Conditionals
A decision (both the human and the computer kind) is little more than the result of a test of conditions. For example: if it is true that the light switch is ON when you leave the room, then you make a small detour to hit the switch on your way out. In other words, you are testing for a certain condition in the course of your normal operation. If the condition is true, then you do something accordingly. If the condition is false, then you carry on with your normal operation as if nothing had happened.
This IF...THEN decision construction
is precisely what goes on inside the computer when your program needs to
test for a specific condition --- like whether a number is odd or
even or whether the program user typed in the correct answer, etc.
In Mops (as in other Forths) the
IF...THEN decision process is a bit
different from some other languages you may know, largely because of the
stack orientation. The formal description of the
IF...THEN construction is as follows:
IF xx THEN zz ( n \-- ) If 'n' is non-zero (true), statement <var>xx</var> is executed, followed by statement <var>zz</var>; if 'n' is zero (false) the program continues with statement <var>zz</var>.
The IF part of the Mops decision process takes a number
off of the stack and tests it to see if it is a non-zero (i.e., any
number but zero). If the number on the stack is indeed non-zero, it
performs the operation(s) immediately following IF.
Execution then proceeds to perform operations written after
THEN. However, if the IF statement
encounters a zero on the stack, it ignores all operations between
IF and THEN and only performs
operation written after the THEN statement. In Mops the
"THEN" means to proceed with the program after the
test, as in first do this, then do that.
You won't be able to experiment with the
IF...THEN construction as easily as
the operations you learned so far. That's because this construction
must be compiled as part of a colon or method definition before it will
run on Mops. For this example, we'll put the code for the
IF...THEN statement inside a colon
definition and compile it before we can run it. Type the following:
`: TEST`\
` IF ." There is a non-zero number on the stack."`\
` THEN cr ;`\This defines TEST as a word that performs a check on
the top number on the stack. If the number is non-zero, then the
statement to that effect shows on the screen. If the top of the stack
contains a zero, then the statement does not appear. Try it by placing
various numbers (including zero and negative numbers) on the stack and
typing TEST. (Remember that an empty stack contains no
numbers, and the IF operation will cause a stack
underflow error message to appear if there are no numbers to test. A
zero, on the other hand, is indeed a number, and occupies space on the
stack.)
Two Alternatives
Some decisions, however, are more complex because they involve two
possible alternatives before proceeding. Take, for example, one of the
most difficult decisions: getting up for work in the morning. After the
alarm has gone off, and you lie in bed deciding whether you should
really get going or grab another half hour, your mind is testing certain
conditions. IF you get up now, THEN
you'll be on time for work, or ELSE you'll risk
losing your job. IF you get up now,
THEN you can get all the hot water, or
ELSE you'll have to rush through the shower to get the
few drops that are left after the rest of the family has showered.
This kind of decision construction has been included in Mops. Its formal description is:
ELSE ( \-- ) If 'n' is non-zero (true), <var>xx</var> statement is executed, followed by <var>zz</var>; if 'n' is zero (false), <var>yy</var> is executed, followed by <var>zz</var>.
If its meaning isn't already clear to you, it is used like this:
IF <var>xx</var> ELSE <var>yy</var> THEN <var>zz</var>
As with the IF...THEN construction,
this decision process looks first to see if the number on the top of the
stack is non-zero before it makes any decision. Now redefine
TEST so it takes into account the ELSE
provision.
`: TEST`\
` IF ." Non-zero number on stack."`\
` ELSE ." Zero on stack."`\
` THEN cr ;`\Place three numbers (1, 0, and 3) on the stack and perform three tests:
1 0 3
test
Non-zero number on stack.
test
Zero on stack.
test
Non-zero number on stack.
Remember that the IF operation takes the top number off
the stack when it performs its check, just like most of Mops'
operations. If you will need the number that is removed by
IF for a subsequent operation, then execute
DUP before IF, or better still,
convert your definition to use named input parameters or local variables
to preserve the value for later calculation.
Truths, Falsehoods, and Comparisons
You may be wondering how the
IF...THEN construction can be useful
if it can only determine whether or not the number on the stack is zero.
You might think that this kind of test would be rather limiting in light
of the "real-world" decisions that a program may have to make, such as
whether two integers are equal to each other, whether one is larger than
the other, or whether a number is positive or negative. Actually, the
IF...THEN construction frequently
operates at the tail end of a fuller decision procedure that makes the
real-world decisions possible. The first part of the procedure consists
of one or more comparison operators whose results are either a zero or
non-zero, depending on the outcome of the comparison.
To simplify the zero and nonzero terminology, Mops adheres to a
programming language convention revolving around the terms TRUE and
FALSE. These are also Mops words, and represent the values that appear
in the stack as a result of the comparison operations.
FALSE represents a zero in the stack;
TRUE represents any non-zero number in the stack,
including negative numbers. The Mops word TRUE returns
a non-zero number, that is, it returns a number which is all ones (in
binary). As we'll see a bit later, this corresponds to the value -1.
Type these words now:
true false
You'll see from the stack display that FALSE is the
same as (zero), and TRUE is -1.
Since these words --- or rather the numbers they represent --- are actually symbolic of a condition that has just been tested, they are sometimes referred to as flags. Flags in programs are something like markers planted in key places that symbolize a certain condition. A "true" flag signifies that a non-zero number is on the stack; a "false" flag signifies that a zero is on the stack. (Another term that is used is boolean�this really means the same as "flag".)
To help ingrain this difference between TRUE and
FALSE n your mind, redefine test yet
again so that it reinforces the way the
IF...THEN...ELSE
construction responds to TRUE and
FALSE flags existing in the stack.
`: TEST`\
` IF ." True!"`\
` ELSE ." False!"`\
` THEN cr ;`\Now, place the numbers (zero) and 4 on the stack, leaving the the true and false flags from before underneath them on the stack. Then run the test four times:
0 4`
test
True!
test
False!
test
False!
test
True!Below is a list of comparison operations that test the values of one or
more numbers on the stack and leave either TRUE or
FALSE flags on the stack. It is operations like these
that you would perform on real-world integers before performing decision
operations like
IF...THEN...ELSE. A
new term appears in the stack notations below:
'boolean'. This means that the result is
either TRUE or FALSE flag on the
stack. (The term boolean is named after George Boole, who developed a
logic system based on TRUE and FALSE
values.)
0< ( n \-- boolean ) Leaves a TRUE flag on the stack if 'n' is less than zero, otherwise leaves a FALSE flag.
0= ( n \-- boolean ) Leaves a TRUE flag on the stack if 'n' equals zero, otherwise leaves a FALSE flag.
0<> ( n \-- boolean ) Leaves a TRUE flag on the stack if 'n' does not equal zero, otherwise leaves a FALSE flag.
0> ( n \-- boolean ) Leaves a TRUE flag on the stack if �n' is greater than zero, otherwise leaves a FALSE flag.
< ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' is less than 'n2', otherwise leaves a FALSE flag.
<= ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' is less than or equal to 'n2', otherwise leaves a FALSE flag.
<> ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' does not equal 'n2', otherwise leaves a FALSE flag.
= ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' equals 'n2', otherwise leaves a FALSE flag.
> ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' is greater than 'n2', otherwise leaves a FALSE flag.
>= ( n1 n2 \-- boolean ) Leaves a TRUE flag on the stack if 'n1' is greater than or equal to 'n2', otherwise leaves a FALSE flag.
All the math in these comparison operations should be familiar to you.
Remember that these operations, like the simple arithmetic ones, are set
up in postfix notation. To remember which order to put numbers on the
stack, simply reconstruct in your mind how the formula would look in
algebraic notation. For example, to find out if
'n1' is greater than
'n2', the algebraic test would be:
<var>n1</var> > <var>n2</var>
In Mops, you simply move the operation sign to the right:
<var>n1</var> <var>n2</var> >
But in this case, Mops is testing the validity of the statement. The
numbers are taken from the stack when they are tested. If the statement
is true, then a TRUE flag goes to the stack; otherwise,
a FALSE flag goes there. Then an
IF¦THEN or
IF...THEN...ELSE
decision can be made on the number(s) in question.
Nested Decisions
It is also possible to have more than one
IF...THEN...ELSE
decision working at one time within a single definition. To accomplish
this, you place
IF...THEN...ELSE
inside one another. For example, you can set up a series of decision
operations that will examine a number in the stack, test it for several
conditions, and then announce on the screen what condition that number
meets. To do this, you'll nest several
IF...THEN statements inside one
another:
`: IFTEST { n -- }`\
` n 0<`\
` IF ." less than "`\
` ELSE n 0>`\
` IF ." greater than "`\
` THEN`\
` THEN ." zero." cr ;`\IFTEST is defined to check whether a number is
positive, negative, or zero. Enter a number in the stack and then
perform an IFTEST on it. Try positive and negative
numbers, as well as zero.
In the definition, the number is assigned to a named input parameter
('n') to preserve the value in case it is
tested by both IF statements; the first
IF removes the number from the stack, which otherwise
would leave nothing for the second IF to test. The
number is then tested whether it is less than zero. If so, less than
zero. is displayed (because the program jumps ahead to the second
THEN). If the number is not negative, it is next
compared to see if it is greater than zero in the second, nested
IF...THEN construction. If the number
is greater than zero, then the TRUE flag is noted by
the second IF statement, and greater than zero. is
displayed. If the number (which has already proven to be not less than
zero) is not greater than zero, then it must be zero, and only zero. is
displayed on the screen.
The key point to remember when nesting
IF...THEN constructions is that every
IF must have a corresponding THEN
somewhere in the same colon definition. They are nested much in the same
way that parenthetical delimiters in math formulas are nested:
( a / ( a - ( b * c ) ) + c )
So, each THEN matches the IF with
which it is lined up. Formatting your code this way,
IF <var>xx</var>
IF <var>ww</var>
IF <var>uu</var>
ELSE <var>zz</var>
THEN
THEN <var>qq</var>
THEN <var>yy</var>
with corresponding IFs, ELSEs and
THENs lining up, is a good idea for readability, not to
mention subsequent debugging should you run into a snag.
Logical Operators
There will probably be occasions in your future programs in which you
will have performed two comparison operations, and the resulting flags
from those operations will be sitting on top of the stack. How the
program proceeds from there depends on the state of those two flags. If
one flag is TRUE and the other FALSE,
they may meet the prerequisite that only one of the comparisons needs to
be true for a certain operation to take place (e.g.,
'n1' is less than
'n2', but 'n1'
is not less than zero). Conversely, you may need both flags to be
TRUE for a certain operation to take place
('n1' is both less than
'n2' and less than zero). In these
special cases, you can use the logical operators, AND
and OR:
AND ( n1 n2 \-- n3 ) Performs a bit-wise AND of 'n1' and 'n2' and leaves the result on the stack.
OR ( n1 n2 \-- n3 ) Performs a bit-wise OR of 'n1' and 'n2' and leaves the result on the stack.
Both of these operations look at the binary makeup of two numbers and
produce a result. For AND, the result will have a 1 in
each position where both the first and the second numbers have a 1.
For OR, the result will have a 1 in each position where
either the first or the second numbers (or both numbers) have a
1.
For example, let's what happens when we AND and
OR the numbers 1 and 3:
1 3 and . cr
1
1 3 or . cr
3
Remember that AND and OR perform these
operations on the binary equivalents of the integers 1 and 3. The
binary form of 1 is
0001
and the binary form of 3 is
0011
When we perform an AND on these numbers, the result is:
0001
because the AND operation returns a 1 (TRUE) for the
rightmost column of bits in these binary numbers because both bits are 1
(TRUE).
For the same two numbers, when we perform an OR
operation (instead of an AND), the result is:
0011
because the OR operation above returns a 1 for the two
rightmost column of bits in the binary numbers because one or both bits
in each column are 1.
The names for these operations,
ANDandOR, are sometimes used as verbs, as in I want to AND 1 and 3.
There is one last logical operator you should know about, the word
XOR (pronounced "exclusive-or"). Here is the formal
description:
XOR ( n1 n2 \-- n3 ) Performs a bit-wise XOR of 'n1' and 'n2' and leaves the result on the stack.
As you can see, you use it exactly like you would use
AND or OR. Try this:
1 3 xor . cr
2
However, unlike a "normal" OR operation (sometimes
referred to as "inclusive-or"), the XOR operation
returns (FALSE) where both respective bit columns in each number are 1
(TRUE). (In other words, we can have one or the other, but not both.)
This is the binary form of our answer, 2:
0010
In the second rightmost column, only one of our two integers was 1
(TRUE), so the answer in that column is 1 (TRUE), just like a "normal"
OR. However, the rightmost bit column of our answer is
(FALSE) because the rightmost columns of both of our two integers (1 and
3) is 1 (TRUE), unlike a "normal" OR.
Experiment a little bit with AND, OR,
and XOR in this fashion. Remember that these operations
are working on the binary equivalent of the decimal numbers you type
into the stack. If you have difficulty understanding an answer, try
working out the problem on paper by converting each number to binary and
then performing the AND, OR, or XOR arithmetic on the numbers as shown
above. Once you understand the concept, you can trust Mops to do these
operations correctly for you at all times.
The CASE Decision
It's not uncommon to have an instance in a program in which the next
step could be one of several possibilities, depending on the actual
number on the stack --- not just whether it's TRUE (non-zero) or
FALSE (zero). For example, a program may ask you to type a number from
zero to nine. For most of the numbers, the subsequent step is the same,
but for numbers 2, 6, and 7, the outcome is different. In other words,
if it is the case of a 2 on the stack, then a unique operation takes
place. Sure, you could run a series of comparison operations and nested
IF...THEN constructions on the number
to narrow it down (e.g., testing if the number is not less than two nor
greater than two), but that gets cumbersome when you're testing for
many numbers.
Mops' shortcut for this multiple-decision making is the
CASE structure. Using the example above, you could
define a word like this:
`: CASETEST ( n -- ) \ Print TWO, SIX, SEVEN, or OTHER`\
` CASE`\
` 2 OF ." TWO" ENDOF`\
` 6 OF ." SIX" ENDOF`\
` 7 OF ." SEVEN" ENDOF`\
` ." OTHER"`\
` ENDCASE cr ;`\This word takes the number on the stack and checks whether it is a
CASE OF 2, 6, or 7. If a particular
CASE is valid, then the branch executes statements
until it encounters an ENDOF delimiter. At that point,
execution jumps to ENDCASE, ignoring all other
statements within the CASE construct. If none of the
cases are valid, then execution continues toward the
ENDCASE delimiter.
If a statement is inserted before ENDCASE (like
'." OTHER"' in the example), then it is
executed whenever the test of all cases fail. This statement is known as
the default statement, since it's the statementwhich gets executed by
default if nothing else does.
Note: The
CASEtest retains the test value on the stack, and it is dropped at the end by theENDCASE. In the default statement, particularly, you might want to make use of the test value. But if you're going to use it (take it off the stack), remember toDUPit first (or just put a dummy value on the stack) to be dropped by theENDCASE.