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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

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satisfiability
btwn
impl

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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$"}

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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}

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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}

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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}