Stack-based algorithms
PostScript is an example of a postfix stack-based language. An expression example in this language is . Calculating the expression involves understanding how stack-orientation works. Stack-orientation can be presented as the following conveyor belt analogy. at the end of a conveyor belt (the ''input''), plates marked2
, 3
, and mul
are placed in sequence. The plate at the end of the conveyor (2
) can be taken, however other plates cannot be accessed until the plate at the end is removed. The plates can only be stored in a stack, and can only be added or removed from atop the stack, not from the middle or bottom. Blank plates (and a marker) can be supplied and plates can be permanently discarded.
2
and put it on the stack, then take plate 3
and put it on the stack. Next, take the mul
plate. This is an instruction to perform. Then, take the top two plates off the stack, multiply their labels (2
and 3
), and write the result (6
) on a new plate. Discard the two old plates (2
and 3
) and the plate mul
, and put the new plate on the stack. With no more plates remaining on the conveyor, the result of the calculation (6
) is shown on the plate atop the stack.
This is a very simple calculation. What if a more complex calculation is needed, such as ? If it is first written in postfix form, that is, ,
the calculation can be performed in exactly the same manner and achieve the correct result. The steps of the calculation are shown in the table below. Each column shows an input element (the plate at the end of the conveyor), and the contents of the stack after processing that input.
After processing all the input, the stack contains 56
, which is the answer.
From this, the following can be concluded: a stack-based programming language has only one way to handle data, by taking one piece of data from atop the stack, termed ''pop''ping, and putting data back atop the stack, termed ''push''ing. Any expression that can be written ''conventionally'', or in another programming language, can be written in postfix (or prefix) form and thus be amenable to being interpreted by a stack-oriented language.
Stack manipulation
Since the stack is the key means to manipulate data in a stack-oriented language, such languages often provide some sort of stack manipulation operators. Commonly provided aredup
, to duplicate the element atop the stack, exch
(or swap
), to exchange elements atop the stack (the first becomes second and the second becomes first), roll
, to cyclically permute elements in the stack or on part of the stack, pop
(or drop
), to discard the element atop the stack (push is implicit), and others. These become key in studying procedures.
Stack effect diagrams
As an aid to understanding the effect of statement, a short comment is used showing the top of the stack before and after the statement. The top of the stack is rightmost if there are multiple items. This notation is commonly used in the Forth language, where comments are enclosed in parentheses.fib
function below is described:
PostScript stacks
PostScript and some other stack languages have other separate stacks for other purposes.Variables and dictionaries
The evaluation of different expressions has already been analysed. The implementation of variables is important for any programming language, but for stack-oriented languages it is of special concern, as there is only one way to interact with data. The way variables are implemented in stack-oriented languages such as PostScript usually involves a separate, specialized stack which holds ''dictionaries'' of key–value pairs. To create a variable, a key (the variable name) must be created first, with which a value is then associated. In PostScript, a name data object is prefixed with a/
, so /x
is a name data object which can be associated with, for example, the number 42
. The define
command is def
, so
/x 42 def
associates with the name x
with the number 42
in the dictionary atop the stack. A difference exists between /x
and x
– the former is a data object representing a name, x
stands for what is defined under /x
.
Procedures
A procedure in a stack-based programming language is treated as a data object in its own right. In PostScript, procedures are denoted between
.
For example, in PostScript syntax,
represents an anonymous procedure to duplicate what is on the top of the stack and then multiply the result - a squaring procedure.
Since procedures are treated as simple data objects, names with procedures can be defined. When they are retrieved, they are executed directly.
Dictionaries provide a means of controlling scoping, as well as storing of definitions.
Since data objects are stored in the top-most dictionary, an unexpected ability arises naturally: when looking up a definition from a dictionary, the topmost dictionary is checked, then the next, and so on. If a procedure is defined that has the same name as another already defined in a different dictionary, the local one will be called.
Anatomy of some typical procedures
Procedures often take arguments. They are handled by the procedure in a very specific way, different from that of other programming languages. To examine afib(4)
:
stack: 4
dup
stack: 4 4
dup
stack: 4 4 4
1 eq
stack: 4 4 false
exch
stack: 4 false 4
0 eq
stack: 4 false false
or
stack: 4 false
not
stack: 4 true
Since the expression evaluates to true, the inner procedure is evaluated.
stack: 4
dup
stack: 4 4
1 sub
stack: 4 3
fib
: ''(recursive call here)''
stack: 4 F(3)
exch
stack: F(3) 4
2 sub
stack: F(3) 2
fib
: ''(recursive call here)''
stack: F(3) F(2)
add
stack: F(3)+F(2)
which is the expected result.
This procedure does not use named variables, purely the stack. Named variables can be created by using the /a exch def
construct. For example,
is a squaring procedure with a named variable n
. Assuming that /sq def
and 3 sq
is called, the procedure sq
is analysed in the following way:
stack: 3 /n
exch
stack: /n 3
def
stack: ''empty'' (it has been defined)
n
stack: 3
n
stack: 3 3
mul
stack: 9
which is the expected result.
Control and flow
As there exist anonymous procedures, flow control can arise naturally. Three pieces of data are required for anAnalysis of the language model
The simple model provided in a stack-oriented language allows expressions and programs to be interpreted simply and theoretically evaluated much faster, since no syntax analysis need be done, onlyshowpage
operator can be overridden with a custom one that applies a certain style to the page, instead of having to define a custom operator or to repeat code to generate the style.
See also
*References