.
              The maximum number of characters printed per line is 70.  Due to
              the interactive nature of bc printing a number cause the side
              effect of assigning the printed value the the special variable
              last. This allows the user to recover the last value printed
              without having to retype the expression that printed the number.
              Assigning to last is legal and will overwrite the last printed
              value with the assigned value.  The newly assigned value will
              remain until the next number is printed or another value is
              assigned to last.  (Some installations may allow the use of a
              single period (.) which is not part of a number as a short hand
              notation for for last.)

       string The string is printed to the output.  Strings start with a double
              quote character and contain all characters until the next double
              quote character.  All characters are take literally, including any
              newline.  No newline character is printed after the string.

       print list
              The print statement (an extension) provides another method of
              output.  The "list" is a list of strings and expressions separated
              by commas.  Each string or expression is printed in the order of
              the list.  No terminating newline is printed.  Expressions are
              evaluated and their value is printed and assigned the the variable
              last. Strings in the print statement are printed to the output and
              may contain special characters.  Special characters start with the
              backslash character (\).  The special characters recognized by bc
              are "a" (alert or bell), "b" (backspace), "f" (form feed), "n"
              (newline), "r" (carriage return), "q" (double quote), "t" (tab),
              and "\" (backslash).  Any other character following the backslash
              will be ignored.

       { statement_list }
              This is the compound statement.  It allows multiple statements to
              be grouped together for execution.

       if ( expression ) then statement1 [else statement2]
              The if statement evaluates the expression and executes statement1
              or statement2 depending on the value of the expression.  If the
              expression is non-zero, statement1 is executed.  If statement2 is
              present and the value of the expression is 0, then statement2 is
              executed.  (The else clause is an extension.)

       while ( expression ) statement
              The while statement will execute the statement while the
              expression is non-zero.  It evaluates the expression before each
              execution of the statement.   Termination of the loop is caused by
              a zero expression value or the execution of a break statement.

       for ( [expression1] ; [expression2] ; [expression3] ) statement
              The for statement controls repeated execution of the statement.
              Expression1 is evaluated before the loop.  Expression2 is
              evaluated before each execution of the statement.  If it is non-
              zero, the statement is evaluated.  If it is zero, the loop is
              terminated.  After each execution of the statement, expression3 is
              evaluated before the reevaluation of expression2.  If expression1
              or expression3 are missing, nothing is evaluated at the point they
              would be evaluated.  If expression2 is missing, it is the same as
              substituting the value 1 for expression2.  (The optional
              expressions are an extension. POSIX bc requires all three
              expressions.)  The following is equivalent code for the for
              statement:
              expression1;
              while (expression2) {
                 statement;
                 expression3;
              }

       break  This statement causes a forced exit of the most recent enclosing
              while statement or for statement.

       continue
              The continue statement (an extension)  causes the most recent
              enclosing for statement to start the next iteration.

       halt   The halt statement (an extension) is an executed statement that
              causes the bc processor to quit only when it is executed.  For
              example, "if (0 == 1) halt" will not cause bc to terminate because
              the halt is not executed.

       return Return the value 0 from a function.  (See the section on
              functions.)

       return ( expression )
              Return the value of the expression from a function.  (See the
              section on functions.)

   PSEUDO STATEMENTS
       These statements are not statements in the traditional sense.  They are
       not executed statements.  Their function is performed at "compile" time.

       limits Print the local limits enforced by the local version of bc.  This
              is an extension.

       quit   When the quit statement is read, the bc processor is terminated,
              regardless of where the quit statement is found.  For example, "if
              (0 == 1) quit" will cause bc to terminate.

       warranty
              Print a longer warranty notice.  This is an extension.

   FUNCTIONS
       Functions provide a method of defining a computation that can be executed
       later.  Functions in bc always compute a value and return it to the
       caller.  Function definitions are "dynamic" in the sense that a function
       is undefined until a definition is encountered in the input.  That
       definition is then used until another definition function for the same
       name is encountered.  The new definition then replaces the older
       definition.  A function is defined as follows:
              define name ( parameters ) { newline
                  auto_list   statement_list }
       A function call is just an expression of the form "name(parameters)".

       Parameters are numbers or arrays (an extension).  In the function
       definition, zero or more parameters are defined by listing their names
       separated by commas.  Numbers are only call by value parameters.  Arrays
       are only call by variable.  Arrays are specified in the parameter
       definition by the notation "name[]".   In the function call, actual
       parameters are full expressions for number parameters.  The same notation
       is used for passing arrays as for defining array parameters.  The named
       array is passed by variable to the function.  Since function definitions
       are dynamic, parameter numbers and types are checked when a function is
       called.  Any mismatch in number or types of parameters will cause a
       runtime error.  A runtime error will also occur for the call to an
       undefined function.

       The auto_listis an optional list of variables that are for "local" use.
       The syntax of the auto list (if present) is "auto name, ... ;".  (The
       semicolon is optional.)  Each name is the name of an auto variable.
       Arrays may be specified by using the same notation as used in parameters.
       These variables have their values pushed onto a stack at the start of the
       function.  The variables are then initialized to zero and used throughout
       the execution of the function.  At function exit, these variables are
       popped so that the original value (at the time of the function call) of
       these variables are restored.  The parameters are really auto variables
       that are initialized to a value provided in the function call.  Auto
       variables are different than traditional local variables in the fact that
       if function A calls function B, B may access function A's auto variables
       by just using the same name, unless function B has called them auto
       variables.  Due to the fact that auto variables and parameters are pushed
       onto a stack, bc supports recursive functions.

       The function body is a list of bc statements.  Again, statements are
       separated by semicolons or newlines.  Return statements cause the
       termination of a function and the return of a value.  There are two
       versions of the return statement.  The first form, "return", returns the
       value 0 to the calling expression.  The second form, "return ( expression
       )", computes the value of the expression and returns that value to the
       calling expression.  There is an implied "return (0)" at the end of every
       function.  This allows a function to terminate and return 0 without an
       explicit return statement.

       Functions also change the usage of the variable ibase.  All constants in
       the function body will be converted using the value of ibase at the time
       of the function call.  Changes of ibase will be ignored during the
       execution of the function except for the standard function read, which
       will always use the current value of ibase for conversion of numbers.

   MATH LIBRARY
       If bc is invoked with the -l option, a math library is preloaded and the
       default scale is set to 20.   The math functions will calculate their
       results to the scale set at the time of their call.  The math library
       defines the following functions:

       s (x)  The sine of x in radians.

       c (x)  The cosine of x in radians.

       a (x)  The arctangent of x.

       l (x)  The natural logarithm of x.

       e (x)  The exponential function of raising e to the value x.

       j (n,x)
              The bessel function of integer order n of x.

   EXAMPLES
       In /bin/sh,  the following will assign the value of "pi" to the shell
       variable pi.
               pi=$(echo "scale=10; 4*a(1)" | bc -l)

       The following is the definition of the exponential function used in the
       math library.  This function is written in POSIX bc.
              scale = 20

              /* Uses the fact that e^x = (e^(x/2))^2
                 When x is small enough, we use the series:
                   e^x = 1 + x + x^2/2! + x^3/3! + ...
              */

              define e(x) {
                auto  a, d, e, f, i, m, v, z

                /* Check the sign of x. */
                if (x<0) {
                  m = 1
                  x = -x
                }

                /* Precondition x. */
                z = scale;
                scale = 4 + z + .44*x;
                while (x > 1) {
                  f += 1;
                  x /= 2;
                }

                /* Initialize the variables. */
                v = 1+x
                a = x
                d = 1

                for (i=2; 1; i++) {
                  e = (a *= x) / (d *= i)
                  if (e == 0) {
                    if (f>0) while (f--)  v = v*v;
                    scale = z
                    if (m) return (1/v);
                    return (v/1);
                  }
                  v += e
                }
              }

       The following is code that uses the extended features of bc to implement
       a simple program for calculating checkbook balances.  This program is
       best kept in a file so that it can be used many times without having to
       retype it at every use.
              scale=2
              print "\nCheck book program!\n"
              print "  Remember, deposits are negative transactions.\n"
              print "  Exit by a 0 transaction.\n\n"

              print "Initial balance? "; bal = read()
              bal /= 1
              print "\n"
              while (1) {
                "current balance = "; bal
                "transaction? "; trans = read()
                if (trans == 0) break;
                bal -= trans
                bal /= 1
              }
              quit

       The following is the definition of the recursive factorial function.
              define f (x) {
                if (x <= 1) return (1);
                return (f(x-1) * x);
              }

   DIFFERENCES
       This version of bc was implemented from the POSIX P1003.2/D11 draft and
       contains several differences and extensions relative to the draft and
       traditional implementations.  It is not implemented in the traditional
       way using dc(1).  This version is a single process which parses and runs
       a byte code translation of the program.  There is an "undocumented"
       option (-c) that causes the program to output the byte code to the
       standard output instead of running it.  It was mainly used for debugging
       the parser and preparing the math library.

       A major source of differences is extensions, where a feature is extended
       to add more functionality and additions, where new features are added.
       The following is the list of differences and extensions.

       LANG   This version does not conform to the POSIX standard in the
              processing of the LANG environment variable and all environment
              variables starting with LC_.

       names  Traditional and POSIX bc have single letter names for functions,
              variables and arrays.  They have been extended to be multi-
              character names that start with a letter and may contain letters,
              numbers and the underscore character.

       Strings
              Strings are not allowed to contain NUL characters.  POSIX says all
              characters must be included in strings.

       last   POSIX bc does not have a last variable.  Some implementations of
              bc use the period (.) in a similar way.

       comparisons
              POSIX bc allows comparisons only in the if statement, the while
              statement, and the second expression of the for statement.  Also,
              only one relational operation is allowed in each of those
              statements.

       if statement, else clause
              POSIX bc does not have an else clause.

       for statement
              POSIX bc requires all expressions to be present in the for
              statement.

       &&, ||, !
              POSIX bc does not have the logical operators.

       read function
              POSIX bc does not have a read function.

       print statement
              POSIX bc does not have a print statement .

       continue statement
              POSIX bc does not have a continue statement.

       array parameters
              POSIX bc does not have array parameters.  Other implementations of
              bc may have call by value array parameters.

       =+, =-, =*, =/, =%, =^
              POSIX bc does not require these "old style" assignment operators
              to be defined.  This version may allow these "old style"
              assignments.  Use the limits statement to see if the installed
              version supports them.  If it does support the "old style"
              assignment operators, the statement "a =- 1" will decrement a by 1
              instead of setting a to the value -1.

       spaces in numbers
              Other implementations of bc allow spaces in numbers.  For example,
              "x=1 3" would assign the value 13 to the variable x.  The same
              statement would cause a syntax error in this version of bc.

       errors and execution
              This implementation varies from other implementations in terms of
              what code will be executed when syntax and other errors are found
              in the program.  If a syntax error is found in a function
              definition, error recovery tries to find the beginning of a
              statement and continue to parse the function.  Once a syntax error
              is found in the function, the function will not be callable and
              becomes undefined.  Syntax errors in the interactive execution
              code will invalidate the current execution block.  The execution
              block is terminated by an end of line that appears after a
              complete sequence of statements.  For example,
              a = 1
              b = 2
       has two execution blocks and
              { a = 1
                b = 2 }
       has one execution block.  Any runtime error will terminate the execution
       of the current execution block.  A runtime warning will not terminate the
       current execution block.

       Interrupts
              During an interactive session, the SIGINT signal (usually
              generated by the control-C character from the terminal) will cause
              execution of the current execution block to be interrupted.  It
              will display a "runtime" error indicating which function was
              interrupted.  After all runtime structures have been cleaned up, a
              message will be printed to notify the user that bc is ready for
              more input.  All previously defined functions remain defined and
              the value of all non-auto variables are the value at the point of
              interruption.  All auto variables and function parameters are
              removed during the clean up process.  During a non-interactive
              session, the SIGINT signal will terminate the entire run of bc.

   LIMITS
       The following are the limits currently in place for this bc processor.
       Some of them may have been changed by an installation.  Use the limits
       statement to see the actual values.

       BC_BASE_MAX
              The maximum output base is currently set at 999.  The maximum
              input base is 16.

       BC_DIM_MAX
              This is currently an arbitrary limit of 65535 as distributed.
              Your installation may be different.

       BC_SCALE_MAX
              The number of digits after the decimal point is limited to INT_MAX
              digits.  Also, the number of digits before the decimal point is
              limited to INT_MAX digits.

       BC_STRING_MAX
              The limit on the number of characters in a string is INT_MAX
              characters.

       exponent
              The value of the exponent in the raise operation (^) is limited to
              LONG_MAX.

       multiply
              The multiply routine may yield incorrect results if a number has
              more than LONG_MAX / 90 total digits.  For 32 bit longs, this
              number is 23,860,929 digits.

       code size
              Each function and the "main" program are limited to 16384 bytes of
              compiled byte code each.  This limit (BC_MAX_SEGS) can be easily
              changed to have more than 16 segments of 1024 bytes.

       variable names
              The current limit on the number of unique names is 32767 for each
              of simple variables, arrays and functions.

FILES
       In most installations, bc is completely self-contained.  Where executable
       size is of importance or the C compiler does not deal with very long
       strings, bc will read the standard math library from the file
       /usr/local/lib/libmath.b.  (The actual location may vary.  It may be
       /lib/libmath.b.)

DIAGNOSTICS
       If any file on the command line can not be opened, bc will report that
       the file is unavailable and terminate.  Also, there are compile and run
       time diagnostics that should be self-explanatory.

BUGS
       Error recovery is not very good yet.

AUTHOR
       Philip A. Nelson
       phil@cs.wwu.edu

ACKNOWLEDGEMENTS
       The author would like to thank Steve Sommars (Steve.Sommars@att.com) for
       his extensive help in testing the implementation.  Many great suggestions
       were given.  This is a much better product due to his involvement.



                                        .                                  bc(1)