In most software topics, the boundary between theory and practice in software is clearly demarcated: theory is for academics who seldom descend from the ivory tower, practice is for industry professionals who have long forgotten the concepts and application of theory. In concurrency, for example, most developers either know or have programmed semaphores, but few remember the conceptual underpinnings devised by Dijkstra. But Business Process Management (BPM) - a key Web services technology with close ties to Web services choreography - belongs to a rarer category, in which theory informs practical design and theoretical jargon is part of the hype with customers.

Somehow the abstruse terms "Pi Calculus" and "Petri net" - as impressive to the ear as database management's "relational calculus" or capacity planning's "Erlang formulae" - have permeated the consciousness of the BPM community. Many BPM onlookers are familiar with, and are interested in, Pi and Petri, but have at best a vague understanding of them.

This article is a guided tour of the scenic route of BPM theory. The Pi Calculus and the Petri net are studied by example, with just enough detail to give a sense of the BPM connection without overdosing on algebra.

Why Theory Matters
Process theory is practically important for several reasons:

• Theory is mentioned frequently in connection with BPM, even in nonacademic material. Countless presentations, for example, state without explanation that the leading BPM language, Business Process Execution Language (BPEL), is influenced by the Pi Calculus and Petri nets. For the many practitioners who are intimidated by the pedantic name-dropping but are curious to uncover its meaning (e.g., What is the Pi Calculus? What are Petri nets? Which parts of which theory are used in BPEL? Why is BPEL based on two theories rather than one?), this article helps elucidate the nature of the connection.
• BPM is relatively immature and benefits from the ideas and rigor of theory. Control flow, for example, is often treated too casually by vendors, who are more likely to emphasize ease of programming than semantic precision. Regrettably, as several papers on process design patterns have demonstrated, most vendors, and even most standards, struggle to support certain common control flow scenarios (e.g., the "multiple instances without runtime knowledge" and "interleaved parallel routing" patterns described in www.workflowpatterns.com). To build successful solutions, practitioners should insist on knowing exactly how a given process will run. To accomplish this, they should choose a good language and understand how the language works. Judged on the basis of control flow, the strongest languages are those based on the Petri net, notably BPEL and Business Process Modeling Notation (BPMN). And, arguably, to understand the nuances of these languages (e.g., dead-path elimination in BPEL) requires an appreciation for the Petri net.
• Contemporary BPM and its cousin Web Services Choreography are obsessed with the construction of complex participant conversations. Choreography especially, because it is mandated to build global collaborative contracts, requires a conceptual framework that can express dynamic communicating processes precisely and concisely. The leading choreography language, Web Services Choreography Description Language (WS-CDL), bases its constructions (e.g., channel passing) on the Pi Calculus. BPEL is also alleged to have Pi underpinnings. By learning the basics of Pi, the practitioner gains insight into the use of these languages.

Family Tree
As Figure 1 shows, four major contemporary standards - WS-CDL, Web Services Choreography Interface (WSCI), Business Process Modeling Language (BPML), and XLANG - betray Pi Calculus influence, three - BPMN, UML activity diagrams, and Web Services Flow Language (WSFL) - derive from the Petri net, and one - BPEL - is a blend, inheriting traits of Pi and Petri from its parent languages XLANG and WSFL, respectively.

The Pi Calculus
Developed by the Scottish mathematician Robin Milner in the 1990s, the Pi Calculus is a formal language for defining concurrent, communicating processes, including, but not restricted to, business processes. In its detail the Pi Calculus is a rather advanced algebraic system requiring a senior level of mathematical training. Milner's presentation of the subject in his landmark paper "The Polyadic Pi-Calculus: A Tutorial" is written in the mathematical idiom of definitions, theorems, and lemmas, inaccessible to most BPM onlookers. And few business analysts or software developers could survive if required to compose their business processes as lines Pi Calculus code.

However somehow, despite its academic roots and its inherent complexity, the Pi Calculus has become one of BPM's most attention-getting cocktail party terms. Popular BPM literature states boldly that major languages such as BPEL and WS-CDL are based on the Pi Calculus. This stunning level of influence, charges a leading BPM commentator (Wil van der Aalst, in his paper "Pi Calculus versus Petri Nets: Let Us Eat Humble Pie Rather Than Further Inflate the Pi Hype"), is dubious, and surely nothing but hype.

Let the people who advocate BPEL4WS, BPMN, ... and WSCI show the precise relation between the language and some formal foundation. People who cannot do this but still claim strong relationships between their language and e.g., Pi-calculus only cause confusion
("Pi Calculus versus Petri Nets," page 2).

Whether or not it is hype, the Pi Calculus-BPM connection merits a serious look. What, in a nutshell, is the Pi Calculus, how does it apply to BPM, and what is the extent and nature of its influence on contemporary popular languages like BPEL and WS-CDL?

The Pi Calculus in a Nutshell
The Pi Calculus is a language used to define concurrent processes that interact with one another dynamically. Each process consists of one or more actions, which can be arranged sequentially, in parallel or conditional paths, or recursively. An action is either the sending or receiving of information on a channel. According to the Pi-Calculus convention, when one process sends to another, it includes the name of the channel to be used for the other process to respond. This name is variable and, as we will see, can change in response to changing conditions.

One of the most distinctive features of the Pi Calculus is mobility, in which the topology of communicating processes changes dynamically in response to changing conditions. An example of mobility is the enrollment of customers with retailers in a deregulated energy market. In part (a) of Figure 2, customer C initially buys energy directly from the supplier (a "standard supply" arrangement), but in part (b) enrolls with retailer A. In part (c), the customer switches to competing retailer B, but then drops the retailer in part (d), thus returning to standard supply.

The source code to model this scenario is remarkably terse:

1	CustomerSS(enroll,switch,drop,rets)=
2	   ?(r:rets). (enroll r,"mike".CustomerR
(r,enroll,switch,drop,rets))
3	CustomerR(r,enroll,switch,drop,rets)=
4	   ?(r2:rets).(switch r,r2,"mike".CustomerR(r2,enroll,switch,drop,rets)) +
5	   drop r,"mike".CustomerSS(enroll,
switch,drop,rets)
6	Supplier(enroll,switch,drop)=
7	   (enroll(r1,c).r1 "addcust",c +
8	   switch (r1,r2,c).r1 "dropcust",c.r2 "addcust",c +
9	   drop(r1,c).r1 "dropcust",c.Supplier(enroll,switch,drop)
10	Retailer(r)=
11	   r(action,c).Retailer(r)
12	Market=
13	   (new chEnroll,chSwitch,chDrop,retSet={retA,retB})
14	   CustomerSS(chEnroll,chSwitch,chDrop,retSet)|
15	   Supplier(chEnroll,chSwitch,chDrop) |
16	   Retailer(retA)|Retailer(retB)

The code defines five processes: CustomerSS (lines 1-2), CustomerR (lines 3-5), Supplier (lines 6-9), Retailer (lines 10-11), and Market (lines 12-16). Each process is written as a mathematical equation, with the left side and right side separated by an equals sign ("="). The left side gives the name of the process and the channels it uses to communicate with the other processes; the right side (indented in the code sample above) is the definition.