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\begin{document}
\thispagestyle{plain}
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\title{Nested Semantics
over Finite Trees \\ are Equationally Hard
}
\author{{Luca Aceto}\inst{1} \and
{Wan Fokkink}\inst{2} \and
{Rob van
Glabbeek}\inst{4} \and {Anna
Ing\'olfsd\'ottir}\inst{1,3}}
\institute{{\bf BRICS} ({\bf B}asic {\bf R}esearch
in {\bf C}omputer {\bf S}cience), Centre of the Danish
National Research Foundation, Department of
Computer Science, Aalborg University, Fr.~Bajersvej 7E, 9220
Aalborg \O, Denmark, \email{luca@cs.auc.dk, annai@cs.auc.dk}
\and
CWI, Department of Software Engineering,
Kruislaan 413, 1098 SJ Amsterdam, The Netherlands, \email{wan@cwi.nl}
\and
deCODE Genetics, Sturlugata 8, 101
Reykjav\'{\i}k, Iceland, \email{annai@decode.is}
\and
INRIA, Sophia Antipolis, France, \email{ rvg@cs.stanford.edu}}
\maketitle
\begin{abstract}
This paper studies nested simulation and nested trace semantics over
the language BCCSP, a basic formalism to express finite process
behaviour. It is shown that none of these semantics affords finite
(in)equational axiomatizations over BCCSP. In particular, for each
of the nested semantics studied in this paper, the collection of
sound, closed (in)equations over a singleton action set is not
finitely based.
\medskip
\noindent
{\sc 2000 Mathematics Subject Classification:} 08A70, 03B45, 03C05, 68Q10,
68Q45, 68Q55, 68Q70.
\noindent
{\sc CR Subject Classification (1991):} D.3.1, F.1.1, F.1.2, F.3.2,
F.3.4, F.4.1.
\noindent
{\sc Keywords and Phrases:} Concurrency, process algebra, BCCSP,
nested simulation, possible futures, nested trace semantics,
equational logic, complete axiomatizations, non-finitely based
algebras, Hennessy-Milner logic.
\end{abstract}
\section{Introduction}\label{Sect:intro}
Labelled transition systems (LTSs) \cite{Ke76} are a fundamental model
of concurrent computation, which is widely used in light of its
flexibility and applicability. In particular, they are the prime model
underlying Plotkin's Structural Operational Semantics \cite{Pl81} and,
following Milner's pioneering work on CCS \cite{Mi89}, are by now the
standard semantic model for various process description languages.
LTSs model processes by explicitly describing their states and their
transitions from state to state, together with the actions that
produced them. Since this view of process behaviours is very detailed,
several notions of behavioural equivalence and preorder have been
proposed for LTSs. The aim of such behavioural semantics is to
identify those (states of) LTSs that afford the same ``observations'',
in some appropriate technical sense. The lack of consensus on what
constitutes an appropriate notion of observable behaviour for reactive
systems has led to a large number of proposals for behavioural
equivalences for concurrent processes. (See the study
\cite{vG2001}, where van Glabbeek presents the linear time-branching
time spectrum---a lattice of known behavioural
equivalences and preorders over LTSs, ordered by inclusion.)
One of the criteria that has been put forward for studying the
mathematical tractability of the behavioural equivalences in the
linear time-branching time spectrum is that they afford elegant,
finite equational axiomatizations over fragments of process algebraic
languages. Equationally based proof systems play an important role in
both the practice and the theory of process algebras. From the point
of view of practice, these proof systems can be used to perform system
verifications in a purely syntactic way, and form the basis of
axiomatic verification tools like, {e.g.}, PAM \cite{Lin92}. From the
theoretical point of view, complete axiomatizations of behavioural
equivalences capture the essence of different notions of semantics for
processes in terms of a basic collection of identities, and this often
allows one to compare semantics which may have been defined in very
different styles and frameworks. A review of existing complete
equational axiomatizations for many of the behavioural semantics in
van Glabbeek's spectrum is offered in \cite{vG2001}. The equational
axiomatizations offered {\em ibidem} are over the language BCCSP, a
common fragment of Milner's CCS~\cite{Mi89} and Hoare's
CSP~\cite{Ho85} suitable for describing finite synchronization trees,
and characterize the differences between behavioural semantics in
terms of a few revealing axioms.
The main omissions in this menagerie of equational axiomatizations for
the behavioural semantics in van Glabbeek's spectrum are
axiomatizations for 2-nested simulation semantics and possible futures
semantics. The relation of 2-nested simulation was introduced by
Groote and Vaandrager \cite{GrV92} as the coarsest equivalence
included in completed trace equivalence for which the tyft/tyxt format
is a congruence format. It thus characterizes the distinctions
amongst processes that can be made by observing their termination
behaviour in program contexts that can be built using a wide array of
operators. (The interested reader is referred to {\em op.~cit}.~for
motivation and the basic theory of 2-nested simulation.) 2-nested
simulation can be decided over finite LTSs in time that is quadratic
in their number of transitions \cite{ShuklaRHS1996}, and can be
characterized by a single parameterized modal logic formula
\cite{MitchellC1996}. However, no equational axiomatization for it has
ever been proposed, even for the language BCCSP. Possible futures
semantics, on the other hand, was proposed by Rounds and Brookes
in~\cite{RB81} as far back as 1981, and it affords an elegant modal
characterization in terms of a subset of Hennessy-Milner logic---in
fact, the modal characterization of possible futures equivalence is a
consequence of a more general, classic result due to Hennessy and
Milner (see~\cite[Theorem~2.2~and page~148]{HM85}) that will find
application in the technical developments of this paper. As shown by
Kannellakis and Smolka in~\cite{KannellakisS1990}, the problem of
deciding possible futures equivalence and all of the other $n$-nested
trace equivalences ($n\geq 1$) from~\cite{HM85} over finite state
processes is PSPACE-complete. However, possible futures equivalence
still lacks a purely equational axiomatization over BCCSP.
In this paper, we offer, amongst other results, a mathematical
justification for the lack of an equational axiomatization for the
2-nested simulation and possible futures equivalence and preorder even
for the language of finite synchronization trees. More precisely, we
show that none of these behavioural relations admits a finite
(in)equational axiomatization over the language BCCSP. These negative
results hold in a very strong form. Indeed, we prove that no finite
collection of inequations that are sound with respect to the 2-nested
simulation preorder can prove all of the inequalities of the form
\begin{eqnarray*} a^{2m} &
\preaxiom & a^{2m} + a^m \quad\quad (m\geq 0) \enspace ,
\end{eqnarray*}
which are sound with respect to the 2-nested simulation preorder.
Similarly, we establish a result to the effect that no finite
collection of (in)equations that are sound with respect to the possible futures preorder or equivalence can be used to derive all of the sound
inequalities of the form
\begin{eqnarray*}
a(a^{m} + a^{2m}) + a a^{3m} & \preaxiom & aa^{2m} + a( a^m + a^{3m})
\quad\quad (m\geq 0) \enspace .
\end{eqnarray*}
We then generalize these negative results to show that none of the
$n$-nested simulation or trace preorders and equivalences
from~\cite{GrV92,HM85} (for $n\geq 2$) afford finite equational
axiomatizations over the language BCCSP.
The import of these results is that not only the equational theory
of the $n$-nested simulation and trace semantics is not finitely
equationally axiomatizable, for $n\geq 2$, but neither is the
collection of (in)equivalences that hold between BCCSP terms over one
action and without occurrences of variables. This state of affairs
should be contrasted with the elegant equational axiomatizations over
BCCSP for most of the other behavioural equivalences in the linear
time-branching time spectrum that are reviewed by van Glabbeek in
\cite{vG2001}. Only in the case of additional, more complex operators,
such as iteration or parallel composition, or in the presence of
infinite sets of actions, are these equivalences known to lack a
finite equational axiomatization; see, {e.g.},
\cite{AcetoFI1998,BFN03,Co71,Gi88,Re64,Sewell1997}. Of special
relevance for concurrency theory are Moller's results to the effect
that the process algebras CCS and ACP without the auxiliary left
merge operator from \cite{BK82} do not have a finite equational
axiomatization modulo bisimulation equivalence \cite{Mo90,Mo90a}.
Fokkink and Luttik have shown in~\cite{FokkinkL2000} that the process
algebra PA \cite{BK84b}, which contains a parallel composition
operator based on pure interleaving without communication and the left
merge operator, affords an $\omega$-complete axiomatization that is
finite if so is the underlying set of actions. Aceto, \'Esik and
Ing\'olfsd\'ottir \cite{AEI03} proved that there is no finite
equational axiomatization that is $\omega$-complete for the max-plus
algebra of the natural numbers, a result whose process algebraic
implications are discussed in \cite{AEI00}.
As shown in~\cite{GrV92,HM85}, the intersection of all of the
$n$-nested simulation or trace equivalences or preorders over
image-finite labelled transition systems, and therefore over the
language BCCSP, is bisimulation equivalence. Hennessy and Milner
proved in~\cite{HM85} that bisimulation equivalence is axiomatized
over the language BCCSP by the four equations in Table~\ref{tab:bccs}.
Thus, in light of the aforementioned negative results, this
fundamental behavioural equivalence, albeit finitely based over BCCSP,
is the intersection of sequences of relations that do not afford
finite equational axiomatizations themselves. This observation begs
the question of whether bisimulation equivalence over BCCSP is the
limit of some sequence of finitely based behavioural equivalences that
have been presented in the literature. In {\em op.~cit}.~Hennessy and
Milner introduced an alternative sequence of relations that
approximate bisimulation equivalence. These relations are based on a
``bisimulation-like'' matching of the {\em single steps} that
processes may perform, whereas the $n$-nested trace equivalences
require matchings of arbitrarily long {\em sequences of steps}. We
prove in this study that, unlike the $n$-nested trace equivalences,
these single-step based approximations of bisimulation equivalence are
all finitely axiomatizable over the language BCCSP, provided that the
set of actions is finite.
The paper is organized as follows. We begin by presenting
preliminaries on the language BCCSP, (in)equational logic, and the
notions of behavioural equivalence and preorder studied in this paper
(Sect.~\ref{Sect:preliminaries}). Our main results on the
non-existence of finite (in)equational axiomatizations for the
$n$-nested simulation and trace equivalence and preorder (for $n\geq
2$) are the topic of Sects.~\ref{Sect:ineq}--\ref{Sect:therest}. In
Sect.~\ref{Sect:ineq} we prove that the 2-nested simulation preorder
has no finite inequational axiomatization over the language BCCSP.
Sect.~\ref{Sect:possible-futures} presents a non-finite
axiomatizability result for the possible futures preorder and
equivalence. We then offer a general result to the effect that all
of the other $n$-nested semantics considered in this study have no
finite inequational axiomatization (Sect.~\ref{Sect:therest}). The
paper concludes with our proof of finite axiomatizability for the
alternative approximations of bisimulation equivalence introduced by
Hennessy and Milner in~\cite{HM85} (Sect.~\ref{Sect:fb}).
The work reported in this paper extends and improves upon the results
presented in~\cite{AFI2001}, where it was shown that 2-nested
simulation semantics and the 3-nested simulation preorder are not
finitely based over the language BCCSP. The aforementioned paper also
offered conditional axiomatizations for the nested simulation
semantics. Since we have been unable to obtain similar results for the
nested trace semantics, we have decided to omit those conditional
axiomatizations from this presentation.
\section{Preliminaries}\label{Sect:preliminaries}
We begin by introducing the basic definitions and results on which the
technical developments to follow are based.
\subsection{The language BCCSP}\label{SEct:BCCSP}
The process algebra BCCSP is a basic formalism to express
finite process behaviour. Its syntax consists of (process) terms that
are constructed from a countably infinite set of (process) variables (with
typical elements $x,y,z$), a constant {\bf 0}, a binary operator $+$
called {\em alternative composition}, and unary {\em prefixing}
operators $a$, where $a$ ranges over some non-empty set $\Act$ of {\em
atomic actions}. We shall use the meta-variables $t,u,v$ to range over
process terms, and write ${\it var}(t)$ for the collection of
variables occurring in the term $t$.
A process term is {\em closed} if it does not contain any variables.
Closed terms will be typically denoted by $p,q,r$. Intuitively,
closed terms represent completely specified finite process behaviours,
where {\bf 0} does not exhibit any behaviour, $p+q$ combines the
behaviours of $p$ and $q$ by offering an initial choice as to whether
to behave like either of these two terms, and $ap$ can execute action
$a$ to transform into $p$. This intuition for the operators of BCCSP
is captured, in the style of Plotkin \cite{Pl81}, by the transition
rules in Table \ref{tab:sos}. These transition rules give rise to
transitions between process terms. The operational semantics for BCCSP
is thus given by the labelled transition system \cite{Ke76} whose
states are terms, and whose $\Act$-labelled transitions are those that
are provable using the rules in Table~\ref{tab:sos}. Based on this
labelled transition system, we shall consider BCCSP terms modulo a
range of behavioural equivalences that will be introduced in
Sect.~\ref{Sect:beh}.
\begin{table}
\caption{Transition rules for BCCSP}
\centering
$
\begin{array} {|ccc|}
\hline
&&\\
~~~~~\frac{\raisebox{.7ex}{\normalsize{$x\mv{a}x'$}}}
{\raisebox{-1.0ex}{\normalsize{$x+y\mv a x'$}}}~~~~~ &
\frac{\raisebox{.7ex}{\normalsize{$y\mv{a}y'$}}}
{\raisebox{-1.0ex}{\normalsize{$x+y\mv a y'$}}}~~~~~ &
ax\mv{a}x~~~~~ \\
&&\\
\hline
\end{array}
$
\label{tab:sos}
\end{table}
A (closed) substitution is a mapping from process variables to
(closed) BCCSP terms. For every term $t$ and
(closed) substitution $\sigma$, the (closed) term obtained by
replacing every occurrence of a variable $x$ in $t$ with the (closed)
term $\sigma(x)$ will be written $\sigma(t)$.
In the remainder of this paper, we let $a^0$ denote {\bf 0}, and
$a^{m+1}$ denote $a(a^m)$. Following standard practice in the
literature on CCS and related languages, trailing {\bf 0}'s will often
be omitted from terms. A {\em term over action $a$} is a BCCSP term
that may only contain occurrences of the prefixing operator $a$. (We
shall restrict our attention to these terms in the technical
developments presented in Section~\ref{Sect:therest}.) For example,
the term $a^{m}$ is over action $a$, for each $m\geq 0$.
\subsection{Inequational Logic}\label{Sect:logic}
An {\em axiom system} is a collection of inequations $t \preox u$ over the
language BCCSP\@. An inequation $p \preox q$ is derivable from $E$,
notation $E \vdash p \preox q$, if it can be proven from the axioms in
$E$ using the rules of inequational logic (viz.~reflexivity, transitivity,
substitution and closure under BCCSP contexts):
$$
t \preox t ~~~ \frac{{t \preox u ~~ u \preox v}}{{t \preox v}} ~~~
\frac{{t \preox u}}{{\sigma(t) \preox \sigma(u)}} ~~~ \frac{t \preox u}{at \preox au}(a \in A) $$
$$~~~
\frac{t \preox
u}{t+r \preox u+r} ~~~ \frac{t \preox
u}{r+t \preox r+u}
\enspace .
$$
Without loss of generality one may assume that
substitutions happen first in inequational proofs, i.e., that the
third rule may only be used when $(t \preox u) \in E$. In this case
$\sigma(t) \preox \sigma(u)$ is called a {\em substitution instance}
of an axiom in $E$.
{\em Equational logic} is like inequational logic, but with the extra
rule of symmetry:
$$\frac{t \preox u}{u \preox t} \enspace .$$
In equational logic, the
formula $t \preox u$ is normally written $t \approx u$. Without loss
of generality, one may assume that applications of symmetry happen
first in equational proofs. Therefore we can see equational logic as a
special case of inequational logic, namely by postulating that for
each axiom in $E$ also its symmetric counterpart is present in $E$. In
the remainder of this paper, we shall always tacitly assume this
property of equational axiom systems.
An example of an (equational) axiom system over the language BCCSP is
given in Table~\ref{tab:bccs}. As shown by Hennessy and Milner
in~\cite{HM85}, that axiom system is sound and complete for
bisimulation equivalence over the language BCCSP.
\begin{table}
\centering
\caption{Axioms for BCCSP}
$
\begin{array}{|rrcl|}
\hline
&&&\\
~~~~~ {\rm A}1~~&x+y &\approx& y+x\\
{\rm A}2~~&(x+y)+z &\approx& x+(y+z)~~~~~\\
{\rm A}3~~&x+x &\approx& x\\
{\rm A}4~~&x+{\mathbf 0} &\approx& x\\
&&&\\
\hline
\end{array}
$
\label{tab:bccs}
\end{table}
In the remainder of this paper, process terms are considered modulo
associativity and commutativity of +, and modulo absorption of {\bf 0}
summands. In other words, we do not distinguish $t+u$ and $u+t$, nor
$(t+u)+v$ and $t+(u+v)$, nor $t+{\mathbf 0}$ and $t$. This is justified
because all of the behavioural equivalences we consider satisfy axioms
A1, A2 and A4 in Table~\ref{tab:bccs}. In what follows, the symbol $=$
will denote syntactic equality modulo axioms A1, A2 and A4. We use a
{\em summation} $\sum_{i\in\{1,\ldots,k\}}t_i$ to denote
$t_1+\cdots+t_k$, where the empty sum represents {\bf 0}. It is easy
to see that, modulo the equations A1, A2 and A4, every BCCSP term $t$
has the form $\sum_{i\in I}x_i+\sum_{j\in J}a_jt_j$, for some finite
index sets $I,J$, terms $a_j t_j$ ($j\in J$) and variables $x_i$ ($i\in
I$). The terms $a_j t_j$ ($j\in J$) and variables $x_i$ ($i\in
I$) will be referred to as the {\em summands} of $t$.
It is well-known (cf., e.g., Sect.~2 in \cite{Gr90}) that if an
(in)equation relating two closed terms can be proven from an axiom
system $E$, then there is a closed proof for it.
In the proofs of some of our main results, it will be convenient to
use a different formulation of the notion of provability of an
(in)equation from a set of axioms. This we now proceed to define for
the sake of clarity.
A {\em context} $C[~]$ is a closed BCCSP term with exactly one
occurrence of a hole $[~]$ in it. For every context $C[~]$ and closed
term $p$, we write $C[p]$ for the closed term that results by placing
$p$ in the hole in $C[~]$. It is not hard to see that
an inequation $p \preox q$ is provable from an inequational axiom system
$E$ iff there is a sequence $p_1 \preox \cdots \preox p_k$ ($k\geq
1$) such that
\begin{itemize}
\item $p=p_1$,
\item $q=p_k$ and
\item $p_i = C[\sigma(t)] \preox C[\sigma(u)] = p_{i+1}$ for some
closed substitution $\sigma$, context $C[~]$ and pair of terms $t,u$
with $t\preox u$ an axiom in $E$ ($1\leq i < k$).
\end{itemize}
In what follows, we shall refer to sequences of the form $p_1
\preaxiom \cdots \preaxiom p_k$ as {\em inequational} {\em
derivations}.
For later use, note that, using axioms A1, A2 and A4 in
Table~\ref{tab:bccs}, every context can be proven equal either to one
of the form $C[b([~]+p)]$ or to one of the form $[~]+p$, for some action
$b$ and closed BCCSP term $p$.
\subsection{Traces of BCCSP Terms}\label{Sect:traces}
The transition relations $\mv{a}$ ($a\in A$) naturally compose to
determine the possible effects that performing a sequence of actions
may have on a BCCSP term.
\begin{definition}\label{Def:traces}
For a sequence $s=a_1\cdots a_k\in\Act^*$ ($k\geq 0$), and BCCSP
terms $t,t'$, we write $t \mv{s} t'$ iff there exists a sequence of
transitions
\[
t=t_0 \mv{a_1} t_1 \mv{a_2} \cdots \mv{a_k} t_k=t' \enspace .
\]
If $t \mv{s} t'$ holds for some BCCSP term $t'$, then $s$ is a {\em
trace} of $t$. We write $\mathit{traces}(t)$ for the set of traces
of a term $t$.
\end{definition}
The following lemma, whose proof is standard, relates the transitions
of a term of the form $\sigma(t)$ to those of $t$ and those of the
terms $\sigma(x)$, with $x$ a variable occurring in $t$.
\begin{lemma}\label{Lem:trans-sub}
For every BCCSP term $t$, substitution $\sigma$, and sequence of
actions $s$, the following statements hold:
\begin{enumerate}
\item if $t \mv{s} u$ for some term $u$, then $\sigma(t) \mv{s} \sigma(u)$;
\item if $\sigma(t) \mv{s} u$ for some term $u$, then
\begin{enumerate}
\item either $t \mv{s} t'$ for some $t'$ with $u = \sigma(t')$,
\item or there are sequences of actions $s_1,s_2$ with $s_2$
non-empty and $s = s_1s_2$, a term $t'$ and a variable $x$ such
that $t \mv{s_1} x+t'$ and $\sigma(x)\mv{s_2} u$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\subsection{Behavioural Semantics}\label{Sect:beh}
Labelled transition systems describe the operational behaviour of
processes in great detail. In order to abstract from irrelevant
information on the way processes compute, a wealth of notions of
behavioural equivalence or approximation have been studied in the
literature on process theory. A systematic investigation of these
notions is presented in \cite{vG2001}, where van Glabbeek presents the
so-called linear time-branching time spectrum, a lattice of known
behavioural equivalences over labelled transition systems ordered by
inclusion. In this study, we shall investigate a fragment of the
notions of equivalence and preorder from {\em op.~cit}., together with
the family of the nested trace equivalences and preorders (see
Definition~\ref{Def:ntrace}). These we now proceed to present.
\begin{definition}
\label{def:simulation}
A binary relation $R$ between closed terms is a {\em simulation}
iff $p \mathrel{R} q$ together with $p \mv{a} p'$ imply that there
is a transition $q \mv{a} q'$ with $p' \mathrel{R} q'$.
\end{definition}
Groote and Vaandrager introduced in~\cite{GrV92} a hierarchy of
$n$-nested simulation preorders and equivalences for $n\geq 2$. These
are defined thus:
\begin{definition}
\label{def:n-nested}
For $n\geq 0$, we define the relation $\presim_{n}$ inductively over
closed BCCSP terms thus:
\begin{itemize}
\item $p \presim_{0} q$ for all $p,q$,
\item $p \presim_{n+1} q$ iff $p \mathrel{R} q$ for some simulation
$R$ with $R^{-1}$ included in $\presim_n$.
\end{itemize}
The kernel of $\presim_{n}$ (i.e., the equivalence $\presim_n \cap~
(\presim_n)^{-1}$) is denoted by $\leftrightarrows_{n}$.
\end{definition}
The relation $\presim_1$ is the well-known {\em simulation preorder}
\cite{Pa81}. The relations $\presim_2$ and $\leftrightarrows_2$ are
the {\em 2-nested simulation preorder} and the {\em 2-nested
simulation equivalence}, respectively. Groote and Vaandrager have
characterized 2-nested semantics as the largest congruence with
respect to the tyft/tyxt format of transition rules which is included
in completed trace semantics---see~\cite{GrV92} for details.
In the remainder of this paper we shall sometimes use, instead of
Definition~\ref{def:n-nested}, the following more descriptive,
fixed-point characterization of the $n$-nested simulation preorder
($n\geq 1$).
\begin{proposition}
\label{theo:nested}
Let $p,q$ be closed BCCSP terms, and $n\geq 0$. Then $p \presim_{n+1} q$ iff
\begin{itemize}
\item[(1)]
for all $p \mv{a} p'$ there is a $q \mv{a} q'$ with $p' \presim_{n+1} q'$, and
\item[(2)]
$q \presim_n p$.
\end{itemize}
\end{proposition}
\begin{proof}
We prove the two implications separately.
\begin{itemize}
\item ($\Rightarrow$) Assume that $p \presim_{n+1} q$. By definition, $p
\mathrel{R} q$ with $R$ a simulation and $R^{-1}$ included in
$\presim_n$. So if $p \mv{a} p'$, then $q \mv{a} q'$ with $p'
\mathrel{R} q'$, which implies
\[
p' \presim_{n+1} q' \enspace .
\]
Moreover,
since $R^{-1}$ is included in $\presim_n$, it follows that $q
\presim_n p$.
\item ($\Leftarrow$) We define $p \mathrel{R}
q$ iff
\begin{itemize}
\item[(1)]
for all $p \mv{a} p'$ there is a $q \mv{a} q'$ with $p' \presim_{n+1} q'$, and
\item[(2)]
$q \presim_n p$.
\end{itemize}
Suppose now that $p \mathrel{R} q$. If $p \mv{a} p'$, then by the
definition of $R$ we have $q \mv{a} q'$ with $p' \presim_{n+1} q'$.
Since we have already proven the `only if' implication, we may
conclude that $p' \mathrel{R} q'$. So $R$ is a simulation.
Furthermore, by (2) above $R^{-1}$ is included in $\presim_n$. Hence,
we have that $p \presim_{n+1} q$, which was to be shown. \hfill
$\square$
\end{itemize}
\end{proof}
\begin{example}\label{Ex:pn-qn}
Let $m\geq 1$. Define, for each $n\in \IN$, the closed BCCSP terms
$p_n$ and $q_n$ thus:
$$\begin{array}{lll@{~~~~~~~~~~~}lll}
p_0 &=& a^{2m-1}{\mathbf 0} & q_0 &=& a^{m-1}{\mathbf 0} \\
p_{n+1} &=& ap_n+ aq_n & q_{n+1} &=& ap_n \enspace .
\end{array}$$
By induction on $n \in \IN$ and using Proposition~\ref{theo:nested},
it is not hard to check that $p_n \presim_{n} q_n$, and thus that $q_n
\presim_{n+1} p_n$.
The terms $p_n$ and $q_n$ ($n\in \IN$) defined above will play a
crucial role in the proof of Theorem~\ref{Thm:incompleteness} to
follow.
\end{example}
Possible futures semantics was introduced by Rounds and Brookes
in~\cite{RB81}, and is defined thus:
\begin{definition}\label{Def:futures}
Let $p$ be a closed BCCSP term. A {\em possible future} of $p$ is a
pair $(s,X)$, where $s$ is a sequence of actions and $X\subseteq
A^*$, such that $p \mv{s} p'$ and $X = \mathit{traces}(p')$, for
some $p'$.
Two closed terms $p$ and $q$ are related by the {\em possible
futures preorder} (respectively, {\em possible futures
equivalence}), written $p \preo_{\it PF} q$ (resp., $p
=_{\it PF} q$), if each possible future of $p$ is also
a possible future of $q$ (resp., if $p$ and $q$ have the same
possible futures).
\end{definition}
The last notions of semantics we shall consider in this paper are the
families of the $n$-nested trace equivalences and preorders. The
$n$-nested trace equivalences were introduced by Hennessy and Milner
in \cite[p.~147]{HM85} as a a tool to define bisimulation equivalence
\cite{Mi89,Pa81}.
\begin{definition}\label{Def:ntrace}
For every $n\geq 0$, the relations of {\em $n$-nested trace
equivalence}, denoted by $=_{n}^T$, and {\em $n$-nested trace
preorder}, denoted by $\preo_{n}^T$, are defined inductively over
closed BCCSP terms thus:
\begin{itemize}
\item $p =_{0}^{T} q$ and $p \preo_{0}^{T} q$ for every $p,q$;
\item $p =_{n+1}^{T} q$ iff for every sequence of actions
$s\in\Act^*$:
\begin{itemize}
\item if $p \mv{s} p'$ then there is a $q'$ such that $q \mv{s} q'$
and $p' =_{n}^{T} q'$, and
\item if $q \mv{s} q'$ then there is a $p'$ such that $p \mv{s} p'$
and $p' =_{n}^{T} q'$;
\end{itemize}
\item $p \preo_{n+1}^{T} q$ iff for every sequence of actions
$s\in\Act^*$:
\begin{itemize}
\item if $p \mv{s} p'$ then there is a $q'$ such that $q \mv{s} q'$
and $p' =_{n}^{T} q'$.
\end{itemize}
\end{itemize}
\end{definition}
Note that the relations $=_{1}^{T}$ and $=_{2}^{T}$ are just trace
equivalence (the equivalence that equates two terms having the same
traces---see~\cite{vG2001,Ho80}) and possible futures equivalence,
respectively, whereas $\preo_2^T$ is the possible futures preorder.
Moreover, it is easy to see that, for every $n\geq 0$, the equivalence
relation $ =_{n}^{T}$ is the kernel of the preorder $\preo_n^T$.
The following result is well-known---see, e.g., the
references~\cite{GrV92,HM85}.
\begin{proposition}\label{Lem:precongruence}
For every $n\geq 0$, the relations $\presim_n$,
$\leftrightarrows_n$, $=_{n}^T$ and $\preo_{n}^T$ are preserved by
the operators of BCCSP.
\end{proposition}
The relations we have previously defined over closed BCCSP terms are
extended to arbitrary BCCSP terms thus:
\begin{definition}\label{Def:sound}
Let $t,u$ be BCCSP terms, and let $\preo$ be any of $\presim_n$,
$\leftrightarrows_n$, $=_{n}^T$ and $\preo_{n}^T$ ($n\geq 0$). The
inequation $t \preaxiom u$ is {\em sound} with respect to $\preo$,
written $t\preo u$, iff $\sigma(t) \preo \sigma(u)$ for every
closed substitution $\sigma$.
\end{definition}
For instance, the inequation $x\preox y$ is sound with respect to all
of the $0$-nested semantics defined above. Examples of (in)equations
that are sound with respect to $\presim_2$ are those in
Table~\ref{tab:bccs} and
\begin{eqnarray*}
a(x+y) & \preaxiom & a(x+y) + ax \enspace .
\end{eqnarray*}
The following result collects some basic properties of nested
simulation and nested trace semantics that will be useful in the
technical developments to follow.
\begin{proposition}\label{Lem:inclusions}
For all BCCSP terms $t,u$ and $n\geq 0$, the following statements hold:
\begin{enumerate}
\item \label{incl0} if $t \presim_{n+1} u$, then $t \leftrightarrows_n u$;
\item \label{incl1} if $t \preo_{n+1}^T u$,
then $t=_n^T u$;
\item \label{incl2} if $t \presim_{n} u$, then $t \preo_{n}^T u$.
\end{enumerate}
\end{proposition}
\begin{proof}
Statement (\ref{incl0}) is due to Groote and Vaandrager
in~\cite{GrV92}, and statement (\ref{incl1}) follows immediately
from the definitions of the relations $\preo_{n+1}^T$ and $=_n^T$.
We therefore limit ourselves to presenting a proof of statement
(\ref{incl2}). To this end, observe, first of all, that in light of
Definition~\ref{Def:sound}, it is sufficient to prove the claim for
closed BCCSP terms. Assume now that $p \presim_{n} q$, where $p,q$
are closed BCCSP terms. We prove $p \preo_{n}^T q$ by induction
on $n$. This is trivial if $n=0$. Suppose therefore that $p
\presim_{n+1} q$. Let $s$ be a sequence of actions in $A$, and
assume that $p \mv{s} p'$ for some $p'$. We aim at showing that $q
\mv{s} q'$ for some $q'$ with $p' =_n^T q'$.
Since $p \presim_{n+1} q$ and $p \mv{s} p'$, using
Proposition~\ref{theo:nested} and a simple induction on the length of
$s$, we have that $q \mv{s} q'$ for some $q'$ with $p' \presim_{n+1}
q'$. By statement (\ref{incl0}) of the proposition, we may infer that $p'
\leftrightarrows_{n} q'$. The inductive hypothesis now yields that
$p' \preo_{n}^T q' \preo_{n}^T p'$. Since the relation $=_n^T$ is
the kernel of $\preo_{n}^T$, we may conclude that $p' =_n^T q'$,
which was to be shown. \hfill $\square$
\end{proof}
\subsection{A Modal Characterization of Nested Trace Equivalence}\label{Sect:modal}
In the proof of our main result in Sect.~\ref{Sect:therest}, we shall
make use of the modal characterization of the $n$-nested trace
equivalences proposed by Hennessy and Milner in~\cite[p.~148]{HM85}.
This we now introduce for the sake of completeness.
\begin{definition}\label{Def:modal}
The set $\mathcal{L}$ of {\em\/ Hennessy-Milner formulae} over
alphabet $A$ is defined by the following grammar:
\[
\varphi :: = \top \mid \varphi \wedge \varphi \mid \neg \varphi
\mid \may{a}\varphi~(a\in A) \enspace .
\]
The {\em satisfaction relation} $\models$ is the binary
relation relating closed BCCSP terms and Hennessy-Milner
formulae defined by structural induction on formulae thus:
\begin{itemize}
\item $p \models \top$, for every closed BCCSP term $p$,
\item $p \models \varphi_1 \wedge \varphi_2$ iff $p \models \varphi_1$
and $p \models \varphi_2$,
\item $p \models \neg \varphi$ iff it is not the case that $p \models
\varphi$, and
\item $p \models \may{a}\varphi$ iff $p \mv{a} p'$ for some $p'$ such
that $p' \models \varphi$.
\end{itemize}
\end{definition}
As an immediate consequence of the characterization theorem for
bisimulation equivalence over image-finite labelled transitions
systems shown by Hennessy and Milner~\cite[Theorem~2.2]{HM85}, two
closed BCCSP terms are bisimulation equivalent if, and only if, they
satisfy the same formulae in $\mathcal{L}$. We now introduce a family
of sub-languages of $\mathcal{L}$ that yield modal characterizations
of the $n$-nested trace equivalences for every $n\geq 0$.
\begin{definition}\label{Def:nested-modal}
For every $n\geq 0$, we define the set $\mathcal{L}_n$ of {\em $n$-nested
Hennessy-Milner formulae} inductively thus:
\begin{itemize}
\item $\mathcal{L}_0$ contains only the formulae $\top$ and $\neg
\top$, and
\item $\mathcal{L}_{n+1}$ is given by the following grammar
\[
\varphi :: = \top \mid \varphi \wedge \varphi \mid \neg \varphi
\mid \may{a_1}\cdots\may{a_k}\psi~(k\geq 0,~a_1\cdots a_k\in A^* \text{
and } \psi\in \mathcal{L}_n) \enspace .
\]
\end{itemize}
\end{definition}
The following result is due to Hennessy and Milner~\cite{HM85}.
\begin{theorem}\label{Thm:HML}
Let $p,q$ be closed BCCSP terms, and let $n\geq 0$. Then $p =_n^T q$
iff $p$ and $q$ satisfy the same formulae in the language
$\mathcal{L}_n$.
\end{theorem}
\begin{remark}\label{Rem:depth}
Note that, for every $n\geq 0$ and closed terms $p,q$, if each
formula in $\mathcal{L}_n$ satisfied by $p$ is also satisfied by
$q$, then $p$ and $q$ satisfy the same formulae in the language
$\mathcal{L}_n$. Indeed, assume that each formula in $\mathcal{L}_n$
satisfied by $p$ is also satisfied by $q$, and that $q$ satisfies
$\varphi\in \mathcal{L}_n$. Using the closure of $\mathcal{L}_n$
with respect to negation, we have that $q\not\models \neg \varphi$,
and therefore that $p\not\models \neg \varphi$. It follows that $p$
satisfies $\varphi$, which was to be shown.
Although tempting, it would therefore be incorrect to assume that,
for every $n\geq 0$ and closed terms $p,q$, it holds that $p
\preo_n^T q$ iff each formula in $\mathcal{L}_n$ satisfied by $p$ is
also satisfied by $q$.
To obtain a modal characterization of the $n$-nested trace
preorders, consider the sub-languages $\mathcal{M}_n$ of
$\mathcal{L}_n$ defined inductively thus:
\begin{itemize}
\item $\mathcal{M}_0$ contains only the formulae $\top$ and $\neg
\top$, and
\item $\mathcal{M}_{n+1}$ is given by the following grammar
\[
\varphi :: = \top \mid \varphi \wedge \varphi
\mid \may{a_1}\cdots\may{a_k}\psi~(k\geq 0,~a_1\cdots a_k\in A^* \text{
and } \psi\in \mathcal{L}_n) \enspace .
\]
\end{itemize}
Following the lines of the proof of Theorem~2.2 in~\cite{HM85}, the
interested reader will have little trouble in establishing that
\begin{quote}
For every $n\geq 0$ and closed terms $p,q$, it holds that $p
\preo_n^T q$ iff each formula in $\mathcal{M}_n$ satisfied by $p$
is also satisfied by $q$.
\end{quote}
\end{remark}
\subsection{Lengths, Norm and Depth of Terms}
We now present some results on the relationships between the lengths
of the completed traces, the depth and the norm of BCCSP terms that
are related by the notions of semantics considered in this paper.
These will find important applications in the proofs of our main
results, and shed light on the nature of the identifications made by
the nested simulation and trace semantics.
\begin{definition}\label{def:termination}
A sequence $s\in\Act^*$ is a {\em completed trace} of a term $t$
iff $t \mv{s} t'$ holds for some term $t'$ without outgoing
transitions. We write $\lengths(t)$ for the set of lengths of the
completed traces of a BCCSP term $t$.
\end{definition}
Note that $\lengths(t)$ is non-empty for each BCCSP term $t$.
Moreover, the only closed BCCSP term that has a completed trace of
length 0 is $\mathbf{0}$. (Recall that we consider terms modulo
absorption of $\mathbf{0}$-summands.)
\begin{definition}\label{def:depth}
The {\em depth} and the {\em norm} of a BCCSP term $t$, denoted by
${\it depth}(t)$ and ${\it norm}(t)$, are the lengths of the longest
and of the shortest completed trace of $t$, respectively.
\end{definition}
The following lemma states the basic relations between the behavioural
semantics studied in this paper and the lengths, depth and norm of
terms that will be needed in the technical developments to follow.
\begin{lemma}\label{Lem:norm}
Let $\preo$ be any of $\preo_n^T$, $=_n^T$, $\leftrightarrows_n$,
and $\presim_{n}$, for $n\geq 2$. If $t\preo u$, then
\begin{itemise}
\item [(a)] $\lengths(t) \subseteq \lengths(u)$,\vspace{-1ex}
\item [(b)] $\depth(t) = \depth(u)$, \vspace{-1ex}
\item [(c)] $\norm(t) \geq \norm(u)$ and\vspace{-1ex}
\item [(d)] $\var(t) = \var(u)$.
\end{itemise}
\end{lemma}
\begin{proof}
In light of Proposition~\ref{Lem:inclusions}, it is sufficient to prove
that the claims hold for the possible futures preorder,
viz.~the relation $\preo_2^T$.
We argue, first of all, that claims (a)--(c) hold when $t \preo_2^T
u$. To this end, note that, by substituting $\mathbf{0}$ for the
variables in $t$ and $u$, we obtain closed terms $p$ and $q$ with
${\it lengths}(t)={\it lengths}(p)$ and $lengths(u)=lengths(q)$. So
it suffices to prove claims (a)--(c) with $p$ and $q$ in place of
$t$ and $u$, respectively. By Definition~\ref{Def:sound}, we have
that $p \preo_2^T q$.
Assume now that $n\in\lengths(p)$. Then there are a sequence $s\in
A^*$ of length $n$ and a closed term $p'$ with no outgoing
transitions such that $p \mv{s} p'$. As $p \preo_2^T q$, there is a
closed term $q'$ such that $q \mv{s} q'$ and $p' =_1^T q'$. Recall
that $p' =_1^T q'$ if, and only if, $p'$ and $q'$ have the same
traces. It therefore follows that $q'$ has no outgoing transitions,
and that $n\in\lengths(q)$, which was to be shown.
Claim (c) follows immediately from (a). To see that claim (b) holds,
observe that if $p \preo_2^T q$ for closed BCCSP terms $p$ and $q$,
then, by Proposition~\ref{Lem:inclusions}(\ref{incl1}), $p$ and $q$ have
the same non-empty finite sets of traces, and thus the same longest
traces.
To prove claim (d), let $t,u$ be BCCSP terms such that $t\preo_2^T
u$. Assume, towards a contradiction, that there is a variable $x$
that occurs in only one of $t$ and $u$. We shall exhibit a closed
substitution $\sigma$ such that $\depth(\sigma(t)) \neq
\depth(\sigma(u))$, contradicting statement (b) of the lemma.
To this end, observe, first of all, that without loss of generality,
we may assume that $x$ occurs in $t$, say.
Let $m$ be a positive
integer larger than $\depth(t)$. By claim (b) of the lemma, we have
that $\depth(t) = \depth(u) < m$ also holds.
Consider now the closed substitution $\sigma$ that maps $x$ to $a^m$,
and all the other variables to $\mathbf{0}$. Using structural
induction, it is a simple matter to prove that
\begin{eqnarray*}
\depth(\sigma(t)) & \geq & m \quad \text{and} \\
\depth(\sigma(u)) & = & \depth(u) ~<~ m \enspace .
\end{eqnarray*}
By statement (b) of the lemma, it follows that $\sigma(t)\preo_2^T
\sigma(u)$ does not hold, contradicting our assumption that
$t\preo_2^T u$. \hfill $\Box$
\end{proof}
\begin{remark}\label{Rem:same-lengths}
Note that $\lengths(t)=\lengths(u)$ and $\norm(t)=\norm(u)$ both
hold, if $t=_2^T u$.
The restriction that $n\geq 2$ is necessary in the statement of
Lemma~\ref{Lem:norm}(a) and (c). In fact, $aa + a \leftrightarrows_1
aa$, but
\[
\begin{array}{l}
\lengths(aa + a) = \{1,2\} \not\subseteq \{2\} = \lengths(aa) \quad\text{and }\\
\norm(aa + a) < \norm(aa) \enspace .
\end{array}
\]
Statements (b) and (d) in Lemma~\ref{Lem:norm} also hold for
$=_1^T$. In fact, it is not hard to see that, for every $t,u$, if $t
\preo_1^T u$ then $\depth(t)\leq \depth(u)$ and
$\var(t)\subseteq \var(u)$.
\end{remark}
\section{Non-finite Axiomatizability of the 2-nested Simulation Preorder}\label{Sect:ineq}
In this section we prove that the 2-nested simulation preorder is not
finitely inequationally axiomatizable. The following lemma will play a
key role in the proof of this statement.
\begin{lemma}
\label{lem:noinbetween}
If $p \presim_2 a^{2m} + a^m$, then either
$p \leftrightarrows_2 a^{2m}$ or $p \leftrightarrows_2 a^{2m} + a^m$.
\end{lemma}
\begin{proof}
The case $m=0$ is trivial; we therefore focus on the case $m>0$. We
note, first of all, that if $q \presim_2 a^k$ for some $k \geq 0$,
then, by Lemma~\ref{Lem:norm}(a), $q$ has only the completed trace
$a^k$; clearly, this implies $a^k \presim_2 q$, and hence $a^k
\leftrightarrows_2 q$.
Consider now a transition $p \mv{a} p'$. Since $p \presim_2 a^{2m} + a^m$,
either $p' \presim_2 a^{2m-1}$ or $p' \presim_2 a^{m-1}$. By
Lemma~\ref{Lem:norm}(b), $p$ has depth $2m$. So there
is at least one transition $p \mv{a} p'$ with $p' \presim_2 a^{2m-1}$.
If for all transitions $p \mv{a} p'$ we have $p' \presim_2 a^{2m-1}$,
then it follows that $p \presim_2 a^{2m}$, and hence $p
\leftrightarrows_2 a^{2m}$. On the other hand, if there exists a
transition $p \mv{a} p''$ with $p'' \presim_2 a^{m-1}$ (and so
$a^{m-1} \presim_2 p''$), then it follows that $a^{2m} + a^m \presim_2
p$, and hence $p \leftrightarrows_2 a^{2m} + a^m$. \hfill $\square$
\end{proof}
The idea behind our proof that the 2-nested simulation preorder is not
finitely inequationally axiomatizable is as follows. Assume a finite
inequational axiomatization $E$ for BCCSP that is sound modulo
$\presim_2$. We show that, if $m$ is sufficiently large, then, for
all closed inequational derivations $a^{2m} \preaxiom p_1 \preaxiom
\cdots \preaxiom p_k$ from $E$ with $p_k \presim_2 a^{2m} + a^m$, we
have that $p_k \leftrightarrows_2 a^{2m}$. Since $ a^{2m} + a^m \not
\presim_2 a^{2m}$, it follows that $a^{2m} \preaxiom a^{2m} + a^m$
cannot be derived from $E$. However, $a^{2m} \presim_2 a^{2m} +
a^m$.
The following lemma is the crux in the implementation of the
aforementioned proof idea.
\begin{lemma}
\label{lem:crux}
Let $t \preaxiom u$ be sound modulo $\presim_2$. Let $m$ be greater
than the depth of $t$. Assume that $C[\sigma(u)] \presim_2 a^{2m} +
a^m$, for some closed substitution $\sigma$. Then $C[\sigma(t)]
\leftrightarrows_2 a^{2m}$ implies $C[\sigma(u)] \leftrightarrows_2
a^{2m}$.
\end{lemma}
\begin{proof}
Let $C[\sigma(t)] \leftrightarrows_2 a^{2m}$; we prove $C[\sigma(u)]
\leftrightarrows_2 a^{2m}$. Since $C[\sigma(u)] \presim_2 a^{2m} +
a^m$, it is sufficient to show that $a^{2m} + a^m \not\presim_2
C[\sigma(u)]$. In fact, if $C[\sigma(u)] \presim_2 a^{2m} + a^m$ and
$a^{2m} + a^m \not\presim_2 C[\sigma(u)]$, by Lemma
\ref{lem:noinbetween} it follows that $C[\sigma(u)]
\leftrightarrows_2 a^{2m}$, which is to be shown. We prove $a^{2m}
+ a^m \not\presim_2 C[\sigma(u)]$ by distinguishing two cases,
depending on the form of the context $C[~]$.
\begin{itemize}
\item {\em Case 1}: Suppose $C[~]$ is of the form $C'[b([~] + r)]$.
In this case, we shall prove $a^{2m} + a^m \not\presim_2
C[\sigma(u)]$ by arguing that $a^{m-1} \not\presim_2 q'$ holds for
each $q'$ such that $C[\sigma(u)] \mv{a} q'$. To this end, consider
a transition
\[
C[\sigma(u)] \mv{a} q' \enspace .
\]
Then
$q'=D[\sigma(u)]$ for some context $D[~]$, and, because of the form
of the context $C[~]$, we may infer that
\[
C[\sigma(t)] \goto{a} p'= D[\sigma(t)] \enspace .
\]
As $\sigma(t) \presim_2 \sigma(u)$ by the soundness of $t\preaxiom
u$ with respect to $\presim_2$, and $p' \presim_2 q'$ by
Proposition~\ref{Lem:precongruence}, Lemma~\ref{Lem:norm}(b) yields
that $p'$ and $q'$ have the same depth. Since $C[\sigma(t)]
\leftrightarrows_2 a^{2m}$, it follows by
Proposition~\ref{theo:nested} that $p' \presim_2 a^{2m-1}$. So by
Lemma~\ref{Lem:norm}(b), we have that
\[
{\it depth}(p') = {\it depth}(q') = 2m-1 \enspace .
\]
As ${\it depth}(a^{m-1}) \not= 2m-1$, another application of
Lemma~\ref{Lem:norm}(b) yields that
\[
a^{m-1} \not\presim_2 q' \enspace .
\]
Since this holds for all transitions $C[\sigma(u)] \mv{a} q'$, and
$a^{2m} + a^m \mv{a} a^{m-1}$, using Proposition~\ref{theo:nested} we
may therefore conclude that $a^{2m} + a^m \not\presim_2 C[\sigma(u)]$.
\item {\em Case 2}: Suppose $C[~]$ is of the form $[~] + r$.
In this case, we shall prove $a^{2m} + a^m \not\presim_2
C[\sigma(u)]$ by arguing that $\norm(C[\sigma(u)])$ is larger than $m$.
To this end, observe, first of all, that, as $t\presim_2 u$ by
our assumptions, statements (b) and (d) in Lemma~\ref{Lem:norm}
imply that ${\it depth}(t) = {\it depth}(u)$, and moreover that $t$
and $u$ contain exactly the same variables. We proceed with the
proof by distinguishing two cases, depending on whether
$\norm(\sigma(t))=0$ or not.
\begin{itemize}
\item {\em Case} $\norm(\sigma(t))=0$.
In this case, $t$ has the form $\sum_{i\in I} x_i$ for some finite
index set $I$, and variables $x_i$ ($i\in I$) with
$\norm(\sigma(x_i))=0$ for each $i\in I$.
Since $t\preaxiom u$ is sound with respect to $\presim_2$,
statements (c)--(d) in Lemma~\ref{Lem:norm} yield that $t = u$
modulo axiom A3. Since axiom A3 is sound with respect to
$\leftrightarrows_2$, using Proposition~\ref{Lem:precongruence} we may
therefore conclude that
\[
a^{2m} + a^m \not\presim_2 a^{2m} \leftrightarrows_2 C[\sigma(t)]
\leftrightarrows_2 C[\sigma(u)] \enspace ,
\]
which was to be shown.
\item {\em Case} $\norm(\sigma(t))>0$.
Since $\sigma(t) + r \leftrightarrows_2 a^{2m}$,
Lemma~\ref{Lem:norm}(c) yields that ${\it norm}(\sigma(t))\geq
2m$, and either ${\it norm}(r)\geq 2m$ or ${\it norm}(r)=0$. By
the soundness of $t\preaxiom u$ with respect to $\presim_2$, and
the assumption that $\norm(\sigma(t))>0$, it follows that
$\depth(\sigma(t))=\depth(\sigma(u))>0$. Hence $\sigma(u)\neq
\mathbf{0}$, and therefore we have that $\norm(\sigma(u))>0$. As
$\sigma(u) + r \presim_2 a^{2m} + a^m$, again using
Lemma~\ref{Lem:norm}(c), we infer that
\[
\norm(\sigma(u))\geq m \enspace .
\]
Since ${\it depth}(t)m$.
By the fact that ${\it depth}(u)={\it depth}(t)m$, it
follows that ${\it norm}(\sigma(u))>m$. Since moreover ${\it
norm}(r)\geq 2m$ or ${\it norm}(r)=0$, we have ${\it
norm}(\sigma(u) + r)>m$. As $a^{2m} + a^m$ has norm $m$, by
Lemma~\ref{Lem:norm}(a) we may conclude that $a^{2m} + a^m
\not\presim_2 \sigma(u) + r$, which was to be shown. \hfill
$\square$
\end{itemize}
\end{itemize}
\end{proof}
\begin{remark}
The inequation $ax \preaxiom ax+a^1$ is sound modulo $\presim_2$.
However, $a^4 \not\leftrightarrows_2 a^4+a^1$. So the proviso in
the statement of Lemma~\ref{lem:crux} that $C[\sigma(u)] \presim_2
a^{2m} + a^m$ cannot be omitted. (Note that $a^4+a^1 \not\presim_2
a^4+a^2$.)
\end{remark}
\begin{theorem}
\label{theo:preorder}
BCCSP modulo the 2-nested simulation
preorder is not finitely inequationally axiomatizable.
\end{theorem}
\begin{proof}
Let $E$ be a finite inequational axiomatization for BCCSP
that is sound modulo $\presim_2$. Let $m > \max\{ {\it depth}(t)
\mid t \preaxiom u\in E \}$.
By Lemma \ref{lem:crux}, and using induction on the length of
derivations, it follows that if the closed inequation $a^{2m}
\preaxiom r$ can be derived from $E$ and $r \presim_2 a^{2m} + a^m$,
then $r \leftrightarrows_2 a^{2m}$. As Lemma~\ref{Lem:norm}(c)
yields that $a^{2m} + a^m \not\presim_2 a^{2m}$, it follows that
$a^{2m} \preaxiom a^{2m} + a^m$ cannot be derived from $E$. Since
$a^{2m} \presim_2 a^{2m} + a^m$, we may conclude that $E$ is not
complete modulo $\presim_2$. \hfill $\square$
\end{proof}
\section{Possible Future Semantics is not Finitely Based}
\label{Sect:possible-futures}
Throughout this section, we let $\preo$ be either the possible futures
preorder, or possible futures equivalence. Our order of business in
this section will be to prove that $\preo$ has no finite
(in)equational axiomatization over BCCSP. The idea behind the proof of
this claim is as follows. Assume that $E$ is a finite inequational
axiomatization for BCCSP that is sound modulo $\preo$. We show that,
if $m$ is sufficiently large, then, for all closed inequations $p
\preox q$ that can be derived from $E$ the following invariant
property holds:
\begin{quote}
If $\lengths(q) \subseteq \{m+1,2m+1,3m+1\}$, and there is a
$p'$ such that $p \goto{a} p'$, $\norm(p')=m$ and $\depth(p') \leq
2m$, then there is a $q'$ such that \plat{q \goto{a} q'},
$\norm(q')=m$ and $\depth(q')\leq 2m$.
\end{quote}
However, we shall exhibit a pair of closed terms that are related by
$\preo$, and do not satisfy the above property. This will allow us to
conclude that $E$ is not complete with respect to $\preo$.
The following lemma characterizes some properties of the inequations
that are sound with respect to $\preo$ that will be useful in the
proof of the main result of this section (Theorem~\ref{Thm:non-ax-pf}
to follow).
\begin{lemma}\label{Lem:variables-pf}
Let the axiom $t \preox u$ be sound modulo $\preo$.
Let $t = \Sigma_{i\in I}x_i + \Sigma_{j\in J}a_jt_j$ and
$u = \Sigma_{k\in K}y_k + \Sigma_{\ell\in L}b_\ell u_\ell$,
and let $x$ be a variable. Then
\begin{itemise}
\item [(a)] $\{x_i \mid i \in I\} \subseteq \{y_k \mid k \in K\}$, and
\vspace{-1ex}
\item [(b)] for each $j \in J$ with $x \in \var(t_j)$ there is an $\ell \in
L$ such that $a_j=b_\ell$, $x \in \var(u_\ell)$ and $\var(u_\ell) \subseteq \var(t_j)$.
\end{itemise}
\end{lemma}
\begin{proof}
Let $t \preox u$ be sound modulo $\preo$, and let $x$ be a variable.
We prove the two statements of the lemma separately.
\begin{itemize}
\item {\em Proof of Claim (a)}: Assume, towards a contradiction, that
the variable $x$ is contained in $\{x_i \mid i \in I\}$, but not in
$\{y_k \mid k \in K\}$. We shall exhibit a closed substitution
$\sigma$ such that $\sigma(t)\not\preo \sigma(u)$, contradicting our
assumption that $t\preox u$ is sound modulo $\preo$.
To this end, pick a positive integer $m> \depth(t)$. Since $t
\preox u$ is sound modulo $\preo$, by Lemma~\ref{Lem:norm}(b) we
have that $m>\depth(u)$ also holds. Consider the closed
substitution $\sigma$ that maps $x$ to $a^m$, and all the other
variables to $\mathbf{0}$. Since $x=x_i$ for some $i\in I$, we have
that $m\in \lengths(\sigma(t))$. On the other hand, $m\not\in
\lengths(\sigma(u))$ because, as $x$ is not contained in $\{y_k \mid
k \in K\}$, every completed trace of $\sigma(u)$ is either one of
$u$ itself (and is thus shorter than $m$) or has $a^m$ has a proper
suffix (and is thus longer than $m$). By Lemma~\ref{Lem:norm}(a),
it follows that $\sigma(t)\preo \sigma(u)$ does not hold,
contradicting our assumption that $t\preox u$ is sound modulo
$\preo$.
\item {\em Proof of Claim (b)}:
Assume, towards a contradiction, that there is a $j \in J$ with $x
\in \var(t_j)$ such that, for each $\ell \in L$ with $a_j=b_\ell$
either $x \not\in \var(u_\ell)$ or $\var(u_\ell) \not\subseteq
\var(t_j)$. We shall exhibit a closed substitution $\sigma$ such
that $\sigma(t) \not\preo_2^T \sigma(u)$, contradicting our
assumption that $t \preox u$ is sound modulo $\preo$.
Let $m$ be a positive integer larger than $\depth(t)$. Since $t
\preox u$ is sound modulo $\preo$, by Lemma~\ref{Lem:norm}(b) we
have that $m>\depth(u)$ also holds. Consider the closed substitution
mapping $x$ to $a^m$, all of the variables not occurring in $t_j$ to
$a^{2m}$, and all the other variables to $\mathbf{0}$. Note that
$\sigma(t) \mv{a_j} \sigma(t_j)$, by Lemma~\ref{Lem:trans-sub}.
Moreover, since $x\in\var(t_j)$ and
\[
\depth(t_j) \leq \depth(t) - 1 \leq m - 2
\enspace ,
\]
it is easy to see that
\begin{eqnarray}\label{Eqn:depthsub}
m \leq \depth(\sigma(t_j)) \leq 2m - 2 \enspace .
\end{eqnarray}
We claim that if $\sigma(u) \mv{a_j} p$, then $\depth(\sigma(t_j))
\neq \depth(p)$. This shows that $\sigma(t) \not\preo_2^T \sigma(u)$
because no $p$ with $\sigma(u) \mv{a_j} p$ can have the same traces
as $ \sigma(t_j)$ (see Remark~\ref{Rem:same-lengths}), contradicting
our assumption that $t \preox u$ is sound modulo $\preo$.
To prove our claim, we consider the possible origins of a transition
$\sigma(u) \mv{a_j} p$.
\begin{itemize}
\item {\em Case 1}: $\sigma(u) \mv{a_j} p$ because $\sigma(y_k)
\mv{a_j} p$, for some $k\in K$. In this case, by the definition of
$\sigma$, we have that $\depth(p)\in\{m-1,2m-1\}$. By
(\ref{Eqn:depthsub}), we may infer that $\depth(\sigma(t_j)) \neq
\depth(p)$, as claimed.
\item {\em Case 2}: $\sigma(u) \mv{a_j} p$ because $p =
\sigma(u_\ell)$ for some $\ell \in L$ such that $a_j=b_\ell$ and
either $x \not\in \var(u_\ell)$ or $\var(u_\ell) \not\subseteq
\var(t_j)$. In this case, by the definition of $\sigma$ and using
that $\depth(u) \mbox{max}\{\depth(t), \depth(u)
\mid (t\preox u) \in E\}$.
We have that
\[
a(a^{m}+a^{2m})+aa^{3m} \preo aa^{2m}+a(a^{m}+a^{3m})
\]
because both processes have the same possible futures. Nevertheless,
$$E \not\,\vdash a(a^{m}+a^{2m})+aa^{3m} \preox aa^{2m}+a(a^{m}+a^{3m}) \enspace .$$
This follows immediately from the following
\begin{claim}
Assume that $E \vdash p \preox q$, $\lengths(q) \subseteq
\{m+1,2m+1,3m+1\}$, and there is a $p'$ such that $p \goto{a} p'$,
$\norm(p')=m$ and $\depth(p') \leq 2m$. Then there is a $q'$ such that
\plat{q \goto{a} q'}, $\norm(q')=m$ and $\depth(q')\leq 2m$.
\end{claim}
{\em Proof of the claim}. Using induction on the length of
inequational derivations, the soundness of $E$ with respect to
$\preo$ and Lemma~\ref{Lem:norm}(a), it suffices to consider the case
that $p=C[\sigma(t)]$ and $q=C[\sigma(u)]$ for a BCCSP context $C[~]$,
a closed substitution $\sigma$, and an axiom $(t \sqsubseteq u) \in
E$. We proceed by distinguishing two sub-cases, depending on the form
of the context $C[~]$.
\begin{itemize}
\item {\it Case 1:} Suppose $C[~]$ is of the form $C'[b([~]+r)]$.\\
Let $p'$ be as in the statement of the claim. Then
$p'=D[\sigma(t)]$ for some context $D[~]$, and, because of the form
of the context $C[~]$, we may infer that
\[
q =C[\sigma(u)] \goto{a} q'= D[\sigma(u)] \enspace .
\]
By the soundness of $E$ and the fact that
$\preo$ is preserved by the operators of BCCSP
(Proposition~\ref{Lem:precongruence}), we have that $p' \preo q'$.
Therefore $\norm(q') \leq \norm (p') = m$ and $\depth(q') =
\depth(p') \leq 2m$ both hold by statements (b) and (c) in
Lemma~\ref{Lem:norm}. As $\norm(q) \geq m+1$ it follows that
$\norm(q')=m$, and we are done.
\item {\it Case 2:} Suppose $C[~]$ is of the form $[~]+r$.\\
Let $t = \Sigma_{i\in I}x_i + \Sigma_{j\in J}a_jt_j$ and $u =
\Sigma_{k\in K}y_k + \Sigma_{\ell\in L}b_\ell u_\ell$. Consider a
transition $\sigma(t)+r \goto{a} p'$ as in the statement of the
claim. We distinguish three possible cases, depending on the origin
of this transition.
\begin{itemize}
\item {\it Case 2.1:} Assume that $r \goto{a} p'$. Then $q \goto{a}
p'$ and we are done.
\item {\it Case 2.2:} Assume that $\sigma(x_i) \goto{a} p'$ for some $i \in
I$. By Lemma~\ref{Lem:variables-pf}(a) and the soundness of $t
\sqsubseteq u$ with respect to $\preo$, we have that $x_i = y_k$
for some $k \in K$. It follows that $q \goto{a} p'$, and we are
done.
\item {\it Case 2.3:} Assume that $p'=\sigma(t_j)$ for some $j \in J$. As
$\norm(\sigma(t_j))=m$ and
\[
\depth(t_j) < depth(t) < m \enspace ,
\]
there must be a variable $x \in \var(t_j)$ such that
$1\leq \norm(\sigma(x))\leq m$. By statement (b) in
Lemma~\ref{Lem:variables-pf}, there is an $\ell \in L$ such that
$a=b_\ell$, $x \in \var(u_\ell)$ and $\var(u_\ell) \subseteq
\var(t_j)$. Take $q' = \sigma(u_\ell)$. Then \plat{q \goto{a}
q'}. Since $x \in \var(u_\ell)$, we have that
\[
\norm(q')\leq
\depth(u_\ell)+\norm(\sigma(x))< 2m \enspace .
\]
Considering that
\[
\lengths(q)\subseteq \{m+1,2m+1,3m+1\} \enspace ,
\]
and thus
$\lengths(q')\subseteq \{m,2m,3m\}$, it must be the case that
$\norm(q')=m$.
As $\depth(\sigma(t_j)) \leq 2m$ by assumption, it follows that
$\depth(\sigma(y)) \leq 2m$ for each $y \in \var(t_j)$. Since
$\var(u_\ell)\subseteq \var(t_j)$, this also holds for each $y \in
\var(u_\ell)$. As $\depth(u_\ell) < depth(u) < m$, this implies
that $\depth(\sigma(u_\ell)) < 3m$. Considering that
$\lengths(q')\subseteq \{m,2m,3m\}$, we may conclude that
$\depth(q')\leq 2m$.
To sum up, we have proven that, also in this case, \plat{q
\goto{a} q'}, $\norm(q')=m$ and $\depth(q')\leq 2m$, which was
to be shown. \hfill$\Box$
\end{itemize}
\end{itemize}
\end{proof}
\section{No Nested Semantics is Finitely Based}\label{Sect:therest}
We now proceed to offer results to the effect that the language BCCSP
modulo $=_{n}^T$ or $\leftrightarrows_{n}$, for $n\geq 2$, or
$\preo_{n}^{T}$ or $\presim_{n}$, for $n \geq 3$, is not finitely
equationally axiomatizable. Rather than considering each of these
behavioural relations in turn, we offer a general proof of non-finite
axiomatizability that applies to all of them at once. The general strategy
underlying such a proof is as follows. We prove that, for each $n\geq
2$, no finite collection of (in)equations that is sound with respect
to $=_{n}^T$ (the coarsest relation amongst $=_{n}^T$,
$\leftrightarrows_{n}$, $\preo_{n+1}^{T}$ and $\presim_{n+1}$) can
prove all of the closed inequations of the form $p \preaxiom q$, with
$p$ and $q$ BCCSP terms over action $a$, that are sound with respect
to $\presim_{n+1}$ (the finest relation amongst $=_{n}^T$,
$\leftrightarrows_{n}$, $\preo_{n+1}^{T}$ and $\presim_{n+1}$).
In the proof of this result, we shall make use of the modal
characterization of the relation $=_{n}^T$ given in
Theorem~\ref{Thm:HML}. More specifically, we shall show that, for each
$n\geq 2$ and finite axiom system $E$ that is sound with respect to
$=_{n}^T$, there is a formula $\psi_n$ in the language
$\mathcal{L}_{n+1}$ (see Definition~\ref{Def:nested-modal}) such that
whenever $E$ proves a closed inequation $p\preaxiom q$, with $p$ and
$q$ BCCSP terms over action $a$, then, subject to some technical
conditions on the lengths of the completed traces of $q$, it holds
that $p$ satisfies $\psi_n$ if, and only if, so does $q$. We shall,
however, show that this property does not hold for the inequation $q_n
\presim_{n+1} p_n$, where the terms $p_n$ and $q_n$ have been defined
in Example~\ref{Ex:pn-qn}. This will allow us to conclude that the
sound inequation $q_n \preaxiom p_n$ cannot be derived from $E$, and
thus that $E$ is incomplete for $=_{n}^T$, $\leftrightarrows_{n}$,
$\preo_{n+1}^{T}$ and $\presim_{n+1}$.
The technical implementation of the above idea will be based upon an
induction on the length of the proof of closed inequations from the
finite axiom system $E$. The crucial step in this proof will be to
show that, subject to technical conditions, the aforementioned formula
$\psi_n$ is satisfied either by both terms in a substitution instance
of an axiom in $E$ or by neither of them. This case will be tackled by
Lemma~\ref{Lem:crux} to follow. We now introduce some technical
notions, and preliminary results, that will be used in the proof of
this crucial lemma.
\begin{definition}\label{substantial}
We call a substitution $\sigma$ {\em substantial} if
$\depth(\sigma(x))>0$ for all variables $x$.
\end{definition}
For reasons of technical convenience, in the proofs of our non-finite
axiomatizability results presented in this section we will only allow
for the use of closed substantial substitutions in the rule of
substitution. This does not limit the generality of those results
because every finite inequational axiomatization $E$ can be converted
into a finite inequational axiomatization $E'$ such that the closed
substitution instances of the axioms of $E$ are the same as the closed
substantial substitution instances of the axioms of $E'$ (when
equating any closed subterm of depth 0 with $\mathbf 0$). This is done
by including in $E'$ any inequation that can be obtained from an
inequation in $E$ by replacing all occurrences of any number of
variables by $\mathbf 0$.
\begin{definition}\label{depth}
Define the {\em depths} at which a subterm occurs in a BCCSP
term as follows:
\begin{itemise}
\item $t$ occurs in $t$ at depth $0$,
\item if $v$ occurs in $t$ or $u$ at depth $d$, then $v$ occurs in
$t+u$ at depth $d$,
\item if $v$ occurs in $t$ at depth $d$ then $v$ occurs in $at$ (with
$a \in A$) at depth $d+1$.
\end{itemise}
A BCCSP term $t$ has a {\em unique depth allocation}
if no variable occurs in $t$ at two different depths.
\end{definition}
For example, the term $ax + x$ does not have a unique depth
allocation, as the variable $x$ occurs both at depth 0 and at depth 1
in it, but $ax + y$ does.
The following lemma describes the interplay between the depths at
which variables occur in a term $t$, and the lengths of terms of the
form $\sigma(t)$, for some substantial substitution $\sigma$.
\begin{lemma}\label{Lem:depth-length}
For every BCCSP term $t$ and $d\geq 0$, the following statements
hold:
\begin{enumerate}
\item \label{dl1} The term $v$ occurs in $t$ at depth $d$ if, and
only if, there are a term $u$ and a sequence of actions $s$ of
length $d$ such that $t \mv{s} v + u$.
\item \label{dl2} Let $x$ be a variable, and let $\sigma$ be a
substitution. For every $n>0$, if
$x$ occurs in $t$ at depth $d$ and $n\in\lengths(\sigma(x))$ then
$d+n\in\lengths(\sigma(t))$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove the two statements separately. Recall that we consider
equality of terms modulo axioms A1, A2 and A4 in
Table~\ref{tab:bccs}.
\begin{itemize}
\item {\em Proof of statement \ref{dl1}}.
We show the two implications separately.
\begin{itemize}
\item ($\Rightarrow$) By induction on the definition of the
{depths} at which $v$ occurs in $t$.
\begin{itemize}
\item Assume that $v$ occurs in $t$ at depth $d$ because $v=t$
and $d=0$. Then, letting $\varepsilon$ denote the empty
string, we have that
\[
t \mv{\varepsilon} v = v + \mathbf{0} \enspace ,
\]
and we are done.
\item Assume that $v$ occurs in $t+t'$ at depth $d$ because $v$
occurs in $t$ or $t'$ at depth $d$. Suppose, without loss of
generality, that $v$ occurs in $t$ at depth $d$. By induction,
we have that there are a term $u$ and a sequence of actions
$s$ of length $d$ such that $t \mv{s} v + u$. If $d$ is
positive, we may immediately conclude that $t+t' \mv{s} v +
u$. If $d=0$, then $t = v+u$. It follows that $t+t'
\mv{\varepsilon} v+u+t'$, and we are done.
\item Assume that $v$ occurs in $at$ (with $a \in A$) at depth
$d+1$ because $v$ occurs in $t$ at depth $d$. By induction we
have that there are a term $u$ and a sequence of actions $s$
of length $d$ such that $t \mv{s} v + u$. It follows that $a t
\mv{as} v + u$, and we are done.
\end{itemize}
\item ($\Leftarrow$) Assume that there are a term $u$ and a
sequence of actions $s$ of length $d$ such that $t \mv{s} v +
u$. We prove that $v$ occurs in $t$ at depth $d$ by induction on
$d$. Throughout the proof, we let $t=\sum_{i\in I}x_i+\sum_{j\in
J}a_jt_j$.
\begin{itemize}
\item {\em Base Case}: $d=0$. Since $t \mv{\varepsilon} v + u$,
we have that
\[
t = \sum_{i\in I}x_i+\sum_{j\in
J}a_jt_j = v + u \enspace .
\]
This means that $v = \sum_{i\in I'}x_i+\sum_{j\in J'}a_jt_j$
for some $I'\subseteq I$ and $J'\subseteq J$. Since $v$ occurs
in $v$ at depth $0$ by the first clause of
Definition~\ref{depth}, using the second clause of
Definition~\ref{depth} we may conclude that $v$ occurs in $t$
at depth $0$.
\item {\em Inductive Step}: $d>0$. Since
\[
t = \sum_{i\in I}x_i+\sum_{j\in
J}a_jt_j \mv{s} v + u \enspace ,
\]
and $s$ is non-empty, we have that $s=a_js'$ and $t_j \mv{s'} v
+ u $, for some $j\in J$. By induction, $v$ occurs in $t_j$ at
depth $d-1$, and therefore in $a_j t_j$ at depth $d$. Using
the second clause of Definition~\ref{depth} we may conclude
that $v$ occurs in $t$ at depth $d$.
\end{itemize}
\end{itemize}
\item {\em Proof of statement \ref{dl2}}.
Assume that $x$ occurs in $t$ at depth $d$,
$n\in\lengths(\sigma(x))$ for some substitution $\sigma$, and $n$
is positive. Since $x$ occurs in $t$ at depth $d$, by
statement~\ref{dl1} of the lemma, we have that $t \mv{s} x + u$
for some sequence of actions $s$ of length $d$ and term $u$. By
Lemma~\ref{Lem:trans-sub}, we have that
\[
\sigma(t) \mv{s} \sigma(x + u) = \sigma(x) + \sigma(u) \enspace .
\]
As $n\in\lengths(\sigma(x))$ by our assumptions, $\sigma(x) \mv{s'}
v$ for some sequence of actions $s'$ of length $n$ and term $v$ with
no outgoing transitions. Since the length of $s'$ is positive, it
follows that $\sigma(t) \mv{ss'} v$ holds, and thus that
$d+n\in\lengths(\sigma(t))$, which was to be shown. \hfill $\square$
\end{itemize}
\end{proof}
\begin{lemma}\label{Lem:depth}
Let $t$ be a BCCSP term with $\depth(t)0$.) As
$|d_1-d_2|0$. Then
\begin{eqnarray*}
p\models\may{a}^m\neg\may{a}\top & \Leftrightarrow &
\exists q\,(p\goto a q\models\may{a}^{m-1}\neg\may{a}\top) \\
&\Leftrightarrow &
\exists q'\,(p;_m a^\ell\goto a q'\models\may{a}^{m+\ell-1}\top) \\
&\Leftrightarrow & p;_m a^\ell\models\may{a}^{m+\ell}\top \enspace ,
\end{eqnarray*}
where the second equivalence follows by (\ref{eq1}) and the inductive
hypothesis, using that $q'=q;_{m-1}a^\ell$ and $\depth(q)0$. Then,
\begin{eqnarray*}
p\models(\may{a}\neg)^n\may{a}^m\neg\may{a}\top & \Leftrightarrow &
\exists q\,(p\goto a q\not\models(\may{a}\neg)^{n-1}\may{a}^m\neg\may{a}\top)\\&\Leftrightarrow &
\exists q'\,(p;_{n+m} a^\ell\goto a q'\not\models(\may{a}\neg)^{n-1}\may{a}^{m+\ell}\top) \\
& \Leftrightarrow & p;_{n+m} a^\ell\models(\may{a}\neg)^n\may{a}^{m+\ell}\top
\enspace ,
\end{eqnarray*}
where the second equivalence follows by (\ref{eq1}) and the inductive
hypothesis, using that $q'=q;_{n+m-1}a^\ell$ and
$\depth(q)0$, then $a^{m+\ell}\not\models\may{a}^m\neg\may{a}\top$. On the other hand,
\[
a^{m+\ell};_m a^\ell = a^{m+\ell} \models\may{a}^{m+\ell}\top
\enspace .
\]
\end{example}
\begin{lemma}\label{Lem:substitution1}
Let $\sigma$ be a closed substitution, and let $t$ be a BCCSP
term with a unique depth allocation and $\depth(t)0$. We begin by proving that
$\sigma'(v)=\sigma(v);_k a^\ell$ for each summand $v$ of $t$.
\begin{itemize}
\item Consider a summand $x$ of $t$. Since $x$ occurs at depth 0 in
$t$, the definition of $\sigma'$ yields that
$\sigma'(x)=\sigma(x);_k a^\ell$.
\item Consider a summand $au$ of $t$. Since
$\sigma'(y)=\sigma(y);_{k-e-1} a^\ell$ for variables $y$ that occur
at depth $e$ in $u$, and $\depth(u) \mbox{max}\{\depth(t), \depth(u) \mid (t\preox u) \in E\} \enspace .
\]
Let $p_n$ and $q_n$ be defined, for each $n\in \IN$, as in
Example~\ref{Ex:pn-qn}. For ease of reference, we recall that:
$$\begin{array}{lll@{~~~~~~~~~~~}lll}
p_0 &=& a^{2m-1}{\mathbf 0} & q_0 &=& a^{m-1}{\mathbf 0} \\
p_{n+1} &=& ap_n+ aq_n & q_{n+1} &=& ap_n
\end{array}$$
As argued in Example~\ref{Ex:pn-qn}, for every $n\geq 1$, we have that
$p_n \presim_{n} q_n$, and thus
\[
q_n \presim_{(n+1)} p_n \enspace .
\]
Let $\psi_1 = \may{a}^{m}\neg\may{a}\top$ and $\psi_{n+1} =
\may{a}\neg\psi_n$. Note that the formula $\psi_n$ is contained in
$\mathcal{L}_{n+1}$, for each $n\geq 1$, and that $\psi_{n+1}$ is the
formula mentioned in the statement of Lemma~\ref{Lem:crux}. By
induction on $n \geq 1$ one checks that $p_n \models \psi_n$ but $q_n
\models \neg\psi_n$.
We now proceed to use the fact that $p_n \models \psi_n$ but $q_n
\models \neg\psi_n$ to argue that the inequation $q_n \preaxiom p_n$
cannot be proven from any finite set of equations that is sound for
$=_{n}^{T}$. To this end, suppose that $E$ is sound for $=_{n}^{T}$
(which, by Proposition~\ref{Lem:inclusions}, is certainly the case if $E$ is
sound for $\leftrightarrows_{n}$, $\preo_{n+1}^{T}$ or
$\presim_{n+1}$), where $n \geq 2$. We show that $E$ is incomplete
for $\presim_{n+1}$ (and thus certainly for $=_{n}^{T}$,
$\leftrightarrows_{n}$ and $\preo_{n+1}^{T}$ by
Proposition~\ref{Lem:inclusions}), because $E \not\,\vdash q_n \preox p_n$.
This follows immediately from the following:
\begin{claim}
Assume that $E \vdash p \preox q$ and $\lengths(q)\subseteq \{n+m-1,~n+2m-1\}$.
Then
\[
p \models \psi_n ~\Leftrightarrow~ q \models \psi_n \enspace .
\]
\end{claim}
In fact, using this claim, we can show that $E \not\,\vdash q_n \preox
p_n$ as follows. Observe, first of all, that $\lengths(p_n)$ is
included in $\{n+m-1,~n+2m-1\}$, for each $n\in\IN$. (In fact,
$\lengths(p_n)$ equals $\{n+m-1,~n+2m-1\}$, for each $n\geq 1$.) We
have already observed that $p_n \models \psi_n$ but $q_n \models
\neg\psi_n$. Thus, by the above claim, the inequation $q_n
\preox p_n$ cannot be derived from $E$.
\medskip
\noindent
{\em Proof of the claim}. We use induction on the length of the
derivation of $p \preox q$ from $E$. The cases of reflexivity and
transitivity are trivial, using the soundness of $E$ with respect to
$=_{n}^{T}$ and that, by Lemma~\ref{Lem:norm}(a), $p =_{n}^{T} q$
implies $\lengths(p)=\lengths(q)$, for each $n\geq 2$. The case that
$p \preox q$ is a closed substantial substitution instance of an axiom
in $E$ has been dealt with by Lemma~\ref{Lem:crux}. What remains to
consider is closure under contexts: if the claim holds for $p \preox
q$ it needs to be shown for $p+r \preox q+r$, for every closed BCCSP
term $r$ over action $a$, and for $ap \preox aq$. The first of these
follows trivially by the observation that
\[
p+r \models \psi_n \text{ iff } p \models \psi_n \text{ or } r \models
\psi_n \enspace .
\]
For the second, the soundness of $E$ yields $p =_{n}^{T} q$. Using
the modal characterization of $=_{n}^{T}$, and that $\psi_{n-1}$ is
contained in $\mathcal{L}_{n}$,
we have that
\[
p \models
\psi_{n-1} \Leftrightarrow q \models \psi_{n-1} \enspace .
\]
Since $\psi_{n} = \may{a}\neg\psi_{n-1}$, it follows that
\[
ap \models
\psi_{n} \Leftrightarrow aq \models \psi_{n} \enspace ,
\]
which was to be shown. \hfill $\square$
\end{proof}
\begin{remark}
If $E$ contains the axiom $ax \preox ax+a$, which is sound for
$\presim_{2}$, we have that $E\vdash a^{2m} \preox
a^{m-1}(a^{m+1}+a)$. As $a^{m-1}(a^{m+1}+a) \models \psi_1$ but
$a^{2m} \not\models \psi_1$, the proof above, and the claim in
particular, does not apply to $\preo_{2}^{T}$ and $\presim_{2}$.
\end{remark}
Indeed, three different proofs appear to be needed to establish all of
our non-finite axiomatizability results. In particular, the proofs of
non-finite axiomatizability for the possible futures and 2-nested
simulation preorders are necessarily distinct, because if the set of
actions $A$ is a singleton, then there is a finite axiom system that
is sound for the possible futures preorder and complete for the
2-nested simulation preorder. This we now proceed to show.
Assume that $a$ is the only action, and consider the axiom system
$E_{\it PF}$ that contains the equations in
Table~\ref{tab:bccs}, and the inequation
\begin{eqnarray}
a(x+y) & \preaxiom & ax + ay \enspace . \label{eqn:pf}
\end{eqnarray}
It is not too hard to see that $E_{\it PF}$ is sound for the possible
futures preorder. In fact, for all closed BCCSP terms $p,q$,
\begin{itemize}
\item the terms $a(p+q)$ and $ap +aq$ have the same traces, and
\item if $a$ is the only action, then $p+q$ has the same set of traces
as either $p$ or $q$.
\end{itemize}
It follows that equation (\ref{eqn:pf}) is sound with respect to the
possible futures preorder, if $a$ is the only action.
We shall now show that $E_{\it PF}$ is complete for the
2-nested simulation preorder over the collection of closed BCCSP terms
over action $a$. The following lemma will play a key role in the proof
of this result.
\begin{lemma}\label{Lem:abs}
Let $p,q$ be closed BCCSP terms over action $a$. Assume that
$\depth(p) \leq \depth(q)$. Then
\[
E_{\it PF} \vdash q \preox q + p \enspace .
\]
\end{lemma}
\begin{proof}
By induction on the sum of the ``sizes'' of the closed BCCSP terms
$p,q$. We proceed by a case analysis on the form $p$ may take.
\begin{itemize}
\item {\em Case $p = \mathbf{0}$}. In this case, $E_{\it
PF} \vdash q \approx q + p$ follows immediately from axiom A4 in
Table~\ref{tab:bccs}.
\item {\em Case $p = ap'$, for some $p'$}. Assume that $q=\sum_{j\in
J}a q_j$, for some finite index set $J$ and closed terms $q_j$
over action $a$ ($j\in J$). Since $\depth(p) \leq \depth(q)$ by
our assumptions, there is an index $j\in J$ such that $\depth(p')
\leq \depth(q_j)$. By the inductive hypothesis, we have that
\[
E_{\it PF} \vdash q_j \preox q_j + p' \enspace .
\]
Hence,
\[
\begin{array}{llcll}
E_{\it PF} \vdash & aq_j & \preox & a(q_j + p') & \\
& & \preox & aq_j + ap' &
\quad\text{(By (\ref{eqn:pf}))} \enspace .
\end{array}
\]
The claim now follows using closure with respect to BCCSP contexts.
\item {\em Case $p = p_1 + p_2$, for some $p_1,p_2$ different from $\mathbf{0}$}. Since $\depth(p) \leq \depth(q)$ by
our assumptions, we have $\depth(p_i) \leq \depth(q)$ for $i=1,2$. By the inductive hypothesis, we may infer that
\[
E_{\it PF} \vdash q \preox q + p_i \enspace ,
\]
for $i=1,2$. Thus,
\[
E_{\it PF} \vdash q \preox q + p_2 \preox q + p_1 + p_2 \enspace ,
\]
which was to be shown.
\hfill $\square$
\end{itemize}
\end{proof}
We are now ready to prove that the axiom system $E_{\it PF}$
is complete for the 2-nested simulation preorder over closed BCCSP
terms over action $a$.
\begin{theorem}\label{Thm:compl-for-2n}
Let $p,q$ be closed BCCSP terms over action $a$. Assume that
$p \presim_2 q$. Then
\[
E_{\it PF} \vdash p \preox q \enspace .
\]
\end{theorem}
\begin{proof}
We prove the claim by induction on the depth of $p$. Let
$p=\sum_{i\in I}a p_i$ and $q=\sum_{j\in J}a q_j$, for some finite
index sets $I$ and $J$ and closed terms $p_i$ ($i\in I$) and $q_j$
($j\in J$) over action $a$. Note that, as $p \presim_2 q$, the
depth of $q$ is equal to that of $p$ (Lemma~\ref{Lem:norm}(b)).
Let $i\in I$. Then, since $p \presim_2 q$, there is an index ${j_i}$
such that $p_i \presim_2 q_{j_i}$ (Proposition~\ref{theo:nested}). Since
the depth of $p_i$ is smaller than that of $p$, by our inductive
hypothesis it follows that the inequation $p_i \preox q_{j_i}$ can be
proven from $E_{\it PF}$. Since this holds for each $i\in
I$, we have that
\[
E_{\it PF} \vdash p \preox \sum_{i\in I}a q_{j_i} \enspace .
\]
To conclude the proof, it suffices only to show that
\[
E_{\it PF} \vdash \sum_{i\in I}a q_{j_i} \preox q \enspace .
\]
To this end, note that, since $E_{\it PF}$ is sound with respect to
the possible futures preorder, and the inequation $p \preox
\sum_{i\in I}a q_{j_i}$ is derivable from it, the terms $p$ and
$\sum_{i\in I}a q_{j_i}$ have the same depth
(Lemma~\ref{Lem:norm}(b)). As previously observed, $p$ and $q$ also
have the same depth. Write now
\[
q = \sum_{i\in I}a q_{j_i} + r \enspace ,
\]
where $r$ is the sum of all the summands of $q$ not occurring in
$\sum_{i\in I}a q_{j_i}$. By the previous observations, we have that
\[
\depth(r) \leq \depth(q) = \depth(\sum_{i\in I}a q_{j_i}) \enspace .
\]
Lemma~\ref{Lem:abs} now yields that
\[
E_{\it PF} \vdash \sum_{i\in I}a q_{j_i} \preox \sum_{i\in
I}a q_{j_i} + r = q \enspace ,
\]
completing the proof.
\hfill $\square$
\end{proof}
\section{Finitely Based Approximations of Bisimulation Equivalence}\label{Sect:fb}
The results presented in the previous sections show that none of the
nested simulation and trace equivalences afford finite equational
axiomatizations over the language BCCSP, even in the presence of a
singleton action set. The only exceptions to this rule are the
$0$-nested and $1$-nested simulation and trace equivalences, which
happen to be the universal relation, simulation and trace equivalence.
Interestingly, however, as shown in~\cite{GrV92,HM85}, the
intersection of all of the $n$-nested simulation or trace
equivalences or preorders over image-finite labelled transition
systems, and therefore over the language BCCSP, is bisimulation
equivalence. Hennessy and Milner proved in~\cite{HM85} that
bisimulation equivalence is axiomatized over the language BCCSP by the
equations in Table~\ref{tab:bccs}. It follows that this fundamental
behavioural equivalence, albeit finitely based over BCCSP, is the
limit of sequences of relations that do not afford finite equational
axiomatizations themselves. This is by no means the only example from
process theory of a ``discontinuous'' property of a behavioural
equivalence---i.e., of a property that ``appears at the limit'', but
is not afforded by its finite approximations. Other examples of this
phenomenon may be found in, e.g., the study of decidability properties
of behavioural equivalences over classes of infinite state processes.
For instance, as shown
in~\cite{BBK1993,ChristensenHM1993,ChristensenHS1995}, bisimulation
equivalence is decidable over the languages BPA and BPP, but none of
the other notions of behavioural equivalence in the linear
time-branching time spectrum is---see, e.g., the
references~\cite{GrooteH1994,Huttel1994}.
It is a natural question to ask at this point whether bisimulation
equivalence over BCCSP is the limit of some sequence of finitely based
behavioural equivalences that have been presented in the literature.
We shall now argue that this does hold, provided that the set of
actions is finite.
As stated in Sect.~\ref{Sect:beh}, the $n$-nested trace equivalences were
introduced in \cite[p.~147]{HM85} as a a tool to define bisimulation
equivalence \cite{Mi89,Pa81}. In {\em op.~cit}.~Hennessy and Milner
introduced another sequence of relations that approximate bisimulation
equivalence. These were defined thus:
\begin{definition}\label{Def:naction}
For every $n\geq 0$, the relations $=_{n}^A$ are defined inductively over
closed BCCSP terms thus:
\begin{itemize}
\item $p =_{0}^{A} q$ for every $p,q$;
\item $p =_{n+1}^{A} q$ iff for every action
$a\in A$:
\begin{itemize}
\item if $p \mv{a} p'$ then there is a $q'$ such that $q \mv{a} q'$
and $p' =_{n}^{A} q'$, and
\item if $q \mv{a} q'$ then there is a $p'$ such that $p \mv{a} p'$
and $p' =_{n}^{A} q'$.
\end{itemize}
\end{itemize}
\end{definition}
Note that, unlike the $n$-nested trace equivalences $=_n^T$, the
relations $=_{n}^{A}$ explore the behaviour of BCCSP terms only up to
``depth $n$''. As shown by Hennessy and Milner, over image-finite
labelled transition systems, bisimulation equivalence is the
intersection of all of the relations $=_{n}^{A}$. Moreover, each of
the $=_{n}^{A}$ is preserved by the operators of Milner's CCS, and
{\em a fortiori} by those of BCCSP.
Our order of business will now be to offer a complete axiomatization
of the relations $=_{n}^{A}$ over closed BCCSP terms. Let
$\mathit{Ax}$ denote the axiom system in Table~\ref{tab:bccs}. We
shall now show how to inductively construct a family of axiom systems
$E_n$, for $n\geq 0$, with the following property:
\begin{theorem}\label{Thm:completeness}
Let $p,q$ be closed BCCSP terms. Then $p =_{n}^{A} q$ if, and only
if, $\mathit{Ax} \cup E_n \vdash p \approx q$.
\end{theorem}
The axiom systems $E_n$, for $n\geq 0$, will be finite, if so is
the set of actions $A$. In what follows we assume that the set of
variables is $\{x_1,x_2,\ldots\}$.
\begin{definition}\label{Def:En}
For each $n\geq 0$, we define the axiom system $E_n$ thus:
\begin{eqnarray*}
E_0 & = & \{ x_1 \approx x_2 \} \quad \text{and} \\
E_{n+1} & = & \{ a(t + x_{n+3}) \approx a(u + x_{n+3}) \mid a\in A,~(t\approx u)\in E_n \} \enspace .
\end{eqnarray*}
\end{definition}
Note that, if $A$ is a finite set set containing, say, $k$ actions,
then the axiom system $E_n$ contains $k^n$ equations, for
each $n\geq 0$. Moreover, observe for later use that, for each $n\geq
0$, the axioms in $E_n$ only use variables
$x_1,\ldots,x_{n+2}$.
We shall now show that Theorem~\ref{Thm:completeness} does hold for
the previously defined axiom systems $E_n$. Since the
soundness of each of the axioms in $E_n$
can easily be shown by induction on $n$, using the aforementioned
congruence properties of the relations $=_{n}^{A}$, we shall limit
ourselves to presenting a proof of the completeness of $\mathit{Ax}
\cup E_n$ with respect to $=_{n}^{A}$ over closed BCCSP
terms. The following lemma will be useful in such a proof.
\begin{lemma}\label{Lem:aux}
Let $n\geq 0$, and let $p,q$ be closed BCCSP terms. Assume that
$\mathit{Ax} \cup E_n\vdash p\approx q$. Then $\mathit{Ax} \cup E_{n+1}\vdash ap \approx aq$, for
each action $a\in A$.
\end{lemma}
\begin{proof}
Assume that $\mathit{Ax} \cup E_n\vdash p \approx q$, for some closed BCCSP terms
$p,q$. Recall that this means that there is a sequence $p_1 \approx \cdots \approx p_k$ ($k\geq
1$) such that
\begin{itemize}
\item $p =p_1$,
\item $q=p_k$ and
\item $p_i = C[\sigma(t)] \approx C[\sigma(u)] = p_{i+1}$ for some
closed substitution $\sigma$, context $C[~]$ and pair of terms $t,u$
with $t\approx u$ or $u\approx t$ an axiom in $\mathit{Ax} \cup E_n$
($1\leq i < k$).
\end{itemize}
We prove that $\mathit{Ax} \cup E_{n+1}\vdash ap \approx
aq$, for each action $a\in A$, by induction on $k$.
\begin{itemize}
\item {\em Base Case}: $k=1$. In this case we have that $p=q$. Thus
the equation $p \approx q$ is provable from $\mathit{Ax}$, and so is
$ap \approx aq$.
\item {\em Inductive Step}: $k>1$. By the inductive hypothesis, the
equation $ap \approx ap_{k-1}$ is provable from the axiom system
$\mathit{Ax} \cup E_{n+1}$. Since $ap_k = aq$, to complete
the proof, we are therefore left to prove that
\begin{equation}\label{Eqn:temp}
\mathit{Ax} \cup
E_{n+1} ~\vdash~ap_{k-1} \approx a p_k \enspace .
\end{equation}
To this end, recall that
\begin{itemize}
\item $p_{k-1} = C[\sigma(t)]$ and
\item $p_k = C[\sigma(u)]$,
\end{itemize}
for some closed substitution $\sigma$, context $C[~]$ and pair of
terms $t,u$ with $t\approx u$ or $u\approx t$ an axiom in $\mathit{Ax}
\cup E_n$. In case an axiom from $\mathit{Ax}$ or its symmetric
counterpart was used, (\ref{Eqn:temp}) follows immediately from the
rule of closure under BCCSP contexts. The proof for the case when
$t\approx u$ is an axiom in $E_n$ proceeds by a case analysis on the
form of the context $C[~]$.
\begin{itemize}
\item {\em Case 1}: Suppose $C[~]$ is of the form $C'[b([~] + r)]$,
for some action $b$ and closed term $r$.
In this case, it is sufficient to show that
\[
\mathit{Ax} \cup E_{n+1}~\vdash~ b(\sigma(t) + r) \approx b(\sigma(u) + r)
\]
as (\ref{Eqn:temp}) will then follow by applying the rule of closure
under BCCSP contexts repeatedly.
To this end, let $\sigma'$ be the closed substitution that maps
variable $x_{n+3}$ to $r$, and acts like $\sigma$ on all of the
other variables. Using the axioms in $\mathit{Ax} \cup
E_{n+1}$, we have that
\begin{eqnarray*}
b(\sigma(t) + r) & \approx & \sigma'(b (t + x_{n+3})) \quad \text{(as $x_{n+3}\not\in\var(t)$)} \\
&\approx & \sigma'(b (u + x_{n+3})) \quad \text{(as $b(t + x_{n+3}) \approx b(u + x_{n+3}) \in E_{n+1}$)}\\
&\approx & b(\sigma(u) + r)\quad \text{(as $x_{n+3}\not\in\var(u)$)} \enspace ,
\end{eqnarray*}
which was to be shown.
\item {\em Case 2}: Suppose $C[~]$ is of the form $[~] + r$, for some closed term $r$.
In this case, letting $\sigma'$ be defined as above, and using the
axioms in $\mathit{Ax} \cup E_{n+1}$, we have that
\begin{eqnarray*}
ap_{k-1} & \approx & a(\sigma(t) + r) \\
& \approx & \sigma'(a(t + x_{n+3})) \quad \text{(as $x_{n+3}\not\in\var(t)$)} \\
&\approx & \sigma'(a (u + x_{n+3})) \quad \text{(as $a(t + x_{n+3}) \approx a(u + x_{n+3}) \in E_{n+1}$)}\\
&\approx & a(\sigma(u) + r)\quad \text{(as $x_{n+3}\not\in\var(u)$)} \\
&\approx & ap_k \enspace ,
\end{eqnarray*}
which was to be shown.
\end{itemize}
The remaining case, viz.~when $u\approx t$ an axiom in $E_n$, is
similar. \hfill $\square$
\end{itemize}
\end{proof}
We are now ready to establish the completeness of $\mathit{Ax} \cup
E_n$ with respect to $=_{n}^{A}$ over closed BCCSP terms,
for each $n\geq 0$.
The proof is by induction on $n$. The base case is trivial since the
equation $x_1 \approx x_2$ can be used to prove every (closed) equation.
For the inductive step, assume that $\mathit{Ax} \cup E_n$
is complete with respect to $=_{n}^{A}$ over closed BCCSP terms, and
that $p =_{n+1}^{A} q$ holds for closed terms $p,q$. We shall now
argue that the equation $p\approx q$ can be derived from the axiom
system $\mathit{Ax} \cup E_{n+1}$. Let $p=\sum_{i\in I}a_i
p_i$ and $q=\sum_{j\in J}b_j q_j$, for some finite index sets $I$ and
$J$ and closed terms $a_i p_i$ ($i\in I$) and $b_j q_j$ ($j\in J$).
Our order of business will now be to show that
\[
\mathit{Ax} \cup E_{n+1} \vdash p \approx p + q \approx q
\enspace .
\]
By symmetry, it is sufficient to show that the equation $p + q \approx
q$ is derivable from $\mathit{Ax} \cup E_{n+1}$. To this
end, let $i\in I$. Then, since $p =_{n+1}^{A} q$, there is an index
${j_i}$ such that $a_i=b_{j_i}$ and $p_i =_{n}^{A} q_{j_i}$. Since the
axiom system $\mathit{Ax} \cup E_n$ is complete with respect
to $=_{n}^{A}$ by our inductive hypothesis, it follows that the
equation $p_i \approx q_{j_i}$ can be proven from $\mathit{Ax} \cup
E_n$. By Lemma~\ref{Lem:aux}, the equation $a_i p_i \approx
b_{j_i} q_{j_i}$ can be derived from $\mathit{Ax} \cup
E_{n+1}$. As this holds for each index $i\in I$, it
follows that $p + q \approx q$ is derivable from $\mathit{Ax} \cup
E_{n+1}$, which was to be shown.
The proof of Theorem~\ref{Thm:completeness} is now complete.
\paragraph{Acknowledgments}
The work reported in this paper was partly carried out while Luca
Aceto was an invited professor at Reykjav\'{\i}k University, Rob van
Glabbeek was visiting researcher at CWI and associate professor at the
National ICT Australia, and Anna Ing\'olfsd\'ottir was at Iceland
Genomics Corporation. They thank these institutions for their
hospitality and the excellent working conditions. Luca Aceto is
grateful to Sverrir Thorvaldsson and his group at deCODE Genetics for
their logistic support during the final stages in the preparation of
this paper.
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\end{document}
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to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
\let\ts\,
\setlength\leftmargini {17\p@}
\setlength\leftmargin {\leftmargini}
\setlength\leftmarginii {\leftmargini}
\setlength\leftmarginiii {\leftmargini}
\setlength\leftmarginiv {\leftmargini}
\setlength \labelsep {.5em}
\setlength \labelwidth{\leftmargini}
\addtolength\labelwidth{-\labelsep}
\def\@listI{\leftmargin\leftmargini
\parsep 0\p@ \@plus1\p@ \@minus\p@
\topsep 8\p@ \@plus2\p@ \@minus4\p@
\itemsep0\p@}
\let\@listi\@listI
\@listi
\def\@listii {\leftmargin\leftmarginii
\labelwidth\leftmarginii
\advance\labelwidth-\labelsep
\topsep 0\p@ \@plus2\p@ \@minus\p@}
\def\@listiii{\leftmargin\leftmarginiii
\labelwidth\leftmarginiii
\advance\labelwidth-\labelsep
\topsep 0\p@ \@plus\p@\@minus\p@
\parsep \z@
\partopsep \p@ \@plus\z@ \@minus\p@}
\renewcommand\labelitemi{\normalfont\bfseries --}
\renewcommand\labelitemii{$\m@th\bullet$}
\setlength\arraycolsep{1.4\p@}
\setlength\tabcolsep{1.4\p@}
\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
{{\contentsname}}}
\def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
\def\lastand{\ifnum\value{auco}=2\relax
\unskip{} \andname\
\else
\unskip \lastandname\
\fi}%
\def\and{\stepcounter{@auth}\relax
\ifnum\value{@auth}=\value{auco}%
\lastand
\else
\unskip,
\fi}%
\@starttoc{toc}\if@restonecol\twocolumn\fi}
\def\l@part#1#2{\addpenalty{\@secpenalty}%
\addvspace{2em plus\p@}% % space above part line
\begingroup
\parindent \z@
\rightskip \z@ plus 5em
\hrule\vskip5pt
\large % same size as for a contribution heading
\bfseries\boldmath % set line in boldface
\leavevmode % TeX command to enter horizontal mode.
#1\par
\vskip5pt
\hrule
\vskip1pt
\nobreak % Never break after part entry
\endgroup}
\def\@dotsep{2}
\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
{chapter.\thechapter}\fi}
\def\addnumcontentsmark#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
{\thechapter}#3}{\thepage}\hyperhrefextend}}
\def\addcontentsmark#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
\def\addcontentsmarkwop#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
\def\@adcmk[#1]{\ifcase #1 \or
\def\@gtempa{\addnumcontentsmark}%
\or \def\@gtempa{\addcontentsmark}%
\or \def\@gtempa{\addcontentsmarkwop}%
\fi\@gtempa{toc}{chapter}}
\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
\vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
\parindent \z@ \rightskip \@tocrmarg
\advance\rightskip by 0pt plus 2cm
\parfillskip -\rightskip \pretolerance=10000
\leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
{\large\bfseries\boldmath#1}\ifx0#2\hfil\null
\else
\nobreak
\leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
\@dotsep mu$}\hfill
\nobreak\hbox to\@pnumwidth{\hss #2}%
\fi\par
\penalty\@highpenalty \endgroup}
\def\l@title#1#2{\addpenalty{-\@highpenalty}
\addvspace{8pt plus 1pt}
\@tempdima \z@
\begingroup
\parindent \z@ \rightskip \@tocrmarg
\advance\rightskip by 0pt plus 2cm
\parfillskip -\rightskip \pretolerance=10000
\leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
#1\nobreak
\leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
\@dotsep mu$}\hfill
\nobreak\hbox to\@pnumwidth{\hss #2}\par
\penalty\@highpenalty \endgroup}
\def\l@author#1#2{\addpenalty{\@highpenalty}
\@tempdima=\z@ %15\p@
\begingroup
\parindent \z@ \rightskip \@tocrmarg
\advance\rightskip by 0pt plus 2cm
\pretolerance=10000
\leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
\textit{#1}\par
\penalty\@highpenalty \endgroup}
\setcounter{tocdepth}{0}
\newdimen\tocchpnum
\newdimen\tocsecnum
\newdimen\tocsectotal
\newdimen\tocsubsecnum
\newdimen\tocsubsectotal
\newdimen\tocsubsubsecnum
\newdimen\tocsubsubsectotal
\newdimen\tocparanum
\newdimen\tocparatotal
\newdimen\tocsubparanum
\tocchpnum=\z@ % no chapter numbers
\tocsecnum=15\p@ % section 88. plus 2.222pt
\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt
\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt
\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt
\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt
\def\calctocindent{%
\tocsectotal=\tocchpnum
\advance\tocsectotal by\tocsecnum
\tocsubsectotal=\tocsectotal
\advance\tocsubsectotal by\tocsubsecnum
\tocsubsubsectotal=\tocsubsectotal
\advance\tocsubsubsectotal by\tocsubsubsecnum
\tocparatotal=\tocsubsubsectotal
\advance\tocparatotal by\tocparanum}
\calctocindent
\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
\fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
\@starttoc{lof}\if@restonecol\twocolumn\fi}
\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
\fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
\@starttoc{lot}\if@restonecol\twocolumn\fi}
\let\l@table\l@figure
\renewcommand\listoffigures{%
\section*{\listfigurename
\@mkboth{\listfigurename}{\listfigurename}}%
\@starttoc{lof}%
}
\renewcommand\listoftables{%
\section*{\listtablename
\@mkboth{\listtablename}{\listtablename}}%
\@starttoc{lot}%
}
\ifx\oribibl\undefined
\ifx\citeauthoryear\undefined
\renewenvironment{thebibliography}[1]
{\section*{\refname}
\def\@biblabel##1{##1.}
\small
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\if@openbib
\advance\leftmargin\bibindent
\itemindent -\bibindent
\listparindent \itemindent
\parsep \z@
\fi
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\if@openbib
\renewcommand\newblock{\par}%
\else
\renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
\fi
\sloppy\clubpenalty4000\widowpenalty4000%
\sfcode`\.=\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
{\let\protect\noexpand\immediate
\write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
\newcount\@tempcntc
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
\@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
{\@ifundefined
{b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
?}\@warning
{Citation `\@citeb' on page \thepage \space undefined}}%
{\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
\ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
\@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
\else
\advance\@tempcntb\@ne
\ifnum\@tempcntb=\@tempcntc
\else\advance\@tempcntb\m@ne\@citeo
\@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
\@citea\def\@citea{,\,\hskip\z@skip}%
\ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
{\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
\def\@citea{--}\fi
\advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
\else
\renewenvironment{thebibliography}[1]
{\section*{\refname}
\small
\list{}%
{\settowidth\labelwidth{}%
\leftmargin\parindent
\itemindent=-\parindent
\labelsep=\z@
\if@openbib
\advance\leftmargin\bibindent
\itemindent -\bibindent
\listparindent \itemindent
\parsep \z@
\fi
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{}}%
\if@openbib
\renewcommand\newblock{\par}%
\else
\renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
\fi
\sloppy\clubpenalty4000\widowpenalty4000%
\sfcode`\.=\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\def\@cite#1{#1}%
\def\@lbibitem[#1]#2{\item[]\if@filesw
{\def\protect##1{\string ##1\space}\immediate
\write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
\fi
\else
\@cons\@openbib@code{\noexpand\small}
\fi
\def\idxquad{\hskip 10\p@}% space that divides entry from number
\def\@idxitem{\par\hangindent 10\p@}
\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
\noindent\hangindent\wd0\box0}% index entry
\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
\noindent\hangindent\wd0\box0}% order index entry
\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
\renewenvironment{theindex}
{\@mkboth{\indexname}{\indexname}%
\thispagestyle{empty}\parindent\z@
\parskip\z@ \@plus .3\p@\relax
\let\item\par
\def\,{\relax\ifmmode\mskip\thinmuskip
\else\hskip0.2em\ignorespaces\fi}%
\normalfont\small
\begin{multicols}{2}[\@makeschapterhead{\indexname}]%
}
{\end{multicols}}
\renewcommand\footnoterule{%
\kern-3\p@
\hrule\@width 2truecm
\kern2.6\p@}
\newdimen\fnindent
\fnindent1em
\long\def\@makefntext#1{%
\parindent \fnindent%
\leftskip \fnindent%
\noindent
\llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
\long\def\@makecaption#1#2{%
\vskip\abovecaptionskip
\sbox\@tempboxa{{\bfseries #1.} #2}%
\ifdim \wd\@tempboxa >\hsize
{\bfseries #1.} #2\par
\else
\global \@minipagefalse
\hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
\fi
\vskip\belowcaptionskip}
\def\fps@figure{htbp}
\def\fnum@figure{\figurename\thinspace\thefigure}
\def \@floatboxreset {%
\reset@font
\small
\@setnobreak
\@setminipage
}
\def\fps@table{htbp}
\def\fnum@table{\tablename~\thetable}
\renewenvironment{table}
{\setlength\abovecaptionskip{0\p@}%
\setlength\belowcaptionskip{10\p@}%
\@float{table}}
{\end@float}
\renewenvironment{table*}
{\setlength\abovecaptionskip{0\p@}%
\setlength\belowcaptionskip{10\p@}%
\@dblfloat{table}}
{\end@dblfloat}
\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
ext@#1\endcsname}{#1}{\protect\numberline{\csname
the#1\endcsname}{\ignorespaces #2}}\begingroup
\@parboxrestore
\@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
\endgroup}
% LaTeX does not provide a command to enter the authors institute
% addresses. The \institute command is defined here.
\newcounter{@inst}
\newcounter{@auth}
\newcounter{auco}
\newdimen\instindent
\newbox\authrun
\newtoks\authorrunning
\newtoks\tocauthor
\newbox\titrun
\newtoks\titlerunning
\newtoks\toctitle
\def\clearheadinfo{\gdef\@author{No Author Given}%
\gdef\@title{No Title Given}%
\gdef\@subtitle{}%
\gdef\@institute{No Institute Given}%
\gdef\@thanks{}%
\global\titlerunning={}\global\authorrunning={}%
\global\toctitle={}\global\tocauthor={}}
\def\institute#1{\gdef\@institute{#1}}
\def\institutename{\par
\begingroup
\parskip=\z@
\parindent=\z@
\setcounter{@inst}{1}%
\def\and{\par\stepcounter{@inst}%
\noindent$^{\the@inst}$\enspace\ignorespaces}%
\setbox0=\vbox{\def\thanks##1{}\@institute}%
\ifnum\c@@inst=1\relax
\gdef\fnnstart{0}%
\else
\xdef\fnnstart{\c@@inst}%
\setcounter{@inst}{1}%
\noindent$^{\the@inst}$\enspace
\fi
\ignorespaces
\@institute\par
\endgroup}
\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
{\star\star\star}\or \dagger\or \ddagger\or
\mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
\or \ddagger\ddagger \else\@ctrerr\fi}}
\def\inst#1{\unskip$^{#1}$}
\def\fnmsep{\unskip$^,$}
\def\email#1{{\tt#1}}
\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
\@ifpackageloaded{babel}{%
\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
}{\switcht@@therlang}%
}
\def\homedir{\~{ }}
\def\subtitle#1{\gdef\@subtitle{#1}}
\clearheadinfo
\renewcommand\maketitle{\newpage
\refstepcounter{chapter}%
\stepcounter{section}%
\setcounter{section}{0}%
\setcounter{subsection}{0}%
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{equation}{0}
\setcounter{footnote}{0}%
\begingroup
\parindent=\z@
\renewcommand\thefootnote{\@fnsymbol\c@footnote}%
\if@twocolumn
\ifnum \col@number=\@ne
\@maketitle
\else
\twocolumn[\@maketitle]%
\fi
\else
\newpage
\global\@topnum\z@ % Prevents figures from going at top of page.
\@maketitle
\fi
\thispagestyle{empty}\@thanks
%
\def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
\def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
\instindent=\hsize
\advance\instindent by-\headlineindent
\if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
\addcontentsline{toc}{title}{\the\toctitle}\fi
\if@runhead
\if!\the\titlerunning!\else
\edef\@title{\the\titlerunning}%
\fi
\global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
\ifdim\wd\titrun>\instindent
\typeout{Title too long for running head. Please supply}%
\typeout{a shorter form with \string\titlerunning\space prior to
\string\maketitle}%
\global\setbox\titrun=\hbox{\small\rm
Title Suppressed Due to Excessive Length}%
\fi
\xdef\@title{\copy\titrun}%
\fi
%
\if!\the\tocauthor!\relax
{\def\and{\noexpand\protect\noexpand\and}%
\protected@xdef\toc@uthor{\@author}}%
\else
\def\\{\noexpand\protect\noexpand\newline}%
\protected@xdef\scratch{\the\tocauthor}%
\protected@xdef\toc@uthor{\scratch}%
\fi
\addcontentsline{toc}{author}{\toc@uthor}%
\if@runhead
\if!\the\authorrunning!
\value{@inst}=\value{@auth}%
\setcounter{@auth}{1}%
\else
\edef\@author{\the\authorrunning}%
\fi
\global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
\ifdim\wd\authrun>\instindent
\typeout{Names of authors too long for running head. Please supply}%
\typeout{a shorter form with \string\authorrunning\space prior to
\string\maketitle}%
\global\setbox\authrun=\hbox{\small\rm
Authors Suppressed Due to Excessive Length}%
\fi
\xdef\@author{\copy\authrun}%
\markboth{\@author}{\@title}%
\fi
\endgroup
\setcounter{footnote}{\fnnstart}%
\clearheadinfo}
%
\def\@maketitle{\newpage
\markboth{}{}%
\def\lastand{\ifnum\value{@inst}=2\relax
\unskip{} \andname\
\else
\unskip \lastandname\
\fi}%
\def\and{\stepcounter{@auth}\relax
\ifnum\value{@auth}=\value{@inst}%
\lastand
\else
\unskip,
\fi}%
\begin{center}%
\let\newline\\
{\Large \bfseries\boldmath
\pretolerance=10000
\@title \par}\vskip .8cm
\if!\@subtitle!\else {\large \bfseries\boldmath
\vskip -.65cm
\pretolerance=10000
\@subtitle \par}\vskip .8cm\fi
\setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
\def\thanks##1{}\@author}%
\global\value{@inst}=\value{@auth}%
\global\value{auco}=\value{@auth}%
\setcounter{@auth}{1}%
{\lineskip .5em
\noindent\ignorespaces
\@author\vskip.35cm}
{\small\institutename}
\end{center}%
}
% definition of the "\spnewtheorem" command.
%
% Usage:
%
% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
%
% New is "cap_font" and "body_font". It stands for
% fontdefinition of the caption and the text itself.
%
% "\spnewtheorem*" gives a theorem without number.
%
% A defined spnewthoerem environment is used as described
% by Lamport.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@thmcountersep{}
\def\@thmcounterend{.}
\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
% definition of \spnewtheorem with number
\def\@spnthm#1#2{%
\@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
{\@definecounter{#1}\@addtoreset{#1}{#3}%
\expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
\csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
\expandafter\xdef\csname #1name\endcsname{#2}%
\global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
\global\@namedef{end#1}{\@endtheorem}}}
\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
{\@definecounter{#1}%
\expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
\expandafter\xdef\csname #1name\endcsname{#2}%
\global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
\global\@namedef{end#1}{\@endtheorem}}}
\def\@spothm#1[#2]#3#4#5{%
\@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
{\expandafter\@ifdefinable\csname #1\endcsname
{\global\@namedef{the#1}{\@nameuse{the#2}}%
\expandafter\xdef\csname #1name\endcsname{#3}%
\global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
\global\@namedef{end#1}{\@endtheorem}}}}
\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
\refstepcounter{#1}%
\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
\ignorespaces}
\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
\def\@spbegintheorem#1#2#3#4{\trivlist
\item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
\item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
% definition of \spnewtheorem* without number
\def\@sthm#1#2{\@Ynthm{#1}{#2}}
\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
{\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
\expandafter\xdef\csname #1name\endcsname{#2}%
\global\@namedef{end#1}{\@endtheorem}}}
\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
{#4}{#2}{#3}\ignorespaces}
\def\@Begintheorem#1#2#3{#3\trivlist
\item[\hskip\labelsep{#2#1\@thmcounterend}]}
\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
\item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
\if@envcntsect
\def\@thmcountersep{.}
\spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
\else
\spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
\if@envcntreset
\@addtoreset{theorem}{section}
\else
\@addtoreset{theorem}{chapter}
\fi
\fi
%definition of divers theorem environments
\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
\if@envcntsame % alle Umgebungen wie Theorem.
\def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
\else % alle Umgebungen mit eigenem Zaehler
\if@envcntsect % mit section numeriert
\def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
\else % nicht mit section numeriert
\if@envcntreset
\def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
\@addtoreset{#1}{section}}
\else
\def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
\@addtoreset{#1}{chapter}}%
\fi
\fi
\fi
\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
%\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
\spn@wtheorem{example}{Example}{\bfseries}{\rmfamily}
\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
%\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
\spn@wtheorem{remark}{Remark}{\bfseries}{\rmfamily}
\def\@takefromreset#1#2{%
\def\@tempa{#1}%
\let\@tempd\@elt
\def\@elt##1{%
\def\@tempb{##1}%
\ifx\@tempa\@tempb\else
\@addtoreset{##1}{#2}%
\fi}%
\expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
\expandafter\def\csname cl@#2\endcsname{}%
\@tempc
\let\@elt\@tempd}
\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
\item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
\def\@Opargbegintheorem##1##2##3##4{##4\trivlist
\item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
}
\renewenvironment{abstract}{%
\list{}{\advance\topsep by0.35cm\relax\small
\leftmargin=1cm
\labelwidth=\z@
\listparindent=\z@
\itemindent\listparindent
\rightmargin\leftmargin}\item[\hskip\labelsep
\bfseries\abstractname]}
{\endlist}
\newdimen\headlineindent % dimension for space between
\headlineindent=1.166cm % number and text of headings.
\def\ps@headings{\let\@mkboth\@gobbletwo
\let\@oddfoot\@empty\let\@evenfoot\@empty
\def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
\leftmark\hfil}
\def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
\llap{\thepage}}
\def\chaptermark##1{}%
\def\sectionmark##1{}%
\def\subsectionmark##1{}}
\def\ps@titlepage{\let\@mkboth\@gobbletwo
\let\@oddfoot\@empty\let\@evenfoot\@empty
\def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
\hfil}
\def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
\llap{\thepage}}
\def\chaptermark##1{}%
\def\sectionmark##1{}%
\def\subsectionmark##1{}}
\if@runhead\ps@headings\else
\ps@empty\fi
\setlength\arraycolsep{1.4\p@}
\setlength\tabcolsep{1.4\p@}
\endinput
%end of file llncs.cls