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Zhengyi Chen
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7 changed files with 453 additions and 7 deletions
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@ -232,7 +232,7 @@ Finally, deadlocks \textbf{does not display any behavior} -- that is, a $\delta$
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\textbf{Algebraic structure} describes a set with additional properties -- e.g., an additive group over $\mathbb{N}$, a ring of integers modulo $x$, etc.
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}
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\item {
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Two structures of the same \textbf{type} refers to structures with the same name -- e.g., two groups, two rings, etc.
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Two \textbf{structures} of the same \textbf{type} refers to structures with the same name (i.e., class of property) -- e.g., two groups, two rings, etc.
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}
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\item {
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A \textbf{structure-preserving} map $f$ between two structures intuitively describes a structure such that, for properties $p \in X$, $q \in Y$ between same-type structures $X, Y$, any tuples $X^n \in p$ accepted by $p$ (e.g., $3 + 5 = 8 \implies (3, 5, 8) \in \mathbb{R}.(+)$) satisfies $\mathsf{map}(f, X^n) \in q$.
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@ -16,12 +16,200 @@
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Define a lattice over which \textit{semantic equivalence relations} for spec. and impl. verification is defined.
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\begin{background}[reflexivity]
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\begin{equation*}
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\forall x \in X: x \circ x \iff \circ\ \mathrm{reflexive\ on}\ X
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\end{equation*}
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\end{background}
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\begin{background}[symmetry]
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\begin{equation*}
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\forall x, y \in X:
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\begin{prooftree}
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\hypo{x \circ y}
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\infer1{y \circ x}
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\end{prooftree}
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\iff \circ\ \mathrm{symmetric\ on}\ X
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\end{equation*}
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\end{background}
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\begin{background}[transitivity]
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\begin{equation*}
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\forall x, y, z \in X:
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\begin{prooftree}
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\hypo{x \circ y}
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\hypo{y \circ z}
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\infer2{x \circ z}
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\end{prooftree}
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\iff \circ\ \mathrm{transitive\ on}\ X
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\end{equation*}
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\end{background}
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\begin{background}[equivalence relation]
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\textbf{Equivalence relation} on set $X$ satisfies reflexivity, symmetry, and transitivity on $X$.
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\end{background}
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\begin{definition}[discrimination measure]
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One equivalence relation $\equiv$ is \textbf{finer} / \textbf{more discriminating} than another $\sim$ if each $\equiv$-eq. class is a subset of a $\sim$-eq. class. In other words,
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\begin{align*}
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\ &p \equiv q \implies p \sim q \\
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\iff\ &\equiv\ \mathrm{finer\ than}\ \sim
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\end{align*}
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In other words, $\equiv$ creates finer partitions on its domain compared to $\sim$.
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\end{definition}
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\paragraph*{Trace Equivalence}
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\begin{definition}[path]
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A \textbf{path} of a process $p$ is an alternating sequence of states and transitions starting from state $p$. It can be infinite or ending in a state.
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A path is \textbf{complete} if it is either infinite or ends in a state where no further transitions are possible -- a maximal path.
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\end{definition}
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\begin{definition}[complete trace]
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A \textbf{complete trace} of process $p$ is the sequence of labels of transitions in a complete path.
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The set of finite complete traces of process $p$ is denoted as ${CT}^{fin}(p)$, while the set of all finite/infinite complete traces of $p$ is ${CT}^{\infty}(p)$ -- aka. $CT(p)$ from now on.
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\end{definition}
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\begin{example}[${CT}^{\infty}$]
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\begin{equation*}
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{CT}^{\infty}(a.(\varepsilon + b.\delta)) = \{a\checkmark, ab\}
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\end{equation*}
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\end{example}
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\begin{definition}[partial trace]
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Likewise, a \textbf{partial trace} of a process $p$ is the sequence of labels of transitions in any partial path.
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We also likewise define ${PT}^{fin}(p)$ and ${PT}^{\infty}(p)$ for some process $p$. Define $PT(p)$ as ${PT}^{fin}(p)$.
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\end{definition}
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\begin{definition}[$=_{PT}$]
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Processes $p, q$ are \textbf{partial trace equivalent} ($p =_{PT} q$) if they have the same partial traces:
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\begin{equation*}
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p =_{PT} q \iff PT(p) = PT(q)
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\end{equation*}
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Mirroring the differences between ${PT}^{fin}$ and ${PT}^{\infty}$, define \textbf{finitary partial trace equivalence} ($=_{{PT}^{fin}}$) and \textbf{infinitary partial trace equivalence} ($=_{{PT}^{\infty}}$).
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\end{definition}
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\begin{definition}[$=_{CT}$]
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Processes $p, q$ are \textbf{complete trace equivalent} ($p =_{CT} q$) if \underline{moreover} they have the same complete traces:
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\begin{equation*}
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p =_{CT} q \iff CT(p) = CT(q)
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\end{equation*}
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Mirroring the differences between ${CT}^{fin}$ and ${CT}^{\infty}$, define \textbf{finitary complete trace equivalence} ($=_{{CT}^{fin}}$) and \textbf{infinitary complete trace equivalence} ($=_{{CT}^{\infty}}$).
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\end{definition}
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\paragraph*{Weak Equivalences and $\tau$-actions}
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\begin{definition}[strong equivalence]
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A \textbf{strong equivalence} relation treats $\tau$ like any other (observable) action.
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We assume above definitions for e.g., $=_{PT}$ to be assuming strong equivalence.
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\end{definition}
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\begin{definition}[weak equivalence]
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In its mirror case, a \textbf{weak equivalence} treats $\tau$ as if it is omitted from the input processes.
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We additionally define weak variants of the above 4 equivalences: $=_{{WPT}^{fin}}$, $=_{{WPT}^{\infty}}$, $=_{{WCT}^{fin}}$, $=_{{WCT}^{\infty}}$.
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\end{definition}
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\paragraph*{Bisimulation Equivalence}
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\begin{definition}[bisimulation]
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Let $A, P$ define the actions and predicates of an LTS (in addition to states, etc.). A \textbf{bisimulation} is a binary relation $\circ \subseteq S \times S$ satisfying:
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\begin{itemize}
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\item $
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s \circ t \implies
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(\forall p \in P: s \models p \iff t \models p)
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$
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\item $
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s \circ t \wedge (\exists a \in A: s \xrightarrow{a} s^{'}) \implies
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(\exists t^{'}: t \xrightarrow{a} t^{'}) \wedge s^{'} \circ t^{'}
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$
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\item $
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s \circ t \wedge (\exists a \in A: t \xrightarrow{a} t^{'}) \implies
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(\exists s^{'}: s \xrightarrow{a} s^{'}) \wedge s^{'} \circ t^{'}
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$
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\end{itemize}
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Bisimulation (aka. \textbf{bisimulation equivalence}) is an equivalence relation. In general, bisimulation differentiates branching structure of processes.
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\end{definition}
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\begin{definition}[bisimilarity]
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Two states $s, t$ are bisimilar ($s \underline{\leftrightarrow} t$) if such a bisimulation $\circ$ exists between $s, t$.
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\end{definition}
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\begin{definition}[branching bisimulation]
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Given $A, P$ upon LTS, weaken \underline{bisimulation} as follows: a \textbf{branching bisumlation} is a binary relation $\circ \subseteq S \times S$ satisfying:
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\begin{enumerate}
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\item {$
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s \circ t \wedge (\exists p \in P: s \models p) \implies
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\exists t_1: t \rightsquigarrow t_1 \models p \wedge s \circ t_1
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$ \label{def.bbeq.sat.1}}
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\item {$
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s \circ t \wedge (\exists p \in P: t \models p) \implies
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\exists s_1: s \rightsquigarrow s_1 \models p \wedge s_1 \circ t
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$ \label{def.bbeq.sat.2}}
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\item {$
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\begin{aligned}
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&s \circ t \wedge (\exists a \in A_{\tau}: s \xrightarrow{a} s^{'}) \\
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\implies &\exists t_1, t_2, t^{'}: t \rightsquigarrow t_1 \xrightarrow{(a)} t_2 = t^{'} \wedge s \circ t_1 \wedge s^{'} \circ t^{'}
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\end{aligned}
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$ \label{def.bbeq.sat.3}}
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\item {$
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\begin{aligned}
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&s \circ t \wedge (\exists a \in A_{\tau}: t \xrightarrow{a} t^{'}) \\
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\implies &\exists s_1, s_2, s^{'}: s \rightsquigarrow s_1 \xrightarrow{(a)} s_2 = s^{'} \wedge s_1 \circ t \wedge s^{'} \circ t^{'}
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\end{aligned}
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$ \label{def.bbeq.sat.4}}
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\end{enumerate}
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where:
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\begin{itemize}
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\item $
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\begin{aligned}
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&s \rightsquigarrow s^{'} \\
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\iff &\exists n \ge 0: \exists s_0, \dots, s_n:
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s = s_0 \xrightarrow{\tau} \dots \xrightarrow{\tau} s_n = s^{'}
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\end{aligned}
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$
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\item $
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A_{\tau} \coloneq A \cup \{\tau\}
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$
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\item $
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s \xrightarrow{(a)} s^{'} \coloneq
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\begin{cases*}
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s \xrightarrow{a} s^{'} & if $a \in A$ \\
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s \xrightarrow{\tau} s^{'} \vee s = s^{'} & if $a = \tau$
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\end{cases*}
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$
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\end{itemize}
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Two processes $p, q$ are branching bisimilar ($p \underline{\leftrightarrow}_{b} t$) if such a binary relation $\circ$ exists.
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\end{definition}
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\begin{definition}[delay bisimulation]
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Given $\underline{\leftrightarrow}_{b}$, drop requirements $s \circ t_1$ and $s_1 \circ t$, thus producing $\underline{\leftrightarrow}_{d}$.
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\end{definition}
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\begin{definition}[weak bisimulation]
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Given $\underline{\leftrightarrow}_{b}$,
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\begin{itemize}
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\item Drop requirements $s \circ t_1$, $s_1 \circ t$;
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\item Relax $t_2 = t^{'}$ and $s_2 = s^{'}$ to $t_2 \rightsquigarrow t^{'}$ and $s_2 \rightsquigarrow s^{'}$, respectively.
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\end{itemize}
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Thus producing $\underline{\leftrightarrow}_{w}$.
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\end{definition}
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\paragraph*{Language Equivalence} This paragraph is moved here for ergonomics.
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\begin{definition}[language equivalence]
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Processes $p, q$ are \textbf{language equivalent} if they have the same traces leading to terminating states -- i.e., equal subset of terminating partial traces.
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Intuitively (and indeed) this is coarser than partial trace equivalence.
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\end{definition}
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\paragraph*{Overview: The Hasse Diagram}
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\dots
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\end{document}
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137
03-ccs.tex
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137
03-ccs.tex
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@ -0,0 +1,137 @@
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\documentclass[99-notes-packed.tex]{subfiles}
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\begin{document}
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\paragraph*{CCS}
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Define the set of operations and semantics:
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\begin{itemize}
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\item {
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$0$ (inaction):
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$0$ represents a graph with 1 (initial) state, 0 transitions.
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}
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\item {
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$a, \bar{a}$ (complementary actions):
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Complementary actions are assumed to communicate / synchronize with one another.
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}
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\item {
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$a.P$ (action prefix) for each action $a$, process $P$, which:
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\begin{enumerate}
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\item Define new initial state $\mathsf{i}$.
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\item Creates transition $\mathsf{i} \xrightarrow{a} I_P$.
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\end{enumerate}
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}
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\item {
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$P + Q$ (summation / choice / alternative composition), where:
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\begin{itemize}
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\item Define new initial state $\mathsf{root}$.
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\item $\mathrm{States}(P + Q) \coloneqq \mathrm{States}(P) \cup \mathrm{States}(Q) \cup \{\mathsf{root}\}$
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\item Replace all $I_P \xrightarrow{a} s$ with $\mathsf{root} \xrightarrow{a} s$.
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\item Replace all $I_Q \xrightarrow{a} s$ with $\mathsf{root} \xrightarrow{a} s$.
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\end{itemize}
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}
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\item {
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$P | Q$ (parallel composition).
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This takes the cartesian product of the states of $P, Q$, such that:
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\begin{itemize}
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\item $s \xrightarrow{a} s^{'} \in P \implies \forall t \in Q: (s, t) \xrightarrow{a} (s^{'}, t)$
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\item $t \xrightarrow{a} t^{'} \in Q \implies \forall s \in P: (s, t) \xrightarrow{a} (s, t^{'})$
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\item $(s \xrightarrow{a} s^{'} \in P) \wedge (t \xrightarrow{\bar{a}} t^{'} \in Q) \implies (s, t) \xrightarrow{\tau} (s^{'}, t^{'})$
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\end{itemize}
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Note that \textbf{CCS adheres strictly to a handshaking communication format} -- this differs from ACP which gives greater leeway to implementation, via the use of $\gamma$ operator.
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}
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\item {
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$P{\backslash}a$ (restriction) for each action $a$.
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This produces copy of $P$ such that all actions $a, \bar{a}$ are omitted. This is useful to remove unsuccessful communication.
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}
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\item {
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$P[f]$ (relabelling) for each function $f: A \rightarrow A$.
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This replaces each label $a, \bar{a}$ by $f(a), \overline{f(a)}$.
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}
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\end{itemize}
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\paragraph*{Recursion}
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\begin{definition}[process names and expressions]
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Suppose we bind names $X, Y, Z, \dots$ to some expression in the CCS language:
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\begin{equation*}
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X = P_X
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\end{equation*}
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Here, $P_X$ represents ANY expression in the language, possibly including $X$.
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It is trivial to see this can cause recursive definitions:
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\begin{equation*}
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X = a.X
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\end{equation*}
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\end{definition}
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\begin{definition}[recursive specification]
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Define \textbf{recursive specification} as partial function $s: X \rightarrow E$:
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\begin{itemize}
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\item $X$: \textbf{recursion variables}.
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\item $E$: \textbf{recursion equations} of form $x = P_x$.
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\end{itemize}
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In general, recursive spec.s are written as follows:
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\begin{equation*}
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\langle x | s \rangle
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\end{equation*}
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which reads as ``process $x$ satisfying equation $s$''.
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\end{definition}
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\begin{definition}[guarded recursion]
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A recursion is \textbf{guarded} if each occurrence of a process name in $P_X$ occurs within the scope of a subexpression $a.P^{'}_X$.
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Think of it as being unwind-able such that progress is guaranteed.
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\end{definition}
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\paragraph*{Structural Operational Semantics (CCS)}
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\begin{center}
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\begin{tabular}{cc}
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$
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\begin{prooftree}
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\infer0{a.E \xrightarrow{a} E}
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\end{prooftree}
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$ &
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$
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\begin{prooftree}
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\hypo{E_j \xrightarrow{a} E_j^{'}}
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\infer1{\sum_{i \in I}E_i \xrightarrow{a} E_j^{'}}
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\end{prooftree}\ (j \in I)
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$ \\
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\\
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$
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\begin{prooftree}
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\hypo{E \xrightarrow{a} E^{'}}
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\infer1{E | F \xrightarrow{a} E^{'} | F}
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\end{prooftree}
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$ &
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$
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\begin{prooftree}
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\hypo{E \xrightarrow{a} E^{'}}
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\hypo{F \xrightarrow{\overline{a}} F^{'}}
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\infer2{E | F \xrightarrow{\tau} E^{'} | F^{'}}
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\end{prooftree}
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$ \\
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\\
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$
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\begin{prooftree}
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\hypo{E \xrightarrow{a} E^{'}}
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\hypo{a \notin L \cup \overline{L}}
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\infer2{E{\backslash}L \xrightarrow{a} E^{'}{\backslash}L}
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\end{prooftree}
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$ &
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$
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\begin{prooftree}
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\hypo{E \xrightarrow{a} E^{'}}
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\infer1{E[f] \xrightarrow{f(a)} E^{'}[f]}
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\end{prooftree}
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$ \\
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\\
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\end{tabular}
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\end{center}
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\end{document}
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116
04-congruence.tex
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116
04-congruence.tex
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\documentclass[99-notes-packed.tex]{subfiles}
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\begin{document}
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\paragraph*{Congurence}
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If an equivalence relation is a \textit{congurence} for an operator -- i.e., an operator is \textit{compositional} for the equivalence -- then there exists a sort of isomorphism detailed as follows:
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\begin{definition}[congurence]
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An equivalence $\sim$ is a \textbf{congruence} for a language $\mathcal{L}$ if:
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\begin{equation*}
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\forall C[\ ] \in \mathcal{L}: P \sim Q \implies C[P] \sim C[Q]
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\end{equation*}
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where:
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\begin{itemize}
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\item $C[\ ]$ (context) represents a $\mathcal{L}$-expression with a hole in it, plugged (e.g., with $P$) as $C[P]$.
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\end{itemize}
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For example, let $P = a.[\ ]$:
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\begin{equation*}
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\begin{prooftree}
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\hypo{P = Q}
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\infer1{a.P = a.Q}
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\end{prooftree}
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\end{equation*}
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Equivalently, we can say that $CCP.(.)$ is compositional under equality ($=$).
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\end{definition}
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\begin{example}[$=_{CT}$ and $\partial_{H}$]
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This is a counterexample for showing why $=_{CT}$ is NOT a congurence over ACP. Obviously:
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\begin{equation*}
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a.b + a.c =_{CT} a.(b + c)
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\end{equation*}
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However:
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\begin{equation*}
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\partial_{\{c\}}(a.b + a.c) \neq_{CT} \partial_{\{c\}}(a.(b + c))
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\end{equation*}
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\end{example}
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\begin{definition}[congurence closure]
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A \textbf{congurence closure} $\sim^c$ of $\sim$ wrt. language $\mathcal{L}$ is defined by:
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\begin{equation*}
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P \sim^{c} Q \iff \forall C[\ ] \in \mathcal{L}: C[P] \sim C[Q]
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\end{equation*}
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\end{definition}
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\paragraph*{Equational Axiomisation}
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In terms of e.g., real addition we describe the operator as possessing e.g., associativity and commutativity, which in turn allows us to do some transformation during analysis, etc.
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Same goes for operators in e.g, CCS:
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\begin{center}
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\begin{tabular}{cc}
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$(P + Q) + R = P + (Q + R)$ & (associativity) \\
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$P + Q = Q + P$ & (commutativity) \\
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$P + P = P$ & (idempotence) \\
|
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$P + 0 = P$ & ($0$ as neutral element of $+$)
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{definition}[CCS: expansion theorem]
|
||||
Suppose:
|
||||
\begin{align*}
|
||||
P \coloneqq& \sum_{i \in I} a_i.P_i \\
|
||||
Q \coloneqq& \sum_{j \in J} b_i.Q_j
|
||||
\end{align*}
|
||||
|
||||
Then,
|
||||
\begin{align*}
|
||||
P | Q &= \sum_{i \in I} a_i(P_i | Q) \\
|
||||
&+ \sum_{i \in I, j \in J} \tau(P_i | Q_j)\ (\mathrm{given\ }a_i = \overline{b_j}) \\
|
||||
&+ \sum_{j \in J} b_i(P | Q_j)
|
||||
\end{align*}
|
||||
|
||||
Expressions of the form $\sum a.P$ are aka. \textbf{head normal form}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Recursive Definition Principle]
|
||||
\begin{equation*}
|
||||
i \in [1, n]: \langle X_i | E \rangle \in \mathsf{Expr}(X_1 \coloneqq \langle X_1 | E \rangle, \dots, X_n \coloneqq \langle X_n | E \rangle)
|
||||
\end{equation*}
|
||||
|
||||
Basically, some series of expressions for $X_1, \dots, X_n$ exists as solution for $E$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Recursive Specification Principle]
|
||||
If there \underline{exists}
|
||||
\begin{equation*}
|
||||
i \in [1, n]: y_i \leftarrow \mathsf{Expr}(y_1, \dots, y_n)
|
||||
\end{equation*}
|
||||
then:
|
||||
\begin{equation*}
|
||||
i \in [1, n]: y_i = \langle X_i | E \rangle
|
||||
\end{equation*}
|
||||
|
||||
In other words, any $y_{1 \dots n}$ that exists is the sole solution for $E$ modulo bisimulation equivalence.
|
||||
\end{definition}
|
||||
|
||||
\paragraph*{Rooted Bisimilarity}
|
||||
We note that depending on semantics of $\mathcal{L}$, equivalences may (and in fact likely) fail to be a congurence over $\mathcal{L}$. This also is the case for e.g., branching bisimilarity: $\tau.a =_{BB} a$ but $\tau.a + b \ne_{BB} a + b$.
|
||||
|
||||
ACP and CCS fixes this by changing the equivalence operator.
|
||||
|
||||
\begin{definition}[Rooted Branching Bisimilarity]
|
||||
\begin{align*}
|
||||
P =_{rBB} Q \iff &(P \xrightarrow{a} P^{'} \implies Q \xrightarrow{a} Q^{'} \wedge P^{'} =_{BB} Q^{'})\ \wedge \\
|
||||
&(Q \xrightarrow{a} Q^{'} \implies P \xrightarrow{a} P^{'} \wedge P^{'} =_{BB} Q^{'})
|
||||
\end{align*}
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Rooted Weak Bisimilarity]
|
||||
\begin{align*}
|
||||
P =_{rWB} Q \iff &(P \xrightarrow{a} P^{'} \implies Q \xrightarrow{a} Q^{'} \wedge P^{'} =_{WB} Q^{'})\ \wedge \\
|
||||
&(Q \xrightarrow{a} Q^{'} \implies P \xrightarrow{a} P^{'} \wedge P^{'} =_{WB} Q^{'})
|
||||
\end{align*}
|
||||
\end{definition}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
|
@ -2,7 +2,8 @@
|
|||
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[a4paper, margin=1cm]{geometry}
|
||||
\usepackage[a4paper, margin=0.25cm, landscape]{geometry}
|
||||
|
||||
|
||||
\usepackage{multicol} % 2-col formatting
|
||||
\usepackage{hyperref}
|
||||
|
|
@ -21,19 +22,23 @@
|
|||
|
||||
% environment `definition` prints as "Definition" and has counter reset per section.
|
||||
\newtheorem{definition}{Definition}[section]
|
||||
|
||||
\newtheorem{background}{Background}[section]
|
||||
\newtheorem{example}{Example}[section]
|
||||
|
||||
\begin{document}
|
||||
\begin{multicols*}{3}
|
||||
|
||||
\section{LTS; ACP}
|
||||
\begin{multicols*}{2}
|
||||
\subfile{01-spec-and-impl}
|
||||
\end{multicols*}
|
||||
|
||||
\section{Semantic Equivalences}
|
||||
\begin{multicols*}{2}
|
||||
\subfile{02-semantic-equivalences}
|
||||
\end{multicols*}
|
||||
|
||||
\section{CCS; SOS}
|
||||
\subfile{03-ccs.tex}
|
||||
|
||||
\section{Equational Axiomisation}
|
||||
\subfile{04-congruence.tex}
|
||||
|
||||
\end{multicols*}
|
||||
\end{document}
|
||||
Loading…
Add table
Add a link
Reference in a new issue