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%T On Congruence Property of Scope Equivalence for Concurrent Programs with Higher\-Order Communication
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%A  Masaki Murakami
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%E Peter H. Welch,  Herman Roebbers,  Jan F. Broenink,  Frederick R. M. Barnes,  Carl G. Ritson,  Adam T. Sampson,  G. S. Stiles,  Brian Vinter
%B Communicating Process Architectures 2009
%X Representation of scopes of names is important for analysis
   and
verification of concurrent systems.  However, it is
   difficult to
represent the scopes of channel names precisely
   with models based on
process algebra.  We introduced a model
   of concurrent systems with
higher\-order communication based
   on graph rewriting in our previous
work.  A bipartite
   directed acyclic graph represents a concurrent
system that
   consists of a number of processes and messages in
   that
model. The model can represent the scopes of local
   names precisely.
We defined an equivalence relation such
   that two systems are
equivalent not only in their behavior
   but in extrusion of scopes of
names. This paper shows that
   the equivalence relation is a
congruence relation wrt
   tau\-prefix, new\-name, replication and
composition even if
   higher\-order communication is allowed.  And we
also show
   the equivalence relation is not congruent wrt
   input\-prefix
though it is congruent wrt input prefix in
   first\-order case.