This chapter has been excerpted from the book "FlowBased Programming: A New Approach to Application Development" (van Nostrand Reinhold, 1994), by J.Paul Morrison. A second edition (2010) is now available from CreateSpace eStore and Amazon.com. The 2nd edition is also available in ebook format from Kindle (Kindle format) and Lulu (epub format). To find out more about FBP, click on FBP.
For definitions of FBP terms, see Glossary.

[References to TR1 in the text refer to an application component which is used in examples earlier in the book  see Chapter 10. TR1 "extends" detail records by multiplying a quantity in a detail record by a unit price obtained from a master record in another file. Processes "upstream" of TR1 have merged masters and details, so TR1 is fed a single merged stream containing masters and details, and with related records delimited by special IPs called "brackets".
Collate refers to a general, "black box", reusable component that merges 2 or more streams on the basis of parameters describing the location of control fields in its incoming IPs, and optionally inserts grouping IPs called "brackets" into its output stream. It is described in some detail in Chapter 8.]
Material from book starts here:
It is easy to see that an FBP component can be regarded as a
function transforming
its input stream into its output stream. These functions can then be
combined
to make complex expressions just as, say, addition, multiplication,
etc.
can be combined in an algebraic expression. A number of languages have
used
this as a base for notations. Let me take a simple example:
Figure 25.1
If we label streams as shown with lower case letters, then the above diagram can be represented succinctly as follows:
c = G(F(a),F(b))
I deliberately used F twice to underline the fact that the same function can be used many times in the same expression. Of course, this relies on the function having no side effects  this is one of the desirable characteristics of functional languages (and one of the qualities which are desirable when designing FBP components). We will return to this point later on.
Now we have seen that streams are made up of patterns of IPs, which in turn have fields or data items. Is it possible to carry functional notation to the point where we can actually build systems processing real data? I believe it is, but in a way that will not be as "mathematical" as most treatments of recursive programming. In the rest of this chapter, I will develop the concepts of recursive stream definitions. But first, for those of you whose math is a bit rusty, we should talk about what exactly are recursive definitions. The rest of you can skip ahead!
Recursive techniques are often taught using the formula for factorials as an example. A factorial is the product of all the integers greater than zero up to and including the number whose factorial you want to calculate. Factorials can be (and often are) calculated iteratively. i.e.
factorial(x):
a = 1
do varying y from x to 1 by 1
a = a * y
enddo
return a
Now, the classic expression for calculating factorials really does the same thing, but it does it recursively, as follows:
factorial(x):
if x = 1
return 1
else
return x * factorial (x  1)
endif
Here we see that "factorial" is actually defined in terms of itself, but used on a "smaller" part of the problem. Since we use factorial on x1 , and the first test is always to see if we have reached 1, we are actually getting one step closer to our goal each time around the definition. If you want to visualize how this might execute, think of a stack holding the environment for each invocation of factorial. Each time we start a factorial calculation, we push the information we will need onto the stack, so the stack gets deeper and deeper. When we eventually get a true condition on the first test, we can calculate a factorial (1) without pushing a new environment on the stack. Now we can finish all the factorial calculations (in reverse sequence), so that eventually we arrive back at the top of the stack and we're finished. Speakers of German will be familiar with a similar phenomenon which occurs in that language, where several "contexts" may be "stored" until the end of the sentence, at which time each context is terminated with the production of an infinitive or past participle.
The advantage of the recursive definition is that it has no local storage variables. We therefore suspect that this characteristic may have a bearing on some of the problems we identified earlier in this book with the "pigeonhole" concept of storage. Functions also ideally have no sideeffects, so they can easily be reused. If we combine these concepts with some other concepts which have appeared in the literature, we can actually describe (a small piece of) business processing in a way which is free from a number of the problems which bedevil the more conventional approaches. I also believe that if you look back at Chapter 18, you will find that there are similarities between that notation and what will be pursued more rigorously in this chapter. Admittedly this will be a tiny example, but my hope is that someone will be sufficiently intrigued that they will carry it further.
Recursive functions are attractive to mathematicians because they have no sideeffects, and therefore are easier to analyze and understand. In W.B. Ackerman's paper (1979) he states: "the language properties that a data flow computer requires are beneficial in their own right, and are very similar to some of the properties that are known to facilitate understandable and maintainable software, ...." Some of these beneficial properties are as follows:

locality of effect

freedom from side effects

"call by value"
Ackerman defines an applicative language as one which does all of its processing by means of operators applied to values. The earliest known applicative language was LISP.
By now you should be familiar with the use of the term "stream" in FBP to describe the set of IPs passing across a particular connection. The stream does not all exist at the same time, but is continuously being generated at one end and consumed at the other. However, it has a "real" existence, and it can be manipulated in various ways. W.H. Burge (1975) showed how stream expressions can be developed using a recursive, applicative style of programming. D.P. Friedman and D.S. Wise contributed a number of papers relating applicative programming to streams by adding the concept of "lazy evaluation" (1976) to Burge's work. This style has the desired freedom from side effects and has another useful characteristic: the equals sign is really a definition statement, and can be used for proving the validity of programs as well as for doing the actual data processing. Such a language is called "definitional".
Quoting W.B. Ackerman (1979) again: "Such languages are well suited to program verification because the assertions one makes in proving correctness are exactly the same as the definitions appearing in the program itself." One can put a restriction on assignment statements to the effect that the definition should not assign a value to a given symbol more than once in a single scope. Thus a statement like
J:=J+1
is ruled out because "J" would have to be given an initial value within the scope, resulting in two assignments within that scope. Also, viewed as a definition, it is obviously a contradiction! A number of writers on programming have described their feelings of shock on their first encounter with this kind of statement, only to become so used to it over the years that they eventually don't notice anything strange about it!
We shall now talk a little bit about the possibility of writing programs in a way that avoids the problem of rebinding variables within a scope, following (Burge 1975) and (Friedman and Wise 1976) in the area of stream functions.
Typically, like the definition of factorial shown above, the desired output is defined in terms of two functions:

a function of the first item in the stream

a function relating this item to the rest of the stream

and a termination rule specifying a value to be returned when the function bottoms out.
The following example resembles the definition of factorial shown above, but moves closer to business applications. Even though it may not much resemble the type of business applications that you are accustomed to, you will have to admit that it is very compact! Suppose we want to count all the IPs in a stream. Then, analogously to the factorial calculation above, we could write a "counter" function F as follows:
F(S) = if S is null,where the result is specified directly by a value, e.g. 0, or 1 + F(rest(S)).
then 0,
else 1 + F(rest(S))
With respect to the rest of this notation, functions are expressed using the conventional bracket notation, and sublists will be specified by means of curly braces. For instance, {w,x,y,z} is a sublist consisting of w, x, y and z. null tests for a stream with no IPs in it, i.e. {}; first(S) is the first IP of a stream, i.e. w in the above example, and rest(S) is a list comprising all the rest of the IPs in the stream, i.e. {x,y,z}. Notice that, while rest returns a list, first only returns a single IP.
In this first example, F is called recursively to return a value based on processing a stream S. At each invocation of F, its environment (the part of the stream that it can see) is pushed down on a runtime stack. When an invocation of F finds itself looking at an empty stream, the null test returns true, F bottoms out, and the stacked environments are progressively popped up, until the original one is reached, at which point the process stops. Notice that there is a family resemblance between recursive definitions which only "recurse" at the righthand end, and right linear grammars (described in the previous chapter). This kind of recursive definition can also have special processing applied to it which maintains the stack at a constant depth.
Now so far we haven't actually used the data in the stream IPs. Suppose therefore that we want to sum all the quantity fields in a stream of IPs. We will introduce the convention a:x to mean "field a of x". This notation and the "miniconstructor" notation described below are based on the Vienna Definition Language, developed some years ago by the IBM Laboratory in Vienna. The desired calculation can now be expressed recursively as follows:
F(S) = if S is null,
then 0,
else q:first(S) + F(rest(S))
where q is the quantity field of an IP.
So far, we have only generated a single quantity from our stream. To do anything more complicated, we will need to be able to generate IPs and string them into streams. To do the former, we will need something which can build an IP given a set of values for its fields. We will use a special function for this which I will call the "miniconstructor" (µ). This takes as its argument a list of selector symbols and values, and returns as a result an IP with those values inserted into the fields designated by the selectors. A selector and its value are separated by commas, while selector/value pairs are separated by semicolons. The miniconstructor is a concise way of specifying how new IPs are to be built.
To combine IPs into a stream, we use a variant of the wellknown listprocessing function cons, which was first used in functional languages to join two lists together. The following equivalence holds:
a = cons(first(a),rest(a))
Friedman and Wise (1976) have extended this concept by removing the requirement that both of the arguments of cons be available at the same instant of time. Their "lazy cons" function does not actually build a stream until both of its arguments are realized  before that it simply records a "promise" to do this. This allows us to imagine a stream being dynamically realized from the front, but with an unrealized back end. The end of the stream stays unrealized until the very end of the process, while the beginning is an everlengthening sequence of items.
Suppose we want to create a stream of extended details (where the quantity field has been extended by the unit price for the product): if the unit price were repeated in every IP, there would be no problem, as all the information that a particular call to the function requires is immediately available. Assigning selectors p, s, d, q, u, e to product number, salesman number, district number, quantity, unit price and extended price, respectively, we would then be able to define the extended details stream as follows:
F(S) = if S is null,
then null
else cons (E(first(S)), F(rest(S)))
E(x) = µ(p,p:x; s,s:x; d,d:x; u,u:x; e,q:x*u:x)
where E is a function which creates a single IP. The expression after the last semicolon in the definition of E(x) would read in English: "set the extended price field of this IP to unit price multiplied by quantity". This would be fine except for the fact that unfortunately the unit price is not in the same IP as quantity!
In fact, in the case of the merged stream being processed by TR1, the unit price for a given product is held in only one IP per substream  namely the master IP for that product. We could define a function "PM" ("previous master"), but that would violate the rule that a function can only "see" an IP optionally followed by its successors. Instead, let us broaden the concept of first and rest to work with lists of lists (just as LISP and the other functional languages do).
This concept of lists of lists allows us in turn to take advantage of the fact that a stream can automatically be structured into substreams by a Collate type of component. Thus, suppose the output of a Collate run is as follows:
((m1,d11,d12),(m2,d21),(m3,d31,d32,d33))
where brackets represent bracket IPs. Then to convert this logically into a structure of lists and sublists, merely replace the open and close bracket IPs with curly brackets, i.e.
S:= {{m1,d11,d12},{m2,d21},{m3,d31,d32,d33}}
Now first and rest at the list level will return sublists, i.e.
first(S):= {m1,d11,d12}
and
rest(S):= {{m2,d21},{m3,d31,d32,d33}}
Note that, analogously to what we saw above with simple lists, first reduces the "nesting level" of a list by one level, while rest leaves it unchanged. The processing for extended details can now be shown succinctly as follows:
F(S) = if S is null,
then null
else cons(G(first(S)), F(rest(S)))
G(x) = G'(first(x),rest(x))
G'(x,y) = if y is null,
then null
else cons (E(x,first(y)), G'(x,rest(y)))
E(x,y) = µ(p,p:y; s,s:y; d,d:y; e,q:y*u:x)
Here the oneargument function G is defined in terms of a twoargument function G', whose first argument is always a master IP, and E now takes two arguments instead of only one.
In the above it can be seen that we will generate one output IP for each incoming detail (all IPs within a lowest level substream are details except the first one, which is the master). We can also see that x in G' acts as a placeholder for the master IP  it remains unchanged throughout the whole evaluation of function G.
Now let's repeat the three tables from Chapter 18 which describe the same calculation, and you will see that essentially the same information is captured, without the use of recursion, but also without using any variables. It seems therefore that these tables must have an underlying relationship with the functional expressions shown above.
The three tables from Chapter 18 respectively describe

the relationships between input and output streams:
INPUT STREAM
1. Merged Input
2. Product Substream: REPEATING
3. Product Master
3. Sales Detail: REPEATING
OUTPUT STREAMS
1. New Masters
2. Product Master: ONE PER Product Substream
1. Sort Input
2. Extended Detail: [ONE PER Sales Detail]
1. Report1
2. Product Summary: ONE PER Product Substream [,NEW] 
the layout of the individual IPs, e.g.:
Extended Detail:
{
Product Number: Ident,
Salesman Number: Ident,
District Number: Ident,
Quantity: Quantity,
Unit Price: MPrice,
Extended Price: Monetary;
} 
and the derivation rules for computed values:
Although we have been able to show some of these calculations in recursive form above, it appears that, in this particular case, the same information content can be expressed almost as succinctly using a quite simple notation which does not require any mathematical expertise at all. We have alluded above to the need to minimize the gap between the business requirement and the means of expressing it. What is the absolutely most concise way of expressing the requirement "build a stream of IPs containing extended quantities calculated as follows: ..." to a machine? In a 1990 article, K. Kahn and V. Saraswat suggest that it may not even have to be text as we know it  they propose that it may be possible to express both definitions and execution using a visual notation with almost no text at all, except for strings and comments. They point out that a major use of names is to make connections, something they are not particularly well suited for! Now look at the above figures and try to imagine how you could replace all those names by arrows connecting various types of icons (as few as possible of course).
Part of my motivation in this chapter (and in this book as a whole) is to try to shake up some preconceptions about how programming has to be done! We cannot predict where the next breakthrough will be made, so it is important that we remain as open as possible to new ideas and new ways of doing things. The above concepts also suggest some desirable characteristics that these new ways should have, so we can measure whether we are moving in the right direction or away from it. As I shall suggest in the last chapter of this book, everyone should have their minds stretched regularly, and this kind of exercise is nowhere more important than in the computer business!