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zzw1=xFDP!4`~=}Sc_k$j6ny)MN^)u|OX(QY$gWGML3BC#mzECFo3J(|Zfvo|Gu!G_~`UPgB2UYhoklCycRjZG4Pajhz7t~uVI6%UD z$1_4M0MbA$(0R!ZnL-eZEf;#I25eO>5I6B@o_tYx2E96D#Bisv{3h#hMe+IRVMX;h ztTU}FICatgL~-F%?*4SKyfF7^%ef$z<{^GQLp~yMaLP<2av^&<3nSArpPT(fkr?xJ z6(D+2w00To%+ ARR910 diff --git a/01-spec-and-impl.tex b/01-spec-and-impl.tex index 57f70bd..5e61398 100644 --- a/01-spec-and-impl.tex +++ b/01-spec-and-impl.tex @@ -232,7 +232,7 @@ Finally, deadlocks \textbf{does not display any behavior} -- that is, a $\delta$ \textbf{Algebraic structure} describes a set with additional properties -- e.g., an additive group over $\mathbb{N}$, a ring of integers modulo $x$, etc. } \item { - Two structures of the same \textbf{type} refers to structures with the same name -- e.g., two groups, two rings, etc. + 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. } \item { 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 $\mathsf{map}(f, X^n) \in q$. diff --git a/02-semantic-equivalences.tex b/02-semantic-equivalences.tex index e120769..26f6d85 100644 --- a/02-semantic-equivalences.tex +++ b/02-semantic-equivalences.tex @@ -16,12 +16,200 @@ Define a lattice over which \textit{semantic equivalence relations} for spec. and impl. verification is defined. +\begin{background}[reflexivity] + \begin{equation*} + \forall x \in X: x \circ x \iff \circ\ \mathrm{reflexive\ on}\ X + \end{equation*} +\end{background} + +\begin{background}[symmetry] + \begin{equation*} + \forall x, y \in X: + \begin{prooftree} + \hypo{x \circ y} + \infer1{y \circ x} + \end{prooftree} + \iff \circ\ \mathrm{symmetric\ on}\ X + \end{equation*} +\end{background} + +\begin{background}[transitivity] + \begin{equation*} + \forall x, y, z \in X: + \begin{prooftree} + \hypo{x \circ y} + \hypo{y \circ z} + \infer2{x \circ z} + \end{prooftree} + \iff \circ\ \mathrm{transitive\ on}\ X + \end{equation*} +\end{background} + +\begin{background}[equivalence relation] + \textbf{Equivalence relation} on set $X$ satisfies reflexivity, symmetry, and transitivity on $X$. +\end{background} + \begin{definition}[discrimination measure] 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, \begin{align*} \ &p \equiv q \implies p \sim q \\ \iff\ &\equiv\ \mathrm{finer\ than}\ \sim \end{align*} + + In other words, $\equiv$ creates finer partitions on its domain compared to $\sim$. \end{definition} +\paragraph*{Trace Equivalence} +\begin{definition}[path] + 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. + + A path is \textbf{complete} if it is either infinite or ends in a state where no further transitions are possible -- a maximal path. +\end{definition} + +\begin{definition}[complete trace] + A \textbf{complete trace} of process $p$ is the sequence of labels of transitions in a complete path. + + 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. +\end{definition} + +\begin{example}[${CT}^{\infty}$] + \begin{equation*} + {CT}^{\infty}(a.(\varepsilon + b.\delta)) = \{a\checkmark, ab\} + \end{equation*} +\end{example} + +\begin{definition}[partial trace] + Likewise, a \textbf{partial trace} of a process $p$ is the sequence of labels of transitions in any partial path. + + We also likewise define ${PT}^{fin}(p)$ and ${PT}^{\infty}(p)$ for some process $p$. Define $PT(p)$ as ${PT}^{fin}(p)$. +\end{definition} + +\begin{definition}[$=_{PT}$] + Processes $p, q$ are \textbf{partial trace equivalent} ($p =_{PT} q$) if they have the same partial traces: + \begin{equation*} + p =_{PT} q \iff PT(p) = PT(q) + \end{equation*} + + Mirroring the differences between ${PT}^{fin}$ and ${PT}^{\infty}$, define \textbf{finitary partial trace equivalence} ($=_{{PT}^{fin}}$) and \textbf{infinitary partial trace equivalence} ($=_{{PT}^{\infty}}$). +\end{definition} + +\begin{definition}[$=_{CT}$] + Processes $p, q$ are \textbf{complete trace equivalent} ($p =_{CT} q$) if \underline{moreover} they have the same complete traces: + \begin{equation*} + p =_{CT} q \iff CT(p) = CT(q) + \end{equation*} + + Mirroring the differences between ${CT}^{fin}$ and ${CT}^{\infty}$, define \textbf{finitary complete trace equivalence} ($=_{{CT}^{fin}}$) and \textbf{infinitary complete trace equivalence} ($=_{{CT}^{\infty}}$). +\end{definition} + +\paragraph*{Weak Equivalences and $\tau$-actions} +\begin{definition}[strong equivalence] + A \textbf{strong equivalence} relation treats $\tau$ like any other (observable) action. + + We assume above definitions for e.g., $=_{PT}$ to be assuming strong equivalence. +\end{definition} + +\begin{definition}[weak equivalence] + In its mirror case, a \textbf{weak equivalence} treats $\tau$ as if it is omitted from the input processes. + + We additionally define weak variants of the above 4 equivalences: $=_{{WPT}^{fin}}$, $=_{{WPT}^{\infty}}$, $=_{{WCT}^{fin}}$, $=_{{WCT}^{\infty}}$. +\end{definition} + +\paragraph*{Bisimulation Equivalence} +\begin{definition}[bisimulation] + 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: + \begin{itemize} + \item $ + s \circ t \implies + (\forall p \in P: s \models p \iff t \models p) + $ + \item $ + s \circ t \wedge (\exists a \in A: s \xrightarrow{a} s^{'}) \implies + (\exists t^{'}: t \xrightarrow{a} t^{'}) \wedge s^{'} \circ t^{'} + $ + \item $ + s \circ t \wedge (\exists a \in A: t \xrightarrow{a} t^{'}) \implies + (\exists s^{'}: s \xrightarrow{a} s^{'}) \wedge s^{'} \circ t^{'} + $ + \end{itemize} + + Bisimulation (aka. \textbf{bisimulation equivalence}) is an equivalence relation. In general, bisimulation differentiates branching structure of processes. +\end{definition} + +\begin{definition}[bisimilarity] + Two states $s, t$ are bisimilar ($s \underline{\leftrightarrow} t$) if such a bisimulation $\circ$ exists between $s, t$. +\end{definition} + +\begin{definition}[branching bisimulation] + Given $A, P$ upon LTS, weaken \underline{bisimulation} as follows: a \textbf{branching bisumlation} is a binary relation $\circ \subseteq S \times S$ satisfying: + \begin{enumerate} + \item {$ + s \circ t \wedge (\exists p \in P: s \models p) \implies + \exists t_1: t \rightsquigarrow t_1 \models p \wedge s \circ t_1 + $ \label{def.bbeq.sat.1}} + \item {$ + s \circ t \wedge (\exists p \in P: t \models p) \implies + \exists s_1: s \rightsquigarrow s_1 \models p \wedge s_1 \circ t + $ \label{def.bbeq.sat.2}} + \item {$ + \begin{aligned} + &s \circ t \wedge (\exists a \in A_{\tau}: s \xrightarrow{a} s^{'}) \\ + \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^{'} + \end{aligned} + $ \label{def.bbeq.sat.3}} + \item {$ + \begin{aligned} + &s \circ t \wedge (\exists a \in A_{\tau}: t \xrightarrow{a} t^{'}) \\ + \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^{'} + \end{aligned} + $ \label{def.bbeq.sat.4}} + \end{enumerate} + where: + \begin{itemize} + \item $ + \begin{aligned} + &s \rightsquigarrow s^{'} \\ + \iff &\exists n \ge 0: \exists s_0, \dots, s_n: + s = s_0 \xrightarrow{\tau} \dots \xrightarrow{\tau} s_n = s^{'} + \end{aligned} + $ + \item $ + A_{\tau} \coloneq A \cup \{\tau\} + $ + \item $ + s \xrightarrow{(a)} s^{'} \coloneq + \begin{cases*} + s \xrightarrow{a} s^{'} & if $a \in A$ \\ + s \xrightarrow{\tau} s^{'} \vee s = s^{'} & if $a = \tau$ + \end{cases*} + $ + \end{itemize} + + Two processes $p, q$ are branching bisimilar ($p \underline{\leftrightarrow}_{b} t$) if such a binary relation $\circ$ exists. +\end{definition} + +\begin{definition}[delay bisimulation] + Given $\underline{\leftrightarrow}_{b}$, drop requirements $s \circ t_1$ and $s_1 \circ t$, thus producing $\underline{\leftrightarrow}_{d}$. +\end{definition} + +\begin{definition}[weak bisimulation] + Given $\underline{\leftrightarrow}_{b}$, + \begin{itemize} + \item Drop requirements $s \circ t_1$, $s_1 \circ t$; + \item Relax $t_2 = t^{'}$ and $s_2 = s^{'}$ to $t_2 \rightsquigarrow t^{'}$ and $s_2 \rightsquigarrow s^{'}$, respectively. + \end{itemize} + Thus producing $\underline{\leftrightarrow}_{w}$. +\end{definition} + +\paragraph*{Language Equivalence} This paragraph is moved here for ergonomics. +\begin{definition}[language equivalence] + 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. + + Intuitively (and indeed) this is coarser than partial trace equivalence. +\end{definition} + +\paragraph*{Overview: The Hasse Diagram} +\dots + + \end{document} \ No newline at end of file diff --git a/03-ccs.tex b/03-ccs.tex new file mode 100644 index 0000000..84576ab --- /dev/null +++ b/03-ccs.tex @@ -0,0 +1,137 @@ +\documentclass[99-notes-packed.tex]{subfiles} + +\begin{document} +\paragraph*{CCS} +Define the set of operations and semantics: +\begin{itemize} + \item { + $0$ (inaction): + + $0$ represents a graph with 1 (initial) state, 0 transitions. + } + \item { + $a, \bar{a}$ (complementary actions): + + Complementary actions are assumed to communicate / synchronize with one another. + } + \item { + $a.P$ (action prefix) for each action $a$, process $P$, which: + \begin{enumerate} + \item Define new initial state $\mathsf{i}$. + \item Creates transition $\mathsf{i} \xrightarrow{a} I_P$. + \end{enumerate} + } + \item { + $P + Q$ (summation / choice / alternative composition), where: + \begin{itemize} + \item Define new initial state $\mathsf{root}$. + \item $\mathrm{States}(P + Q) \coloneqq \mathrm{States}(P) \cup \mathrm{States}(Q) \cup \{\mathsf{root}\}$ + \item Replace all $I_P \xrightarrow{a} s$ with $\mathsf{root} \xrightarrow{a} s$. + \item Replace all $I_Q \xrightarrow{a} s$ with $\mathsf{root} \xrightarrow{a} s$. + \end{itemize} + } + \item { + $P | Q$ (parallel composition). + + This takes the cartesian product of the states of $P, Q$, such that: + \begin{itemize} + \item $s \xrightarrow{a} s^{'} \in P \implies \forall t \in Q: (s, t) \xrightarrow{a} (s^{'}, t)$ + \item $t \xrightarrow{a} t^{'} \in Q \implies \forall s \in P: (s, t) \xrightarrow{a} (s, t^{'})$ + \item $(s \xrightarrow{a} s^{'} \in P) \wedge (t \xrightarrow{\bar{a}} t^{'} \in Q) \implies (s, t) \xrightarrow{\tau} (s^{'}, t^{'})$ + \end{itemize} + + 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. + } + \item { + $P{\backslash}a$ (restriction) for each action $a$. + + This produces copy of $P$ such that all actions $a, \bar{a}$ are omitted. This is useful to remove unsuccessful communication. + } + \item { + $P[f]$ (relabelling) for each function $f: A \rightarrow A$. + + This replaces each label $a, \bar{a}$ by $f(a), \overline{f(a)}$. + } +\end{itemize} + +\paragraph*{Recursion} +\begin{definition}[process names and expressions] + Suppose we bind names $X, Y, Z, \dots$ to some expression in the CCS language: + \begin{equation*} + X = P_X + \end{equation*} + Here, $P_X$ represents ANY expression in the language, possibly including $X$. + + It is trivial to see this can cause recursive definitions: + \begin{equation*} + X = a.X + \end{equation*} +\end{definition} + +\begin{definition}[recursive specification] + Define \textbf{recursive specification} as partial function $s: X \rightarrow E$: + \begin{itemize} + \item $X$: \textbf{recursion variables}. + \item $E$: \textbf{recursion equations} of form $x = P_x$. + \end{itemize} + + In general, recursive spec.s are written as follows: + \begin{equation*} + \langle x | s \rangle + \end{equation*} + which reads as ``process $x$ satisfying equation $s$''. +\end{definition} + +\begin{definition}[guarded recursion] + A recursion is \textbf{guarded} if each occurrence of a process name in $P_X$ occurs within the scope of a subexpression $a.P^{'}_X$. + + Think of it as being unwind-able such that progress is guaranteed. +\end{definition} + +\paragraph*{Structural Operational Semantics (CCS)} +\begin{center} + \begin{tabular}{cc} + $ + \begin{prooftree} + \infer0{a.E \xrightarrow{a} E} + \end{prooftree} + $ & + $ + \begin{prooftree} + \hypo{E_j \xrightarrow{a} E_j^{'}} + \infer1{\sum_{i \in I}E_i \xrightarrow{a} E_j^{'}} + \end{prooftree}\ (j \in I) + $ \\ + \\ + $ + \begin{prooftree} + \hypo{E \xrightarrow{a} E^{'}} + \infer1{E | F \xrightarrow{a} E^{'} | F} + \end{prooftree} + $ & + $ + \begin{prooftree} + \hypo{E \xrightarrow{a} E^{'}} + \hypo{F \xrightarrow{\overline{a}} F^{'}} + \infer2{E | F \xrightarrow{\tau} E^{'} | F^{'}} + \end{prooftree} + $ \\ + \\ + $ + \begin{prooftree} + \hypo{E \xrightarrow{a} E^{'}} + \hypo{a \notin L \cup \overline{L}} + \infer2{E{\backslash}L \xrightarrow{a} E^{'}{\backslash}L} + \end{prooftree} + $ & + $ + \begin{prooftree} + \hypo{E \xrightarrow{a} E^{'}} + \infer1{E[f] \xrightarrow{f(a)} E^{'}[f]} + \end{prooftree} + $ \\ + \\ + \end{tabular} +\end{center} + +\end{document} \ No newline at end of file diff --git a/04-congruence.tex b/04-congruence.tex new file mode 100644 index 0000000..88a9391 --- /dev/null +++ b/04-congruence.tex @@ -0,0 +1,116 @@ +\documentclass[99-notes-packed.tex]{subfiles} + +\begin{document} +\paragraph*{Congurence} +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: + +\begin{definition}[congurence] + An equivalence $\sim$ is a \textbf{congruence} for a language $\mathcal{L}$ if: + \begin{equation*} + \forall C[\ ] \in \mathcal{L}: P \sim Q \implies C[P] \sim C[Q] + \end{equation*} + where: + \begin{itemize} + \item $C[\ ]$ (context) represents a $\mathcal{L}$-expression with a hole in it, plugged (e.g., with $P$) as $C[P]$. + \end{itemize} + + For example, let $P = a.[\ ]$: + \begin{equation*} + \begin{prooftree} + \hypo{P = Q} + \infer1{a.P = a.Q} + \end{prooftree} + \end{equation*} + Equivalently, we can say that $CCP.(.)$ is compositional under equality ($=$). +\end{definition} + +\begin{example}[$=_{CT}$ and $\partial_{H}$] + This is a counterexample for showing why $=_{CT}$ is NOT a congurence over ACP. Obviously: + \begin{equation*} + a.b + a.c =_{CT} a.(b + c) + \end{equation*} + + However: + \begin{equation*} + \partial_{\{c\}}(a.b + a.c) \neq_{CT} \partial_{\{c\}}(a.(b + c)) + \end{equation*} +\end{example} + +\begin{definition}[congurence closure] + A \textbf{congurence closure} $\sim^c$ of $\sim$ wrt. language $\mathcal{L}$ is defined by: + \begin{equation*} + P \sim^{c} Q \iff \forall C[\ ] \in \mathcal{L}: C[P] \sim C[Q] + \end{equation*} +\end{definition} + +\paragraph*{Equational Axiomisation} +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. + +Same goes for operators in e.g, CCS: +\begin{center} + \begin{tabular}{cc} + $(P + Q) + R = P + (Q + R)$ & (associativity) \\ + $P + Q = Q + P$ & (commutativity) \\ + $P + P = P$ & (idempotence) \\ + $P + 0 = P$ & ($0$ 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} \ No newline at end of file diff --git a/99-notes-packed.pdf b/99-notes-packed.pdf index daba53ec3732aab656c9bd8e198e1c1ee302b821..c2188907aef7c0e299146248badcf79e44b0657b 100644 GIT binary patch delta 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-\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} \ No newline at end of file