Symbolic Bisimulations and Proof Systems for The
$\pi$-Calculus
$*$
(Abstract)
Huimin
$\mathrm{L}\mathrm{i}\mathrm{n}^{\uparrow}$Laboratory for Computer Science
Institute ofSoftware, Chinese Academy of Sciences
Axiomatisationsfor basic process algebras such as CCS, CSP, ACP, LOTOS..., are
based on equational logic. But purely equational reasoning is difficultfor the $\pi$-calculus
because here input prefixes introduce bound names, i.e. nameswhich are local within the
scope of the input prefixes and can be instantiated with arbitrary free names. Reflectedin
the axiomatisations this amounts to require when an input prefix is introduced the input
name must not be used before, a concept difficult to be formulated in the framework
of equational logic. The “symbolic” style proof system generalise equational logic by
using conditional equations of the form $\phi\triangleright T=U$ (where $\phi$ is a boolean condition on
names and $T,$ $U$ are process terms) as judgements. For each construct in the calculus
there corresponds an introduction rule in the core proof system. For instance the rule
for input prefix is
INPUT $\frac{\emptyset\triangleright\tau=U}{\phi\triangleright a(X).T=b(x).U}\phi\Rightarrow a=b$, $x\not\in n(\phi)$
Here $\phi$ represents the context condition for deriving $T=U$, and the side condition
ensures the input name $x$ is fresh.
In [Lin94] it is shown that, together with the standard CCS axioms for strong
bisim-ulation, this core proof system is sound and complete for late strong bisimulation in
recursion-free $\pi$-calculus. By adding either an inference rule dealing with early input
$([\mathrm{H}\mathrm{L}96])$ or an axiom schema $([\mathrm{P}\mathrm{S}93])$ proof systems for early bisimulation can be
ob-tained. In $[\mathrm{L}\mathrm{i}\mathrm{n}95\mathrm{a}]$ it is further shown that to extend these results to weak bisimulation
equivalences all what needed is the three $\tau$-laws $([\mathrm{M}\mathrm{i}189])$.
For the $\pi$-calculus with recursion the key inference rule is unique fixpoint induction.
It is syntactically identical to its pure-CCS counterpart $([\mathrm{M}\mathrm{i}184])$, but work at the level
of abstractions:
UFI $\frac{F=G[F/X]}{F=\mu XG}$ $X$ guarded in $G$
*Supportedbyresearch grants from the President Fund ofChineseAcademyofSciencesand National Natural Science Foundation ofChina.
$\uparrow \mathrm{T}\mathrm{h}\mathrm{e}$ author’s address: Institute
of Software, Chinese Academy of Sciences, $\mathrm{P}.\mathrm{O}$.Box 8718, Beijing
100080. Email: lhm@ox.ios.ac.cn.
数理解析研究所講究録
where both $F$ and $G$ ar$e$
name-closed
abstractions.Extended
with this rule, and thestandard
rules for$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}/\mathrm{u}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ recursions, the core proofsystem is sound and com-plete for strongbisimulation
infinite-control
$\pi$-calculus whererecursions
$\mathrm{a}r\mathrm{e}$guarded
$([\mathrm{L}\mathrm{i}\mathrm{n}95\mathrm{b}])$. To achieve complet$e$ proof system for arbitrary recursions, the following
axiom can be adopted to reduce
unguarded
recursions to guarded ones:UNG
fixX$( \vec{x})(\sum i\in I\emptyset i\nu\vec{X}_{i}’X(\vec{X}_{i})+T)=\mathrm{f}\mathrm{i}\mathrm{X}X(\vec{X})(\Sigma_{i\in I}\phi_{i}\nu\vec{x}_{i}’\tau[\vec{X}_{i}/\vec{x}]+^{\tau})$where the
summands
in the recursion body satisfy certain saturationcondition.Theproofs of the above
mentioned
completenessresults rely on thenotion of symbolicbisimulation, first proposed in [HL95] for general
message-passing
process algebras. Asalternative characterisations ofthe
standard
definitionsofbisimulations inthe$\pi$-calculus,symbolic bisimulations overcome the infinity inherited in the standard definitions by
employing finite partitions over the name space, instead of instantiating input names
and name parametersof recursivelydefined processes. Another powerful technical device
upon which the completeness proofs heavily rely is the notion of maximally consistent
name conditions.
Currently work is going on to extend the above results to weak bisimulation
equiva-lences in finite-cont$r\mathrm{o}1\pi$-calculus.
References
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