From 1fb5e982ae14dab97dec12267087199de31ddbe7 Mon Sep 17 00:00:00 2001 From: rensink Date: Fri, 30 Aug 2024 17:42:20 +0200 Subject: [PATCH] Further refinement of the exploration doc --- .gitignore | 1 + doc/exploration/explore.tex | 202 +++++++++++++++++++------------- doc/exploration/figs/figs.pptx | Bin 36543 -> 36951 bytes doc/exploration/figs/s-venn.png | Bin 0 -> 169711 bytes doc/exploration/figs/t-venn.png | Bin 0 -> 53122 bytes doc/exploration/figs/venn.png | Bin 169454 -> 0 bytes doc/exploration/preamble.sty | 47 +++++--- doc/exploration/texput.log | 21 ---- 8 files changed, 151 insertions(+), 120 deletions(-) create mode 100644 doc/exploration/figs/s-venn.png create mode 100644 doc/exploration/figs/t-venn.png delete mode 100644 doc/exploration/figs/venn.png delete mode 100644 doc/exploration/texput.log diff --git a/.gitignore b/.gitignore index b921a0b21..2086f42cc 100644 --- a/.gitignore +++ b/.gitignore @@ -4,3 +4,4 @@ *.log /doc/exploration/explore.pdf /doc/exploration/*.gz +/doc/exploration/texput.log diff --git a/doc/exploration/explore.tex b/doc/exploration/explore.tex index eda8d121c..f98f94286 100644 --- a/doc/exploration/explore.tex +++ b/doc/exploration/explore.tex @@ -11,164 +11,202 @@ \section*{Terminology} \begin{description} \item[Absence] Equivalent to \emph{absent depth}. -\item[Absent depth] The \emph{absent depth} of a state is the minimal transient depth of that state and all its transitive successors. It follows that the \emph{absent depth} is smaller than or equal to the \emph{transient depth}. As long as a state is not \emph{complete}, its \emph{absent depth} may decrease as a consequence of further exploration. +\item[Absent depth (S)] The \emph{absent depth} of a state is the minimal transient depth of that state and all its transitive successors. It follows that the \emph{absent depth} is smaller than or equal to the \emph{transient depth}. As long as a state is not \emph{complete}, its \emph{absent depth} may decrease as a consequence of further exploration. -\item[Absent] A state or transition is \emph{absent} if it should not be considered to be part of the state space. In particular, a state is \emph{absent} if it is \emph{complete} and has positive \emph{absent depth}; a transition is \emph{absent} if its target state is \emph{absent}. It follows that an \emph{absent} state is always \emph{transient}. Whether or not a state or transition is \emph{absent} is independent of whether or not it is \emph{inner}. +\item[Absent (S,T)] A state or transition is \emph{absent} if it should not be considered to be part of the state space. In particular, a state is \emph{absent} if it is \emph{complete} and has positive \emph{absent depth}; a transition is \emph{absent} if its target state is \emph{absent}. It follows that an \emph{absent} state is always \emph{transient}. Whether or not a state or transition is \emph{absent} is independent of whether or not it is \emph{inner}. -\item[Closed] A state is \emph{closed} if it is not \emph{open}. A \emph{closed} state may or may not be \emph{complete}. +\item[Atomic (T)] A transition is \emph{atomic} if it is \emph{outer} and consists of a single step, meaning that its source and target states are \emph{steady}. -\item[Complete] A state is \emph{complete} if it is \emph{closed} and all transitive successors up to (but not necessarily including) the first \emph{steady} state are also \emph{closed}. This means that the status of the state is fully known; in particular, it is possible, on the basis of its successors, to decide whether the state is \emph{absent}. +\item[Closed (S)] A state is \emph{closed} if it is not \emph{open}. A \emph{closed} state may or may not be \emph{complete}. -\item[Public] A state or transition is \emph{public} if it is both \emph{outer} and \emph{present} (or, in other words, if it is neither \emph{inner} nor \emph{absent}). +\item[Complete (S)] A state is \emph{complete} if it is \emph{closed} and all transitive successors up to (but not necessarily including) the first \emph{steady} state are also \emph{closed}. This means that the status of the state is fully known; in particular, it is possible, on the basis of its successors, to decide whether the state is \emph{absent}. -\item[Final] A state is \emph{final} if exploration is considered to terminate after reaching it. For instance, if two control blocks are concatenated, the second comes into play after the exploration of the first has reached a final state; also, if a state space is wrapped in an atomic block, the only \emph{steady} states, apart from the \emph{initial} state, are the \emph{final} ones. In other words, \emph{final} has the typical meaning in (regular) automata. A \emph{final} state is certainly \emph{closed} and \emph{steady}; it may or may not have successors. +\item[Public (S,T)] A state or transition is \emph{public} if it is both \emph{outer} and \emph{present} (or, in other words, if it is neither \emph{inner} nor \emph{absent}). -\item[Initial] A state is \emph{initial} if it is the start state for the exploration. Its \emph{prime} is \emph{steady}. +\item[Final (S)] A state is \emph{final} if exploration is considered to terminate after reaching it. For instance, if two control blocks are concatenated, the second comes into play after the exploration of the first has reached a final state; also, if a state space is wrapped in an atomic block, the only \emph{steady} states, apart from the \emph{initial} state, are the \emph{final} ones. In other words, \emph{final} has the typical meaning in (regular) automata. A \emph{final} state is certainly \emph{closed} and \emph{steady}; it may or may not have successors. -\item[Incomplete] A state is \emph{incomplete} if it is not \emph{complete}. +\item[Initial (S)] A state is \emph{initial} if it is the start state for the exploration. Its \emph{prime} is \emph{steady}. -\item[Inner] A state is \emph{inner} if it is inside the atomic block that makes up the body of a recipe, and a transition is \emph{inner} if it is a step in the execution of a recipe. Hence, an \emph{inner} state is certainly \emph{transient}, but the inverse does not hold. An \emph{inner} state that is not \emph{closed} may still evolve into an \emph{outer} one as a consequence of further exploration. +\item[Incomplete (S)] A state is \emph{incomplete} if it is not \emph{complete}. -\item[Open] A state is \emph{open} if its direct outgoing transitions are not fully known. It follows that an open state can become \emph{closed} through further exploration of its outgoing transitions. +\item[Inner (S,T)] A state is \emph{inner} if it is inside the atomic block that makes up the body of a recipe, and a transition is \emph{inner} if it is a step in the execution of a recipe. Hence, an \emph{inner} state is certainly \emph{transient}, but the inverse does not hold. An \emph{inner} state that is not \emph{closed} may still evolve into an \emph{outer} one as a consequence of further exploration. -\item[Outer] A state or transition is \emph{outer} if it is not \emph{inner}. +\item[Open (S)] A state is \emph{open} if its direct outgoing transitions are not fully known. It follows that an open state can become \emph{closed} through further exploration of its outgoing transitions. -\item[Present.] A state or transition is \emph{present} if it is not \emph{absent}. For a state, this means that it either has \emph{absent depth} zero (in which case it will certainly remain \emph{present} under further exploration} or it is \emph{incomplete} (in which case it may become \emph{absent} if, upon becoming \emph{complete}, it has positive \emph{absent depth}). +\item[Outer (S,T)] A state or transition is \emph{outer} if it is not \emph{inner}. -\item[Prime] The \emph{prime} of a state is the version of that state when it is just discovered, and no outgoing transitions have been explored. Hence, the \emph{prime} is \emph{open}, and may be \tmph{transient} even though (because of further exploration) the state itself has become \emph{steady} because its \emph{transient depth} has decreased to zero. +\item[Partial (T)] A transition is \emph{partial} if it is part of an atomic block. Apart from the \emph{inner} transitions, the \emph{partial} ones include single-step recipe executions. -\item[Public] A state or transition is \emph{public} if it is both \emph{outer} and \emph{present}. +\item[Present (S,T)] A state or transition is \emph{present} if it is not \emph{absent}. For a state, this means that it either has \emph{absent depth} zero (in which case it will certainly remain \emph{present} under further exploration) or it is \emph{incomplete} (in which case it may become \emph{absent} if, upon becoming \emph{complete}, it has positive \emph{absent depth}). -\item[Steady] A state is \emph{steady} if it is not \emph{transient}. +\item[Prime (S)] The \emph{prime} of a state is the version of that state when it is just discovered, and no outgoing transitions have been explored. Hence, the \emph{prime} is \emph{open}, and may be \emph{transient} even though (because of further exploration) the state itself has become \emph{steady} because its \emph{transient depth} has decreased to zero. -\item[Transience] Equivalent to \emph{transient depth}. +\item[Public (S,T)] A state or transition is \emph{public} if it is both \emph{outer} and \emph{present}. -\item[Transient] A state is \emph{transient} if it is inside an atomic block, meaning that it has a positive \emph{transient depth}. A \emph{transient} state may or may not be \emph{inner}. A transient state that is \emph{open} may still evolve into a \emph{steady} state as a consequence of further exploration. +\item[Steady (S)] A state is \emph{steady} if it is not \emph{transient}. -\item[Transient depth] The \emph{transient depth} of a state is the number of (nested) atomic blocks that it is inside of. A state is \emph{steady} if and only if its \emph{transient depth} is zero; otherwise it is \emph{transient}. As long as a state is not \emph{closed}, its \emph{transient depth} may decrease as a consequence of further exploration. +\item[Transience (S)] Equivalent to \emph{transient depth}. + +\item[Transient (S)] A state is \emph{transient} if it is inside an atomic block, meaning that it has a positive \emph{transient depth}. A \emph{transient} state may or may not be \emph{inner}. A transient state that is \emph{open} may still evolve into a \emph{steady} state as a consequence of further exploration. + +\item[Transient depth (S)] The \emph{transient depth} of a state is the number of (nested) atomic blocks that it is inside of. A state is \emph{steady} if and only if its \emph{transient depth} is zero; otherwise it is \emph{transient}. As long as a state is not \emph{closed}, its \emph{transient depth} may decrease as a consequence of further exploration. \end{description} Figure~\ref{fig:venn} shows these different concepts in relation to each other. \begin{figure} \centering -\includegraphics[scale=.5]{figs/venn} -\caption{Properties of states} +\begin{subfigure}{0.45\textwidth} +\centering +\includegraphics[scale=.4]{figs/s-venn} +\caption{Stats} +\end{subfigure}% +\begin{subfigure}{0.3\textwidth} +\centering +\includegraphics[scale=.4]{figs/t-venn} + +\bigskip +\caption{Transitions} +\end{subfigure}% +% +\begin{subfigure}{.25\textwidth} +\scalebox{.8}{$\begin{array}[b]{@{}r@{{\;\equiv\;}}l@{}} +\mathit{inner} & \neg\mathit{outer} \\ +\mathit{incomplete} & \neg\mathit{complete} \\ +\mathit{open} & \neg\mathit{closed} \\ +\mathit{partial} & \neg\mathit{atomic} \\ +\mathit{present} & \neg\mathit{absent} \\ +\mathit{public} & \mathit{outer} \\ +\multicolumn{2}{r}{{}\wedge\mathit{present}} \\ +\mathit{transient} & \neg\mathit{steady} +\end{array}$} + +\bigskip +\phantomcaption +\end{subfigure} +\caption{Properties of states and transitions} \label{fig:venn} \end{figure} -\section*{Formalisation} + +\section*{State spaces} \medskip\noindent We globally assume a set of labels $A$. \medskip\noindent -A state space is a tuple $\cS=\tupof{S,\iota,{\ra},{\up},\TDepth}$ with +A \emph{state space fragment} is a tuple $\cS=\tupof{S,\iota,{\ra},{\up},\TLevel,\Open}$ with \begin{itemize} \item $S$ a set of states; \item $\iota\in S$ the initial state; \item ${\ra}\subseteq S\times A\times S$ a transition relation; \item ${\up}\subseteq S$ a termination predicate; -\item ${\TDepth}:S\to \natN$ a \emph{transient depth} (or \emph{transience}) function. +\item ${\TLevel}:S\to \natN$ a \emph{transience level} function; +\item $\Closed\subseteq S$ a set of \emph{closed states}, such that $\up\subseteq \Closed$. \end{itemize} % -State $s\in S$ is called \emph{final} if $p\up$, \emph{transient} (denoted $\Transient(s)$) if $\TDepth(s)>0$ and \emph{steady} (denoted $\Steady(s)$) if $\TDepth(s)=0$. There is a derived \emph{transaction relation}, which holds between all (pairs of) steady states for which there is a (non-empty) transition sequence that only passes through transient states. +$\cS$ is called \emph{complete}, or just a \emph{state space}, if $\Closed=S$. A state $s\in S$ is called \emph{final} if $s\up$. We also use a number of auxiliary concepts for states $s\in S$: + +\begin{itemize} +\item $\Transient(s)$ expresses that $s$ is \emph{transient}. It is defined by +% +\[ \Transient = \gensetof{s\in S}{\TLevel(s)>0} \enspace. \] + +\item $\Steady(s)$ expresses that $s$ is \emph{steady}, which is the inverse of $\Transient(s)$. + +\item $\Complete$ expresses that $s$ is \emph{complete}, meaning that $s$ as well as all its $\trans{}$-successors up to and including the first steady state are closed. $\Complete$ is defined as the smallest set such that % -\[ s\ttrans s' \enspace\iffdef\enspace - s=s_0\trans{} s_1\trans{} \cdots \trans{} s_n=s' \:\wedge\: - \setof{s_0,\ldots,s_n} \cap \Steady=\setof{s,s'} \enspace. +\[ \Complete = \Closed\cap \bigl(\Steady \cup \gensetof{s\in S}{\forall s\trans{} s'.\, s'\in \Complete\cup \Steady}\bigr) \enspace. \] +It follows that, if $\cS$ is complete, $\Complete=S$. + +\item $\ALevel_C(s)$ is the \emph{absence level} of $s$, meaning the minimum transient depth of $s$ and all its $\trans{}$-successors. It is defined by +% +\[ \ALevel: s\mapsto \min\gensetof{\TLevel(s')}{s\trans{}^* s'} \enspace. \] + +\item $\Absent(s)$ expresses that $s$ is \emph{absent}, which is the case if there is no steady state reachable from $s$. It is defined by % +\[ \Absent= \gensetof{s}{\ALevel(s)>0} \enspace. \] +\end{itemize} + +\section*{Pseudo-state spaces} + State spaces are generated from a \emph{pseudo-state spaces}. \medskip\noindent -A pseudo-state space is a tuple $\cP=\tupof{P,\iota,{\mapsto},{\goesto},{\up},\TDepth}$ with +A pseudo-state space is a tuple $\cP=\tupof{P,\iota,{\goesto},{\mapsto},{\up},\TLevel,\Recipe}$ with \begin{itemize} \item $P$ a finite set of pseudo-states; \item $\iota\in P$ the initial pseudo-state; -\item ${\step{}}\subseteq P\times A\times P$ a step relation; -\item ${\goesto}: P\times P$ an acyclic evolution relation; +\item ${\goesto}\subseteq P\times P$ a partial order relation called \emph{evolution}; +\item ${\step{}}\subseteq P\times A\times P$ a ternary \emph{step relation}; \item ${\up} \subseteq P$ a termination predicate; -\item ${\TDepth}:P\to \natN$ a transient depth function. +\item ${\TLevel}:P\to \natN$ a transience level function; +\item $\Recipe:A\pto A$ a partial \emph{recipe mapping}. \end{itemize} -% -Pseudo-state $p\in P$ is called \emph{prime} (denoted $\Prime(p)$) if $p\ncomesfrom$, \emph{closed} (denoted $\Closed(p)$) if $p\ngoesto$ and \emph{open} (denoted $\Open(p)$) if it is not closed. +A pseudo-state $p\in P$ is called \emph{prime} (denoted $\Prime(p)$) if $p$ is a $\goesto$-minimum, \emph{closed} (denoted $\Closed(p)$) if $p$ is a $\goesto$-maximum and \emph{open} (denoted $\Open(p)$) if it is not closed. $\Transient$ and $\Steady$ are defined using $\TLevel$ as for state space fragments. % A pseudo-state space is \emph{well-formed} if it satisfies the following additional properties: \begin{itemize} +\item Evolution is piecewise linear; i.e., every pseudo-state has at most one direct $\prec$-predecessor and at most one direct $\prec$-successor; \item Stepping is deterministic; i.e., $\step{}$ is a partial function from $P$ to $A\times P$; -\item Evolution is deterministic; i.e., $\goesto$ is a partial function from $P$ to $P$; -\item Evolution is injective; i.e., $\comesfrom$ is a partial function from $P$ to $P$; \item All steps go from open to prime pseudo-states; i.e., $p\step{}q$ implies $\Open(p)$ and $\Prime(q)$; \item All final pseudo-states are steady and closed; i.e., $p\up$ implies $\Steady(p)$ and $\Closed(p)$; -\item Stepping cannot decrease transience; i.e., $p\step{}q$ implies $\TDepth(q)\geq \TDepth(p)$; -\item Evolution cannot increase transience; i.e., $p\goesto q$ implies $\TDepth(q)\leq \TDepth(p)$. +\item Stepping cannot decrease transience; i.e., $p\step{}q$ implies $\TLevel(q)\geq \TLevel(p)$; \end{itemize} % From now on, we only deal with well-formed pseudo-states spaces. The \emph{prime of} and \emph{closure of} a pseudo-state $p$ are defined as % \begin{align*} - \prm p & = q \quad \text{where $\Prime (q)$ and $q\goesto^* p$} \\ - \cls p & = q \quad \text{where $p\goesto^* q$ and $\Closed(q)$} \enspace. + \prm p & = q \quad \text{where $\Prime(q)$ and $q\goesto p$} \\ + \cls p & = q \quad \text{where $p\goesto q$ and $\Closed(q)$} \enspace. \end{align*} % -Note that these are well-defined because $P$ is finite and $\goesto$ is acyclic, deterministic and injective (implying that it is a union of finite chains of pseudo-states). A pseudo-state space effectively represents a state space in which the states are $\goesto$-related sets of pseudo-states --- which, as remarked above, are actually chains. Rather than formalising the relation in this way, however, we let a state be represented by the initial element of such a chain, i.e., by the (unique) prime pseudo-state. +Note that these are well-defined because $P$ is finite and $\goesto$ is piecewise linear. + +The recipe mapping of a pseudo-state space encodes that an $a$-labelled step for $a\in\dom\Recipe$ kicks off the execution of recipe $\Recipe(a)$ (of which $a$ is the initial partial step). That is, a step $p\step a q$ for which $a\in \dom\Recipe$ gives rise to a $\Recipe(a)$-labelled \emph{recipe transition} in the derived state space, which is considered to be finished upon reaching the first successor $q'$ of $q$ (possibly $q'=q$) for which $\TLevel(q')\leq \TLevel(p)$; unless, that is, $p\step a q$ is itself part of an ongoing recipe transition --- in other words, recipe transitions are not nested. If $a\notin\dom \Recipe$, on the other hand, $p\step a q$ gives rise to an $a$-labelled ``simple'' transition in the derived state space. To capture recipe transitions, we define: +% +\[ p \ttrans{a} \bar q \;\iffdef\; p\step a q_1\leadsto \cdots \leadsto q_n \text{ with } \forall i\TLevel(p) \text{ and } \TLevel(q_n)\leq \TLevel(p) +\] +% +in which $\bar q=q_1\cdots q_n$ is a sequence of pseudo-states that the recipe goes through, and $\leadsto$ stands for the union of $\bigcup_{b\in A} \step{b}$ and $\goestostep$ (the direct predecessor relation in $\goesto$). \medskip\noindent -Given a pseudo-state space $\cP$ as above, a \emph{configuration} is set $C\subseteq P$ of pseudo-states with $\iota\in C$ that is $\goesto$-left-closed (implying, among other things, that $\prm p\in C$ for all $p\in C$) and, moreover, $p\in C$ with $p\comesfrom\:\step q$ implies $q\in C$. (It follows that $P$ is itself a configuration of $\cP$.) Every configuration $C$ generates a state space $\cS_C= \tupof{S_C,\iota_C,\trans{}_C, \up_C, \TDepth_C}$ as follows: +A pseudo-state space represents a state space in which the states effectively correspond to $\goesto$-ordered sets of pseudo-states (which are actually chains because $\goesto$ is piecewise linear). Rather than formalising the relation in this way, however, we let a state be represented by the initial element of such a chain, i.e., by the (unique) prime pseudo-state. + +\medskip\noindent +Given a pseudo-state space $\cP$ as above, a \emph{configuration} is set $C\subseteq P$ of pseudo-states with $\iota\in C$ that is $\goesto$-left-closed (implying, among other things, that $\prm p\in C$ for all $p\in C$) and such that, moreover, $p\in C$ with $p\comesfrom\:\step{} q$ implies $q\in C$. (It follows that $P$ is itself a configuration of $\cP$.) Every configuration $C$ generates a state space $\cS_C= \tupof{S_C,\iota_C,\trans{}_C, \up_C, \TLevel_C}$ as follows: % \begin{align*} S_C & = \gensetof{\prm p}{p\in C} \\ -\iota_C & = \iota \\ -{\trans{}_C} & = \gensetof{(\prm p,a,q)}{p,q\in C, p\step a q} \\ -\up_C & = \gensetof{\prm p}{p\in C, p\up} \\ -\TDepth_C & : p \mapsto \min \gensetof{\TDepth(q)}{q\in C,p=\prm q} \enspace \text{for all $p\in S_C$} \enspace. +\iota_C & = \iota_\cP \\ +{\trans{}_C} & = \gensetof{(\prm p,a,q)}{p,q\in C, a\notin\dom\Recipe, p\step a_\cP q} \\ +&\phantom{=} {}\cup \gensetof{(\prm p,\Recipe(a),\prm{q_n})}{p,q_1,\ldots,q_n\in C, a\in\dom\Recipe, p\ttrans a_\cP \bar q} \\ +\up_C & = \gensetof{\prm p}{p\in C, p\up_\cP} \\ +\TLevel_C & : p \mapsto \min \gensetof{\TLevel_\cP(q)}{p\goesto q\in C} \enspace \text{for all $p\in S_C$} \\ +\Closed_C & = \gensetof{\prm p}{p\in C\cap\Closed_\cP} \enspace. \end{align*} % -The notion of closure and stability are also extended to configurations (so as to coincide with their counterparts for $S_C$): -\begin{align*} -\Closed_C & = \gensetof{\prm p}{p\in C, p\ngoesto} \\ -\Steady_C & = \gensetof{\prm p}{p\in C, \TDepth(p)=0} \enspace. -\end{align*} -% -We define some further auxiliary notions. -% -\begin{itemize} -\item $\Complete_C(s)$ for $s\in S_C$ expresses that $s$ as well as all its discovered $\trans{}_C$-successors are closed, up to and including the first steady state. It is defined as the smallest set such that -% -\[ \Complete_C = \Closed_C\cap \bigl(\Steady_C \cup \gensetof{p\in S_C}{\forall p\trans{}_C q.\, q\in \Complete_C\cup \Steady_C}\bigr) \enspace. -\] +We also use $\Steady_C$, $\Complete_C$, $\ALevel_C$ and $\Absent_C$ to denote $\Steady_{\cS_C}$ etc. -\item $\ETDepth_C(s)$ for $s\in S_C$ is the \emph{eventual transient depth} in $C$, meaning the minimum transient depth of $s$ and all its $\trans{}_C$-successors. It is defined by -% -\[ \ETDepth_C: p\mapsto \min\gensetof{\TDepth(q)}{p\trans{}_C^* q} \text{ for all $p\in S_C$} \enspace. \] - -\item $\ESteady_C(s)$ for $s\in S_C$ expresses that $s$ is \emph{eventually steady} in $C$, meaning that it or one of its discovered $\trans{}_C$-successors is steady (i.e., has transient depth 0). It is defined by -% -\[ \ESteady_C=\gensetof{p\in S_C}{\ETDepth_C(p)=0} \enspace. \] - -\item $\Absent_C(s)$ for $s\in S_C$ expresses that $s$ is \emph{absent} in $C$, which is the case if it is complete (all reachable steady states have been discovered) but not eventually steady (no reachable steady state has been discovered). It is defined by -\[ \Absent_C=\Complete_C\setminus \ESteady_C \enspace. \] -\end{itemize} -% -We want to construct $S_P$ incrementally by approaching $P$ through a sequence of configurations, starting with $\setof{\iota}$ and adding, at each iteration, an evolution $p\goesto p'$ with $p\in C$ and $p'\notin C$. The latter is defined by -\[ C\oplus(p\goesto p') = C\cup \setof{p'} \cup \gensetof{q}{p\step a q} \enspace. \] -Incremental construction means that, if $D=C\oplus(p\goesto p')$, all components of $S_D$ can be constructed from $S_C$ and $\cP$. Indeed, we have +\medskip\noindent +We want to construct $S_P$ incrementally by approaching $P$ through a sequence of configurations, starting with $\setof{\iota}$ and adding, at each iteration, an evolution step $p\goestostep p'$ (denoting that $p'$ is a direct successor of $p$ according to $\goesto$) from $p\in C$ and $p'\notin C$. The latter is defined by +\[ C\oplus(p,p') = C\cup \setof{p'} \cup \gensetof{q}{p\step a q} \enspace. \] +Incremental construction means that, if $D=C\oplus(p,p')$, all components of $S_D$ can be constructed from $S_C$ and $\cP$. Indeed, we have % \begin{align*} -\trans{}_D & = {\trans{}_C} \cup\gensetof{(\prm p,a,q)}{p\step a q} \\ -\up_D & = \up_C \cup \gensetof{p'}{p'\up} \\ -\TDepth_D & : s\mapsto +\up_D & = \up_C \cup \gensetof{\prm{p'}}{p'\up_\cP} \\ +\TLevel_D & : s\mapsto \begin{cases} - \TDepth(p') & \text{if $s=\prm p$} \\ - \TDepth(q) & \text{if $s=q\notin S_C$} \\ - \TDepth_C(s) & \text{otherwise} + \TLevel_\cP(p') & \text{if $s=\prm{p'}$} \\ + \TLevel_\cP(q) & \text{if $s=q\notin S_C$} \\ + \TLevel_C(s) & \text{otherwise} \end{cases} \\ -\Closed_D & = \Closed_C \cup\gensetof{\prm p}{p'\ngoesto} \\ -\Steady_D & = \Steady_C \cup \gensetof{\prm p}{\TDepth(p')=0} \enspace. +\Closed_D & = \Closed_C \cup\gensetof{\prm{p'}}{p'\in \Closed_\cP} \\ +\Steady_D & = \Steady_C \cup \gensetof{\prm{p'}}{p'\in\Steady_\cP} \enspace. \end{align*} % -For $\ttrans_C$, however, the case is less easy. For the purpose of this construction, we introduce several auxiliary data structures. +The incremental construction of $\Complete_C$, $\ALevel_C$ and especially $\trans{}_C$, however, is less straightforward, because all of those involve paths of unbounded length. In particular, the inclusion (or ``exploration'') of a new step $p\step a q$ of the pseudo-state space into a configuration may generate recipe transitions that start in a predecessor of $p$ and end in a successor of $q$. For the purpose of this construction, we therefore introduce several auxiliary data structures. \begin{itemize} \item A transition $t\trans{a}_C t$ is \emph{partial} if it starts in a $\cP$-transient state: diff --git a/doc/exploration/figs/figs.pptx b/doc/exploration/figs/figs.pptx index 0275ce89222b8e4b3923d5106ccf1346c2936f2d..37e058d172ddaf3c6b5c6b775d8f377974413402 100644 GIT binary patch delta 5893 zcmc(j^;6W}_s5rIiA6vzD?vh?gq#LB`1xPn6NQ;z6r^JGSfTVP%NJ{t8 z@bSavhtK=-4}8CO=FELO&z*DbnKN_m%)PDI;Hi(`sx>Iipym5yQfv?iTLrNXr3T<{ z=Nd@^_LcA6$?iu4zH7G1rUt>D3y(SQWS(Gg6H(hKCbBCCdw5>{da$_qhlr(~F;#UF zix`5Ig}#*akk;{XZ7Nems9CkoHrs0B?Get9l|ubb-R+&Ri)&h1hJy9llDJy!taW`? zu~wCmi^|Ho%T$x7S~jBrzr8Y{=pkSu)!itIgKXl6XQp$av_+ohZkK`Uh0Crf>{g%T49>C;;_R@UU z&SL?S%y&i{@B8`Adfq&*nn}aVz{UKFTZdy{q?75zsr`%$=e}8Ls#AP{k#U%uCKo1J zG*<}!m#t*sGW>&FF{jEEH%$A*r$-GXG+Icusf}atd~EiIjA0D4lDP0N`ySq?z!C!U z)xfAb-lj+a_SPBF(mUlf65^hrYG z1!4*LB@&)RG19OR9`t~9jD4oS#v8w9Ce2248!InH29b8f`3o?#2fZJ6K@4?W2KyeA 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