On Confluence in the pi-Calculus

An account of the basic theory of confluence in the 1r-calculus is presented, techniques for showing confluence of mobile systems are given, and the utility of some of the theory presented is illustrated via an analysis of a distributed algorithm. 1 I n t r o d u c t i o n Confluence arises in a variety of forms in computation theory. It was first studied in the context of concurrent systems by Milner in [6]. Its essence, to quote [7], is that "of any two possible actions, the occurrence of one will never preclude the other". As shown in the works cited, for pure CCS agents confluence implies detern'linacy and semantie-invariance under silent actions, and is preserved by several important system-building operations. These facts make it possible to guarantee by construction that certain systems are confluent and to exploit this fact fruitfully when analysing their behaviours. A more general study was made in [1] which in particular clarified the relationships among various notions of confluence and semantic-invariance under silent actions, and illustrated the utility of the ideas for state-space reduction and protocol analysis; see also [1] for further references. Confluence of value-passing CCS agents was studied first in [18] and later in [22] where consideration was given to conditions under which confluent systems result from combinations of 'semi-confluent' agents and the ideas were utilized to show determinacy of programs in a fragment of a concurrent imperative programming language. The elaboration of techniques for reasoning about mobile systems expressed in the zr-calculus [9] and variants of it has involved extension of established methods and development of new concepts specific to the richer setting. Stemming from [8] there have been several works on disciplines of name-use respected by agents, sometimes expressed via type systems; see for instance [2, 15, 23, 25, 20, 16]. Such disciplines contribute much to the effectiveness of 7r-calculi as descriptive formalisms and analytical tools. One promising strand of development concerns varieties of confluence. These have been used in showing determinacy of systems prescribed by concurrent object-oriented programs [13], in justifying optimizations in the Pict compiler [3, 17], and in proving the solmdness of transformation rules for concurrent object-oriented programs [4, 14]. The aims of this paper are to give an account of the basic theory of confluence in the 7r-calculus, to develop techniques for showing that mobile systems are confluent, and to illustrate the utility of some of the theory presented via an analysis of a