on L2

.gitignore

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+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib

.vscode/ltex.dictionary.en-US.txt

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+satisfiability
+btwn
+impl

.vscode/ltex.hiddenFalsePositives.en-US.txt

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+{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qconcurrent components)\\E$"}
+{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qconcurrent components).\\E$"}
+{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qconcurrent components of a system).\\E$"}
+{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qprocess algebra).\\E$"}
+{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qalso holds for impl.\\E$"}

01-spec-and-impl.tex

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+\documentclass[99-notes-packed.tex]{subfiles}
+
+\begin{document}
+
+% \paragraph*{Specification vs. Implementation}
+% \underbar{\textit{Specification}} describes what a system ought to satisfy and perform. \textbf{A \textit{formal specification}}, in particular, \textbf{is a specification derived using formal methods that ensure the required properties of some problem at hand}. A formal specification of a distributed system often comes in (at least) 2 parts:
+
+% \begin{enumerate}
+%     \item \textbf{
+%         \textit{Requirements} imposed on the system -- i.e., a list of properties that the system should satisfy (e.g., safety / liveness properties).
+%         \label{def.spec.requirements}
+%     }
+%     \item \textbf{
+%         \textit{Operations} of the system, which describes the behavior (i.e., satisfiability of predicates) given interactions (e.g., effects of communication).
+%         \label{def.spec.operations}
+%     }
+% \end{enumerate}
+
+% In total, a formal specification is then used to verify the following:
+
+% \begin{enumerate}
+%     \item {
+%         Guarantee that a description of a system really meets the requirements of the same system -- i.e., the specification is correct.
+%     }
+%     \item \textbf{
+%         Ensure that an \underbar{\textit{implementation}} satisfies the specification -- i.e., ensure that any correctness properties that hold for spec. also holds for impl.
+%     }
+% \end{enumerate}
+
+\paragraph*{LTS and Process Graphs}
+Both specifications and implementations could be represented by \underbar{\textit{models of concurrency}}, for example \textit{labelled transition systems (LTS)} or \textit{process graphs}.
+
+\begin{definition}[Process Graph]
+    A \textit{process graph} is a triple $(S, I, \rightarrowtriangle)$ such that:
+    \begin{itemize}
+        \item $S$ a set of states;
+        \item $I \in S$ an \textit{initial state};
+        \item {
+            $\rightarrowtriangle$ a set of triples $(s, a, t)$ each describing a (named) relation $S \rightarrow S$:
+            \begin{itemize}
+                \item $s, t \in S$;
+                \item $a \in {Act}$ -- a set of actions.
+            \end{itemize}
+        }
+    \end{itemize}
+    \label{def.process-graph}
+\end{definition}
+
+\begin{definition}[LTS]
+    Same as \hyperref[def.process-graph]{process graph}, except without an initial state. Sometimes used synonymously with process graphs bc. mathematicians are evil.
+\end{definition}
+
+Alternatively, one may use \textit{process algebraic expressions} to formally represent spec.s and impl.s, for example using \textit{CCS (Calculus of Communicating Systems)}, \textit{CSP (Communicating Sequential Processes)}, and \textit{ACP (Algebra of Communicating Processes)}. \underline{Each semantics is of different expressive power}.
+
+\paragraph*{ACP}
+Define the set of operations:
+\begin{itemize}
+    \item {
+        $\varepsilon$ (successful termination -- $\mathrm{ACP}_{\varepsilon}$ extension).
+    }
+    \item {
+        $\delta$ (deadlock).
+    }
+    \item {
+        $a$ (action constant) for each action $a \in {Act}$.
+
+        Each $a$ describe a \textbf{visible action} -- $\tau \notin {Act}$;
+    }
+    \item {
+        $P \cdot Q$ (sequential composition between processes $P, Q$)
+    }
+    \item {
+        $P + Q$ (summation / choice / alternative composition);
+    }
+    \item {
+        $P || Q$ (parallel composition).
+    }
+    \item {
+        ${\partial}_{H}(P)$ (restriction / encapsulation).
+
+        Given set of (visible) actions $H$, this removes $\forall a \in H$ in $P$.
+
+        Practically this is often used after defining $\gamma(a, b)$ to enforce sync -- via removing non-synced $a.b$ or $b.a$ behaviors;
+    }
+    \item {
+        ${\tau}_{I}(P)$ (abstraction -- $\mathrm{ACP}_{\tau}$ extension).
+
+        Given set of (visible) actions $I$, this converts $\forall a \in I$ into $\tau$ in $P$.
+
+        A $\tau$ action is \textbf{non-observable} -- this will be significant for describing traces \& equivalence relations.
+    }
+    \item {
+        $\gamma: A \times A \rightarrow A$ (partial communication function).
+
+        For example, $\gamma(a, b)$ defines new (synchronized) visible action alongside $a, b$.
+    }
+\end{itemize}
+
+We further define the following transition rules (omitting commutative equivalents). First, transition rules for basic process algebra wrt. termination, sequential composition, and choice:
+\begin{center}
+    \begin{tabular}{ccc}
+        $
+        \begin{prooftree}
+            \infer0{a \xrightarrow{a} \varepsilon}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} \varepsilon}
+            \infer1{a + b \xrightarrow{a} \varepsilon}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} \varepsilon}
+            \infer1{a \cdot b \xrightarrow{a} b}
+        \end{prooftree}
+        $   \\
+        \\
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} a^{'}}
+            \infer1{a + b \xrightarrow{a} a^{'}}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} a^{'}}
+            \infer1{a \cdot b \xrightarrow{a} a^{'} \cdot b}
+        \end{prooftree}
+        $   &
+            \\
+    \end{tabular}
+\end{center}
+
+Then, for parallel processes which may or may not communicate: 
+\begin{center}
+    \begin{tabular}{cc}
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} \varepsilon}
+            \infer1{a || b \xrightarrow{a} b}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} a^{'}}
+            \infer1{a || b \xrightarrow{a} a^{'} || b}
+        \end{prooftree}
+        $   \\
+        \\
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} \varepsilon}
+            \hypo{b \xrightarrow{b} \varepsilon}
+            \infer2{a || b \xrightarrow{\gamma(a, b)} \varepsilon}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} a^{'}}
+            \hypo{b \xrightarrow{b} \varepsilon}
+            \infer2{a || b \xrightarrow{\gamma(a, b)} a^{'}}
+        \end{prooftree}
+        $   \\
+        \\
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} \varepsilon}
+            \hypo{b \xrightarrow{b} b^{'}}
+            \infer2{a || b \xrightarrow{\gamma(a, b)} b^{'}}
+        \end{prooftree}
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{a} a^{'}}
+            \hypo{b \xrightarrow{b} b^{'}}
+            \infer2{a || b \xrightarrow{\gamma(a, b)} a^{'} || b^{'}}
+        \end{prooftree}
+        $   \\
+    \end{tabular}
+\end{center}
+
+Furthermore, for encapsulation ${\partial}_H$: 
+\begin{center}
+    \begin{tabular}{cc}
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{x} \varepsilon}
+            \infer1{\partial_H(a) \xrightarrow{x} \varepsilon}
+        \end{prooftree}\ 
+        x \notin H
+        $   &
+        $
+        \begin{prooftree}
+            \hypo{a \xrightarrow{x} a^{'}}
+            \infer1{{\partial}_H(a) \xrightarrow{x} {\partial}_H(a^{'})}
+        \end{prooftree}\ 
+        x \notin H
+        $   \\
+    \end{tabular}
+\end{center}
+This is to say, $\partial_H(a)$ can execute all transitions of $a$ that are also not in $H$.
+
+Finally, deadlocks \textbf{does not display any behavior} -- that is, a $\delta$ process cannot transition to any other states no matter what (though obviously as a constituent part of e.g., a parallel process the other concurrent constituent can still run). 
+
+\begin{background}[commutativity]
+    \begin{equation*}
+        f(a, b) = f(b, a) \iff f\ \mathrm{commutative}
+    \end{equation*}
+\end{background}
+
+\begin{background}[associativity]
+    \begin{equation*}
+        (a \circ b) \circ c = a \circ (b \circ c) \iff \circ\ \mathrm{associative}
+    \end{equation*}
+\end{background}
+
+\begin{background}[distributivity]
+    \begin{equation*}
+        f(x, a \circ b) = f(x, a) \circ f(x, b) \iff f\ \mathrm{distributes\ over}\ \circ
+    \end{equation*}
+\end{background}
+
+\begin{background}[isomorphism]
+    An isomorphism describes a \textbf{bijective} \textbf{homomorphism}: 
+    \begin{itemize}
+        \item {
+            \textbf{Homomorphism} describes a \textbf{structure-preserving} map between two algebraic \textbf{structures} of the same \textbf{type}: 
+            \begin{itemize}
+                \item {
+                    \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. 
+                }
+                \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 $\mathsf{map}(f, X^n) \in q$. 
+                }
+            \end{itemize}
+        }
+        \item {
+            \textbf{Bijection} describes a 1-to-1 correspondence between elements of two sets -- i.e., invertible. 
+        }
+    \end{itemize}
+\end{background}
+
+\end{document}

02-semantic-equivalences.tex

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+\documentclass[99-notes-packed.tex]{subfiles}
+
+\begin{document}
+
+\begin{background}[lattice]
+    A \textbf{lattice} describes a real coordinate space $\mathbb{R}^n$ that satisfies: 
+    \begin{itemize}
+        \item {
+            Addition / subtraction between two points always produce another point in lattice -- i.e., closed under addition / subtraction.
+        }
+        \item {
+            Lattice points are separated by bounded distances in some range $(0, \mathrm{max}]$.
+        }
+    \end{itemize}
+\end{background}
+
+Define a lattice over which \textit{semantic equivalence relations} for spec. and impl. verification is defined.
+
+\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*}
+\end{definition}
+
+\end{document}

99-notes-packed.tex

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+\documentclass{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[a4paper, margin=1cm]{geometry}
+
+\usepackage{multicol}                       % 2-col formatting
+\usepackage{hyperref}
+\usepackage[english]{babel}                 % theorem environ
+\usepackage{framed}
+\usepackage{mathtools}
+\usepackage{stmaryrd}                       % more arrows
+\usepackage{mathpartir}
+\usepackage{graphicx}
+\usepackage{tabularx}
+\usepackage{ebproof}
+\usepackage{amsmath}
+\usepackage{amssymb}
+
+\usepackage{subfiles}                       % multi-file project
+
+% environment `definition` prints as "Definition" and has counter reset per section.
+\newtheorem{definition}{Definition}[section]
+
+\newtheorem{background}{Background}[section]
+
+\begin{document}
+
+\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*}
+
+\end{document}