From d15d288bec9a03a71a8491d52b06bb7df3a1dce0 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Mon, 1 Jul 2024 11:55:12 -0400 Subject: [PATCH 01/31] Setup tex files --- .gitignore | 11 +++++++ pieo-trees/notes.tex | 74 ++++++++++++++++++++++++++++++++++++++++++++ pieo-trees/refs.bib | 9 ++++++ 3 files changed, 94 insertions(+) create mode 100644 .gitignore create mode 100644 pieo-trees/notes.tex create mode 100644 pieo-trees/refs.bib diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..c05cabc --- /dev/null +++ b/.gitignore @@ -0,0 +1,11 @@ +# latex +*.pdf +*.synctex.gz +*.cls +*.aux +*.fdb_latexmk +*.fls +*.log +*.out +*.bbl +*.blg diff --git a/pieo-trees/notes.tex b/pieo-trees/notes.tex new file mode 100644 index 0000000..dcc614f --- /dev/null +++ b/pieo-trees/notes.tex @@ -0,0 +1,74 @@ +\documentclass{article} + +% font +\usepackage{cmbright} + +% margin +\usepackage[margin=1in]{geometry} + +% references +\usepackage[colorlinks]{hyperref} +\PassOptionsToPackage{colorlinks}{hyperref} +\hypersetup{urlcolor = RedViolet, linkcolor = RoyalBlue, citecolor = NavyBlue} +\urlstyle{same} + +\usepackage{amsmath, amssymb, amsthm} +\usepackage[svgnames, dvipsnames]{xcolor} +\usepackage{mhsetup, mathtools} + +\usepackage[capitalise]{cleveref} +\usepackage[T1]{fontenc} +\usepackage{silence} % for suppressing warnings +\usepackage{cite} + +% PL macros +\usepackage{mathpartir} +\usepackage{stmaryrd} +\newcommand{\inference}[3]{\inferrule*[Right=#1]{#2}{#3}} +\newcommand{\axiom}[2]{\inferrule*[Right=#1]{\;}{#2}} +\DeclareMathOperator{\bigstep}{\Downarrow} +\DeclareMathOperator{\false}{\textbf{false}} +\DeclareMathOperator{\true}{\mathbf{true}} +\newcommand{\hoare}[3]{\{#1\} \; #2 \; \{#3\}} + +% theorems +\newtheorem{thm}{Theorem}[section] +\newtheorem{lem}[thm]{Lemma} +\renewcommand{\thethm}{\arabic{thm}} % Remove subsection from counter +\theoremstyle{definition} +\newtheorem{dfn}{Definition}[section] +\renewcommand{\thedfn}{\arabic{dfn}} % Remove subsection from counter + +% no more indent +\setlength{\parindent}{0pt} + +\begin{document} + +{\LARGE \textbf{PIEO Trees for Fun and Profit}} + +\hrulefill + +This writeup formalizes syntax and semantics of PIEO tree, much like \cite{OG} does with PIFO trees. + +\section*{PIEOs} + +\section*{PIEO Trees} + +Let $\mathrm{Topo}$ be the set of (tree) topologies, as defined in \cite{OG}: +\begin{dfn} + The set of (tree) topologies, denoted $\mathrm{Topo}$, is the smallest set satisfying the rules: + \begin{align*} + \axiom{}{\ast \in \mathrm{Topo}} && + \inference{}{n \in \mathbb N\\ ts \in \mathrm{Topo}^n}{\mathrm{Node}(ts) \in \mathrm{Topo}} + \end{align*} + + Here, $\ast$ is the topology of a single-node tree, and, given a list $ts$ of $n$ topologies, $\mathrm{Node}(ts)$ is the topology of a node with $ts$ as its children. +\end{dfn} + +\newpage + +\renewcommand\refname{\Large References} +\bibliographystyle{alpha} +\bibliography{refs} + +\end{document} diff --git a/pieo-trees/refs.bib b/pieo-trees/refs.bib new file mode 100644 index 0000000..a6edd07 --- /dev/null +++ b/pieo-trees/refs.bib @@ -0,0 +1,9 @@ +@misc{OG, + title={Formal Abstractions for Packet Scheduling}, + author={Anshuman Mohan and Yunhe Liu and Nate Foster and Tobias Kappé and Dexter Kozen}, + year={2023}, + eprint={2211.11659}, + archivePrefix={arXiv}, + primaryClass={cs.NI}, + url={https://arxiv.org/abs/2211.11659}, +} From 593f850a05ee2650beed75ca6dd699abf1b314d1 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Sun, 7 Jul 2024 19:37:22 -0400 Subject: [PATCH 02/31] Fill in initial defn --- pieo-trees/notes.tex | 118 +++++++++++++++++++++++++++++++++++-------- 1 file changed, 98 insertions(+), 20 deletions(-) diff --git a/pieo-trees/notes.tex b/pieo-trees/notes.tex index dcc614f..a49a215 100644 --- a/pieo-trees/notes.tex +++ b/pieo-trees/notes.tex @@ -1,4 +1,4 @@ -\documentclass{article} +\documentclass{amsart} % font \usepackage{cmbright} @@ -31,43 +31,121 @@ \DeclareMathOperator{\true}{\mathbf{true}} \newcommand{\hoare}[3]{\{#1\} \; #2 \; \{#3\}} +% Formal Abstractions macros +\DeclareMathOperator{\pkt}{\mathrm{pkt}} + % theorems \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} -\renewcommand{\thethm}{\arabic{thm}} % Remove subsection from counter \theoremstyle{definition} -\newtheorem{dfn}{Definition}[section] -\renewcommand{\thedfn}{\arabic{dfn}} % Remove subsection from counter +\newtheorem{dfn}[thm]{Definition} +\newtheorem{abuse}[thm]{Abuse of Notation} +\newtheorem{ex}[thm]{Example} % no more indent \setlength{\parindent}{0pt} -\begin{document} - -{\LARGE \textbf{PIEO Trees for Fun and Profit}} - -\hrulefill +% right-justified sections hack +\usepackage{titlesec} +\newcommand{\marginsecnumber}[1]{% + \makebox[0pt][r]{#1\hspace{6pt}}% +} +\titleformat{\section} + {\normalfont\Large\bfseries} + {\marginsecnumber\thesection} + {0pt} + {} +\titleformat{\subsection} + {\normalfont\large\bfseries} + {\marginsecnumber\thesubsection} + {0pt} + {} +\titleformat{\subsubsection} + {\normalfont\normalsize\bfseries} + {\marginsecnumber\thesubsubsection} + {0pt} + {} -This writeup formalizes syntax and semantics of PIEO tree, much like \cite{OG} does with PIFO trees. +\begin{document} -\section*{PIEOs} +\pagestyle{empty} -\section*{PIEO Trees} +{\LARGE \textbf{PIEO Trees for Fun and Profit}} -Let $\mathrm{Topo}$ be the set of (tree) topologies, as defined in \cite{OG}: -\begin{dfn} - The set of (tree) topologies, denoted $\mathrm{Topo}$, is the smallest set satisfying the rules: +\hrulefill\\ + +We assume familiarity with \cite{OG}, adopt its notational conventions, and borrow many of its definitions! + +% \section{Overview} +% +% { +% \color{red} +% TODO! +% \begin{enumerate} +% \item An example of a scheduling transaction that cannot be expressed with PIFO trees: maybe token bucket? +% \item Introduce what a PIEO is and what it does +% \item An overview of what to expect with each section. +% \end{enumerate} +% } +% +% \newpage + +\section{Structure \& Semantics} + +Our PIEO trees are near identical to PIFO trees, with only two modifications: +\begin{enumerate} + \item All nodes hold PIEOs instead of PIFOs + \item Internal PIEOs hold just enough extra data to compute predicates of packets that live in subtrees. +\end{enumerate} + +To make ``just enough extra data'' precise, let $f$ be a predicate on $\mathbf{Pkt}$. +We'll presuppose the existence of a set $\mathbf{Data}_f$ and associated surjective map $\mathrm{data}_f: \mathbf{Pkt} \to \mathbf{Data}_f$. +This map and set are such that, for $\pkt, \pkt^\prime \in \mathbf{Pkt}$, +$$\mathrm{data}_f(\pkt) = \mathrm{data}_f(\pkt^\prime) \implies f(\pkt) = f(\pkt^\prime)$$ +In other words, all preimages of $d \in \mathbf{Data}_f$ evaluate to the same truth value under $f$. +For this reason, + +\begin{abuse} + We'll often write $f(d)$ when we mean $f(\pkt)$ for some $\pkt \in f^{-1}(d)$. +\end{abuse} + +Broadly, this set $\mathbf{Data}_f$ is designed to encode just the parts of $\mathbf{Pkt}$ necessary for computing $f(\pkt)$. + +\begin{ex} + For any $f$, it is legal to set $\mathrm{Data}_f = \mathbf{Pkt}$ and $\mathrm{data}_f$ to be the identity map. + We'd expect this to work since each $\pkt \in \mathbf{Pkt}$ certainly holds all the information necessary to compute $f(\pkt)$. +\end{ex} + +We can now formally define PIEO trees! + +\begin{dfn}[PIEO tree] + Let $t \in \mathbf{Topo}$ be a topology and $f$ a predicate over $\mathbf{Pkt}$. + The set of \emph{PIEO trees} over $t$ and $f$, denoted $\mathbf{PIEOTree}(t, f)$, is defined inductively by \begin{align*} - \axiom{}{\ast \in \mathrm{Topo}} && - \inference{}{n \in \mathbb N\\ ts \in \mathrm{Topo}^n}{\mathrm{Node}(ts) \in \mathrm{Topo}} + \inference{} + { + p \in \mathbf{PIEO}(\mathbf{Pkt}, f) + } + { + \mathrm{Leaf}(p) \in \mathbf{PIEOTree}(\ast) + } + && + \inference{} + { + n \in \mathbb N\\ + ts \in \mathbf{Topo}^n\\ + p \in \mathbf{PIEO}(\{1,\ldots, n\} \times \mathbf{Data}_f(\mathbf{Pkt}), f)\\\\ + \forall i \in [1, n]. \; qs[i] \in \mathbf{PIEOTree}(ts[i]) + } + { + \mathrm{Internal}(qs, p) \in \mathbf{PIEO}(\mathrm{Node}(ts)) + } \end{align*} - - Here, $\ast$ is the topology of a single-node tree, and, given a list $ts$ of $n$ topologies, $\mathrm{Node}(ts)$ is the topology of a node with $ts$ as its children. \end{dfn} \newpage -\renewcommand\refname{\Large References} +\renewcommand\refname{\LARGE References} \bibliographystyle{alpha} \bibliography{refs} From fe3a3d4cf32b65e489ff6ba1404a27780ad48766 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Sun, 7 Jul 2024 19:39:14 -0400 Subject: [PATCH 03/31] Push pdf --- .gitignore | 1 - pieo-trees/notes.pdf | Bin 0 -> 111107 bytes 2 files changed, 1 deletion(-) create mode 100644 pieo-trees/notes.pdf diff --git a/.gitignore b/.gitignore index c05cabc..2e5c757 100644 --- a/.gitignore +++ b/.gitignore @@ -1,5 +1,4 @@ # latex -*.pdf *.synctex.gz *.cls *.aux 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\end{enumerate} \end{dfn} @@ -217,8 +217,21 @@ \section{Structure \& Semantics} We say $q$ is well-formed, denoted $\vdash q$, if for all $f \in \mathcal F$, $\vdash_f q$. \end{dfn} -\newpage +\section{Embedding and Simulation} +\begin{dfn} + Let $t_1, t_2 \in \Topo$. + We call a relation $R \subseteq \PIFOTree(t_1) \times \PIFOTree(t_2)$ a \emph{simulation} if, + for all $\pkt \in \Pkt$, $f \in \mathcal F$, and $q_1 \; R \; q_2$, + \begin{enumerate} + \item If $\pop(q_1, f)$ is undefined, then so is $\pop(q_2, f)$ + \item If $\pop(q_1, f) = (\pkt, q_1^\prime)$, then $\pop(q_2) = (\pkt, q_2^\prime)$ such that $q_1^\prime \; R \; q_2^\prime$. + \item For all $pt_1 \in \Path(t_1)$ and $d \in \Data$, there exists $pt_2 \in \Path(t_2)$ such that + $$\push(q_1, \pkt, d, pt_1) \; R \; \push(q_2, \pkt, d, pt_2)$$ + \end{enumerate} +\end{dfn} + +\newpage \renewcommand\refname{\LARGE References} \bibliographystyle{alpha} From 2b77335944af6ef0cbc0b7f010ac49ef597a6c86 Mon Sep 17 00:00:00 2001 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Semantics} \item admit partial functions \begin{align*} \pop &: \PIEO(S, D, F) \times F \halfto S \times \PIEO(S, D, F) \\ - \push &: \PIEO(S, D, F) \times S \times D \times \Rk \to \PIEO(S, D, F) + \push &: \PIEO(S, D, F) \times S \times D \times \Rk \to \PIEO(S, D, F) \\ + \proj &: \PIEO(S, D, F) \times F \to \PIFO(S) \end{align*} \end{enumerate} + Maps $\push$ and $\pop$ are as usual. + The \emph{projection} $\proj(p, f)$ is the PIFO of entries in $p$ with data satisfying $f$. For $p \in \PIEO(S, D, F)$, $s \in S$, and $f \in F$, we write \begin{enumerate} \item $|p|$ for the number of entries in $p$ @@ -211,7 +215,7 @@ \section{Structure \& Semantics} && \inference{} { - \forall i \in [1, |qs|] \; \vdash_f qs[i] \land |p|_{i, f} = |qs[i]| + \forall i \in [1, |qs|], \; \vdash_f qs[i] \land |p|_{i, f} = |qs[i]| } { \vdash_f \Internal(qs, p) @@ -221,7 +225,66 @@ \section{Structure \& Semantics} We say $q$ is well-formed, denoted $\vdash q$, if there exists $f \in \mathcal F$ such that, for all $f^\prime \geq f$, $\vdash_{f^\prime} q$. \end{dfn} -\section{Embedding and Simulation} +\section{Projection} + +\begin{dfn} + For $f \in \mathcal F$, define $\proj_f: \PIEOTree(t) \to \PIFOTree(t)$ by + \begin{align*} + \inference{} + { + p^\prime = \proj(p, f) + } + { + \proj_f(\Leaf(p)) = \Leaf(p^\prime) + } + && + \inference{} + { + p^\prime = \proj(p, f)\\ + \forall i \in [1, |qs|], \; qs^\prime[i] = \proj_f(qs[i]) + } + { + \proj_f(\Internal(qs, p)) = \Internal(qs^\prime, p^\prime) + } + \end{align*} +\end{dfn} + +\begin{lem} + For $q \in \PIEOTree(t)$ and $f \in \mathcal F$, $\pop(q, f)$ is undefined if and only if $\pop(\proj_f(q))$ is undefined. +\end{lem} + +\begin{proof} + TODO: structural induction on $q$ +\end{proof} + +\begin{lem} + For $q \in \PIEOTree(t)$ and $f \in \mathcal F$, + $$ + \pop(q, f) = (\pkt, q^\prime) \implies \pop(\proj_f(q)) = (\pkt, \proj_f(q^\prime)) + $$ +\end{lem} + +\begin{proof} + TODO! +\end{proof} + +\begin{lem} + For $q \in \PIEOTree(t)$, $\pkt \in \Pkt$, $d \in \Data$, $pt \in \Path(t)$, and $f \in \mathcal F$, + $$ + \proj_f(\push(q, \pkt, d, pt)) = + \begin{cases} + \push(\proj_f(q), \pkt, pt) & f(d) \text{ holds true}\\ + \proj_f(q) & \text{otherwise} + \end{cases} + $$ +\end{lem} + +\begin{proof} + TODO! +\end{proof} + + +\section{Embedding \& Simulation} \begin{dfn} Let $t_1, t_2 \in \Topo$. 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\usepackage[colorlinks]{hyperref} \PassOptionsToPackage{colorlinks}{hyperref} \hypersetup{urlcolor = RedViolet, linkcolor = RoyalBlue, citecolor = NavyBlue} -\urlstyle{same} -\usepackage{amsmath, amssymb, amsthm} +% basics +\usepackage[leqno]{amsmath} +\usepackage{amssymb, amsthm} \usepackage[svgnames, dvipsnames]{xcolor} \usepackage{mhsetup, mathtools} - \usepackage[capitalise]{cleveref} -\usepackage[T1]{fontenc} -\usepackage{silence} % for suppressing warnings -\usepackage{cite} + +% commutative diagram +\usepackage{tikz-cd} % hats \usepackage{yhmath} @@ -29,10 +29,6 @@ \usepackage{stmaryrd} \newcommand{\inference}[3]{\inferrule*[Right=#1]{#2}{#3}} \newcommand{\axiom}[2]{\inferrule*[Right=#1]{\;}{#2}} -\DeclareMathOperator{\bigstep}{\Downarrow} -\DeclareMathOperator{\false}{\textbf{false}} -\DeclareMathOperator{\true}{\mathbf{true}} -\newcommand{\hoare}[3]{\{#1\} \; #2 \; \{#3\}} % (Formal Abstractions macros)++ \DeclareMathOperator{\halfto}{\rightharpoonup} @@ -86,6 +82,15 @@ {0pt} {} +% tag on the right hack +\makeatletter +\newcommand{\leqnomode}{\tagsleft@true} +\newcommand{\reqnomode}{\tagsleft@false} +\makeatother + +% todos +\usepackage[textwidth=0.7in]{todonotes} + \begin{document} \pagestyle{empty} @@ -94,11 +99,14 @@ \hrulefill\\ -We assume familiarity with \cite{OG}, adopt its notational conventions, and borrow its definitions! +\reqnomode + +We assume familiarity with \cite{OG}, adopt its notational conventions, and borrow many of its definitions! \section{Structure \& Semantics} \begin{dfn} + \label{dfn:pieo} For sets $S$, $D$, and predicates $F$ over $D$, let $\PIEO(S, D, F)$ denote the set of \emph{PIEO}s that \begin{enumerate} @@ -114,7 +122,9 @@ \section{Structure \& Semantics} \end{enumerate} Maps $\push$ and $\pop$ are as usual. The \emph{projection} $\proj(p, f)$ is the PIFO of entries in $p$ with data satisfying $f$. - For $p \in \PIEO(S, D, F)$, $s \in S$, and $f \in F$, we write + We consider PIEOs $p, p^\prime$ equal if, for all $f \in F$, $\proj(p,f) = \proj(p^\prime, f)$, i.e. their projections are always equal. + + For PIEO $p$, entry $s \in S$, and predicate $f \in F$, we write \begin{enumerate} \item $|p|$ for the number of entries in $p$ \item $|p|_{s}$ for the number of times $s$ occurs in $p$ @@ -250,14 +260,41 @@ \section{Projection} \end{dfn} \begin{lem} + \label{lem:pieotree_eq} + For $q, q^\prime \in \PIEOTree(t)$, + $$\forall f \in \mathcal F, \; \proj_f(q) = \proj_f(q^\prime) \implies q = q^\prime$$ +\end{lem} + +\begin{proof} + Suppose $\proj_f(q) = \proj_f(q^\prime)$ for all $f \in \mathcal F$. + We'll proceed by induction on $t$ to show $q = q^\prime$. + \begin{itemize} + \item[(Leaf)] For $t = \ast$, let $q = \Leaf(p)$ and $q^\prime = \Leaf(p^\prime)$. + Since $$\proj_f(q) = \Leaf(\proj(p, f)) = \Leaf(\proj(p^\prime, f)) = \proj_f(q^\prime)$$ + we know $\proj(p, f) = \proj(p^\prime, f)$ for all $f \in \mathcal F$. + By \Cref{dfn:pieo}, $p = p^\prime$ and hence $q = q^\prime$. + \item[(Node)] For $t = \Node(ts)$ and $n = |ts|$, let $q = \Internal(qs, p)$ and $q^\prime = \Internal(qs^\prime, p)$. + Notice + \begin{align*} + \proj_f(p) &= \proj_f(p^\prime)\\ + \proj_f(qs[i]) &= \proj_f(qs^\prime[i]) \tag{$i = 1,\ldots, n$} + \end{align*} + for all $f \in \mathcal F$. + Hence, $p = p^\prime$ via \Cref{dfn:pieo} and $qs = qs^\prime$ by the inductive hypothesis, i.e. $q = q^\prime$. + \end{itemize} +\end{proof} + +\begin{lem} + \label{lem:pop_undef} For $q \in \PIEOTree(t)$ and $f \in \mathcal F$, $\pop(q, f)$ is undefined if and only if $\pop(\proj_f(q))$ is undefined. \end{lem} \begin{proof} - TODO: structural induction on $q$ + \textcolor{red}{TODO} \end{proof} \begin{lem} + \label{lem:pop} For $q \in \PIEOTree(t)$ and $f \in \mathcal F$, $$ \pop(q, f) = (\pkt, q^\prime) \implies \pop(\proj_f(q)) = (\pkt, \proj_f(q^\prime)) @@ -265,10 +302,11 @@ \section{Projection} \end{lem} \begin{proof} - TODO! + \textcolor{red}{TODO} \end{proof} \begin{lem} + \label{lem:push} For $q \in \PIEOTree(t)$, $\pkt \in \Pkt$, $d \in \Data$, $pt \in \Path(t)$, and $f \in \mathcal F$, $$ \proj_f(\push(q, \pkt, d, pt)) = @@ -280,13 +318,14 @@ \section{Projection} \end{lem} \begin{proof} - TODO! + \textcolor{red}{TODO} \end{proof} \section{Embedding \& Simulation} \begin{dfn} + \label{dfn:sim} Let $t_1, t_2 \in \Topo$. We call a relation $R \subseteq \PIEOTree(t_1) \times \PIEOTree(t_2)$ a \emph{simulation} if, for all $\pkt \in \Pkt$, $f \in \mathcal F$, and $q_1 \; R \; q_2$, @@ -331,17 +370,113 @@ \section{Embedding \& Simulation} \end{itemize} \end{dfn} +\begin{thm} + \label{thm:cd} + The following diagram commutes + $$ + \begin{tikzcd} + {\PIEOTree(t_1)} &&& {\PIEOTree(t_2)} \\ + \\ + {\PIFOTree(t_1)} &&& {\PIFOTree(t_2)} + \arrow["{\overline{f}}", from=1-1, to=1-4] + \arrow["{\proj_g}"{description}, from=1-1, to=3-1] + \arrow["{\proj_g}"{description}, from=1-4, to=3-4] + \arrow["{\widehat{f}}", from=3-1, to=3-4] + \end{tikzcd} + $$ + In other words, for $q_1 \in \PIEOTree(t_1)$, $q_2 = \overline{f}(q_1) \in \PIEOTree(t_2)$, and $g \in \mathcal F$, $\proj_g(q_2) = \widehat{f}(\proj_g(q_1))$. +\end{thm} + +\begin{proof} + \textcolor{red}{TODO} +\end{proof} + \begin{thm} \label{thm:profit} Let $t_1, t_2 \in \Topo$. If $f$ embeds $t_1$ into $t_2$, then - $$R = \{(q, f(q)) \mid q \in \PIEOTree(t_1)\}$$ + $$R = \{(q, \overline{f}(q)) \mid q \in \PIEOTree(t_1)\}$$ is a simulation. \end{thm} +\begin{proof} + We'll show the conditions from \Cref{dfn:sim} hold. + Fix $g \in \mathcal F$ and $q_1 \in \PIEOTree(t_1)$. + Let $q_2 = \overline{f}(q_1)$. + \begin{enumerate} + \item Suppose $\pop(q_1, g)$ is undefined. + Applying both \Cref{lem:pop_undef} and \cite[Lemma ~5.6]{OG}, notice $\pop(\widehat{f}(\proj_g(q_1)))$ is undefined. + By \Cref{thm:cd}, $\widehat{f}(\proj_g(q_1)) = \proj_g(q_2)$. + Hence, $\pop(\proj_g(q_2))$ is undefined. + Applying \Cref{lem:pop_undef} once more, $\pop(q_2, g)$ is undefined. + + \item Suppose $\pop(q_1, g)$ is defined. + By \Cref{lem:pop_undef} and \cite[Lemma ~5.6]{OG}, $\pop(\widehat{f}(\proj_g(q_1)))$ is defined. + Hence, $\pop(\proj_g(q_2))$ is defined via \Cref{thm:cd}. + By \Cref{lem:pop_undef}, $\pop(q_2, g)$ is defined. + Let's say + \begin{align*} + \pop(q_1, g) = (\pkt_1, q_1^\prime) && \pop(q_2, g) = (\pkt_2, q_2^\prime) + \end{align*} + By \Cref{lem:pop}, + \begin{align*} + \pop(\proj_g(q_1)) = (\pkt_1, \proj_g(q_1^\prime)) + && + \pop(\proj_g(q_2)) = (\pkt_2, \proj_g(q_2^\prime)) + \end{align*} + By \cite[Lemma ~5.7]{OG}, + $$\pop(\widehat{f}(\proj_g(q_1))) = (\pkt_1, \widehat{f}(\proj_g(q_1^\prime)))$$ + By \Cref{thm:cd}, + \begin{align*} + \pkt_1 &= \pkt_2\\ + \widehat{f}(\proj_g(q_1^\prime)) &= \proj_g(q_2^\prime) \label{eq:popsim} \tag{$\dagger$} + \end{align*} + Since our choice of $g$ was arbitrary, notice \Cref{eq:popsim} holds for all $g \in \mathcal F$. + Hence, using \Cref{thm:cd}, + $$\proj_{g^\prime}(\overline{f}(q_1^\prime)) = \widehat{f}(\proj_{g^\prime}(q_1^\prime)) = \proj_{g^\prime}(q_2^\prime)$$ + for all $g^\prime \in \mathcal F$. + At last, \Cref{lem:pieotree_eq} yields $\overline{f}(q_1^\prime) = q_2^\prime$. \todo{this logic is wrong!} + + \leqnomode + + \item Consider $\pkt \in \Pkt$, $d \in \Data$, and $pt \in \Path(t_1)$. + For $g \in \mathcal F$ such that $g(d)$ holds true, + \begin{align*} + \proj_g(\overline{f}(\push(q_1, \pkt, d, pt))) + &= \widehat{f}(\proj_g(\push(q_1, \pkt, d, pt))) \tag{by \Cref{thm:cd}}\\ + &= \widehat{f}(\push(\proj_g(q_1), \pkt, pt)) \tag{by \Cref{lem:push}}\\ + &= \push(\widehat{f}(\proj_g(q_1)), \pkt, \widetilde{f}(pt)) \tag{by \cite[Lemma ~5.9]{OG}}\\ + &= \push(\proj_g(q_2), \pkt, \widetilde{f}(pt)) \tag{by \Cref{thm:cd}}\\ + &= \proj_g(\push(q_2, \pkt, d, \widetilde{f}(pt))) \tag{by \Cref{lem:push}} + \end{align*} + For $g \in \mathcal F$ such that $g(d)$ does not hold true, + \begin{align*} + \proj_g(\overline{f}(\push(q_1, \pkt, d, pt))) + &= \widehat{f}(\proj_g(\push(q_1, \pkt, d, pt))) \tag{by \Cref{thm:cd}}\\ + &= \widehat{f}(\proj_g(q_1)) \tag{by \Cref{lem:push}}\\ + &= \proj_g(q_2) \tag{by \Cref{thm:cd}}\\ + &= \proj_g(\push(q_2, \pkt, d, \widetilde{f}(pt))) \tag{by \Cref{lem:push}} + \end{align*} + Overall, + $ + \proj_g(\overline{f}(\push(q_1, \pkt, d, pt))) + = + \proj_g(\push(q_2, \pkt, d, \widetilde{f}(pt))) + $ + for all $g \in \mathcal F$. + Hence, + $$ + \overline{f}(\push(q_1, \pkt, d, pt)) + = + \push(q_2, \pkt, d, \widetilde{f}(pt)) + $$ + by \Cref{lem:pieotree_eq}, as desired. + \end{enumerate} +\end{proof} + \newpage \renewcommand\refname{\LARGE References} -\bibliographystyle{apalike} +\bibliographystyle{alpha} \bibliography{refs} \end{document} From 31eaa89859d764be6b43f860e517fd91b43e0131 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Fri, 19 Jul 2024 00:00:22 -0400 Subject: [PATCH 10/31] Prove Lemma 2.3 --- pieo-trees/notes.pdf | Bin 138869 -> 139583 bytes pieo-trees/notes.tex | 18 +++++++++++++++++- 2 files changed, 17 insertions(+), 1 deletion(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index aeec73140cb07e65f1b980571194776000ebebeb..358d49b38430b46bc23061b66bcb5f27039ac907 100644 GIT binary patch delta 28976 zcmZUaQ+J>Z&>&;mwry*YOl;e>ojkEInb@|?iEU?MV`3ZoerI>jUhG}}gRWCu{n!oD 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notice \Cref{eq:popsim} holds for all $g \in \mathcal F$. + Since our choice of $g$ was arbitrary, notice \Cref{eq:popsim} holds for all $g \in \mathcal F$ \textcolor{red}{(this is not true!)}. Hence, using \Cref{thm:cd}, $$\proj_{g^\prime}(\overline{f}(q_1^\prime)) = \widehat{f}(\proj_{g^\prime}(q_1^\prime)) = \proj_{g^\prime}(q_2^\prime)$$ for all $g^\prime \in \mathcal F$. - At last, \Cref{lem:pieotree_eq} yields $\overline{f}(q_1^\prime) = q_2^\prime$. \todo{this logic is wrong!} + At last, \Cref{lem:pieotree_eq} yields $\overline{f}(q_1^\prime) = q_2^\prime$. \leqnomode From 978f26d347b5a15eb394833573e639c9dff8eb38 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Fri, 19 Jul 2024 15:28:36 -0400 Subject: [PATCH 12/31] Prove diagram commutes --- pieo-trees/notes.pdf | Bin 139282 -> 142616 bytes pieo-trees/notes.tex | 51 +++++++++++++++++++++++++++++++++++++++---- 2 files changed, 47 insertions(+), 4 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index dde19e02fb95af79be5dfec8b73d76359e271241..a868b12eaa8008cc6a18fcf563e3c80b660e7335 100644 GIT binary patch delta 28889 zcmZshQ*fpY(5ByH;)!kBwk9?wb|#+KeqwWCPi#+Y+qP{x*>C@<-96c(uC6}mgYK@o 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\Topo$, let $f$ be an embedding from $t_1$ to $t_2$. We lift $f$ to a map $\overline{f}$ from $\PIEOTree(t_1)$ to $\PIEOTree(t_2)$ inductively. \begin{itemize} @@ -383,6 +384,8 @@ \section{Embedding \& Simulation} \end{itemize} \end{dfn} +\reqnomode + \begin{thm} \label{thm:cd} The following diagram commutes @@ -397,11 +400,51 @@ \section{Embedding \& Simulation} \arrow["{\widehat{f}}", from=3-1, to=3-4] \end{tikzcd} $$ - In other words, for $q_1 \in \PIEOTree(t_1)$, $q_2 = \overline{f}(q_1) \in \PIEOTree(t_2)$, and $g \in \mathcal F$, $\proj_g(q_2) = \widehat{f}(\proj_g(q_1))$. + In other words, for $q \in \PIEOTree(t_1)$ and $g \in \mathcal F$, $\proj_g(\overline{f}(q)) = \widehat{f}(\proj_g(q))$. \end{thm} \begin{proof} - \textcolor{red}{TODO} + We'll proceed by induction on $t_1$. + Suppose $t_1 = \ast$ and $q = \Leaf(p)$. + By \cite[Lemma ~5.3]{OG}, $t_2 = \ast$ as well. + By \Cref{dfn:f_bar} and \cite[Definition ~5.4]{OG}, both $\overline{f}$ and $\widehat{f}$ are the identity. + Hence, + $$\proj_g(\overline{f}(q)) = \proj_g(q) = \widehat{f}(\proj_g(q))$$ + + Suppose $t_1 = \Node(ts)$ and $q = \Internal(qs, p)$. + For any prefix $\alpha$ of $f(i)$ for $i \in [1, |ts|]$, we'll show + \begin{equation} + \label{eq:cd1} + \proj_g(\overline{f}(q)_{\alpha}) = \widehat{f}(\proj_g(q))_{\alpha} + \tag{$\ast$} + \end{equation} + by inverse induction on $\alpha$. + Instantiating \Cref{eq:cd1} with $\alpha = \epsilon$ yields the desired result. + \begin{itemize} + \item For $\alpha = f(i)$, \Cref{eq:cd1} holds by the outer inductive hypothesis because + $$\proj_g(\overline{f}(q)_{\alpha}) = \proj_g(\overline{f_i}(q)) = \widehat{f_i}(\proj_g(q)) = \widehat{f}(\proj_g(q))_{\alpha}$$ + + \item Suppose $\alpha$ is some strict prefix of $f(i)$, pointing to a node with $m$ children. Let + \begin{align*} + \proj_g(\overline{f}(q)) = \Internal(qs^\prime, p^\prime) + &&\text{and}&& + \widehat{f}(\proj_g(q)) = \Internal(qs^{\prime\prime}, p^{\prime\prime}) + \end{align*} + + There's two parts to showing \Cref{eq:cd1}, namely $qs^{\prime} = qs^{\prime\prime}$ and $p^\prime = p^{\prime\prime}$. + + \begin{itemize} + \item For all $j \in [1,m]$, + $$\proj_g(\overline{f}(q)_{\alpha \cdot j}) = \widehat{f}(\proj_g(q))_{\alpha \cdot j}$$ + For $j$ such that $\alpha \cdot j$ is a prefix of some $f(i)$, this follows from the inner inductive hypothesis. + For all other $j$, notice the LHS and RHS are both PIFO trees of topology $t_2$, with empty PIFOs on all leaves and internal nodes. + Hence, $qs^{\prime}[j] = qs^{\prime\prime}[j]$ for all $j \in [1,m]$, i.e. $qs^{\prime} = qs^{\prime\prime}$. + \item By inspection, it's clear following the construction for $p_\alpha$ from \Cref{dfn:f_bar} and then computing the projection $\proj(p_\alpha, g)$ + yields the same result as following the recipe for $p_\alpha$ from \cite[Definition ~5.4]{OG} on the projection $\proj_g(q)$: + that is, exactly when we filter out elements not satisfying $g$ does not matter. + Hence, $p^{\prime} = p^{\prime\prime}$. + \end{itemize} + \end{itemize} \end{proof} \begin{thm} From 5c3a46a25741eaed710a20d6185da82dbaf9e73b Mon Sep 17 00:00:00 2001 From: polybeandip Date: Fri, 19 Jul 2024 17:37:43 -0400 Subject: [PATCH 13/31] Prove Lemma 2.5 --- pieo-trees/notes.pdf | Bin 142616 -> 144244 bytes pieo-trees/notes.tex | 69 ++++++++++++++++++++++++++++++++++++++----- 2 files changed, 61 insertions(+), 8 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index a868b12eaa8008cc6a18fcf563e3c80b660e7335..fdd66742935819ba866a67fecb9a1e0237439dbe 100644 GIT binary patch delta 25971 zcmZU4LwF`yv}LS{ZQC{~wr$&H<&SOKwr$(CZ99Em_nY->pV1!e^(~#W4XCFRs5lZ( z7EY$b{BJ72wvJ{ZPAih{u6}=4QCm#d9{ddq>w7pZa-ikYyK5nn%Zrzo`EOf6^=$91&i8XWItzJtT*j=Ooq5PCY+(=# 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The \emph{projection} $\proj(p, f)$ is the PIFO of entries in $p$ with data satisfying $f$. + These three maps play nicely together: + \begin{align} + \pop(p, f) \text{ is undefined} &\iff \pop(\proj(p, f)) \text{ is undefined} \label{eq:pieo1}\\ + \pop(p, f) = (\pkt, p^\prime) &\implies \pop(\proj(p, f)) = (\pkt, \proj(p^\prime, f)) \label{eq:pieo2}\\ + \proj(\push(p, s, d, r), f) &= + \begin{cases} + \push(\proj(p, f), s, r) & f(d) \text{ holds true}\\ + \proj(p, f) & \text{otherwise} + \end{cases} + \label{eq:pieo3} + \end{align} We consider PIEOs $p, p^\prime$ equal if, for all $f \in F$, $\proj(p,f) = \proj(p^\prime, f)$, i.e. their projections are always equal. For PIEO $p$, entry $s \in S$, and predicate $f \in F$, we write @@ -290,19 +301,20 @@ \section{Projection} We'll do induction on $t$. \begin{itemize} \item[(Leaf)] For $t = \ast$, let $q = \Leaf(p)$ and $\proj_f(q) = \Leaf(p^\prime)$. + By \Cref{eq:pieo1} in \Cref{dfn:pieo}, \begin{align*} - \pop(q, f) \text{ is undefined} &\iff \text{$p$ has no items with meta-data satisfying $f$}\\ - &\iff p^\prime \text{ is empty}\\ + \pop(q, f) \text{ is undefined} &\iff \pop(p, f) \text{ is undefined} \\ + &\iff \pop(p^\prime) \text{ is undefined} \\ &\iff \pop(\proj_f(q)) \text{ is undefined} \end{align*} \item[(Node)] For $t = \Node(ts)$, let $q = \Internal(qs, p)$ and $\proj_f(q) = \Internal(qs^\prime, p^\prime)$. Notice - $$\text{$p$ has no items with meta-data satisfying $f$} \iff p^\prime \text{ is empty}$$ - and + $$\pop(p, f) \text{ is undefined} \iff \pop(p^\prime) \text{ is undefined}$$ + by \Cref{eq:pieo1} in \Cref{dfn:pieo} and $$\pop(qs[i], f) \text{ is undefined} \iff \pop(qs^\prime[i]) \text{ is undefined $\forall i \in [1, |ts|]$}$$ by the inductive hypothesis. - Hence, $\pop(q, f)$ is undefined if and only if $\pop(\proj_f(q))$ is, as desired. + Hence, $\pop(q, f)$ is undefined if and only if $\pop(\proj_f(q))$ is. \end{itemize} \end{proof} @@ -315,7 +327,11 @@ \section{Projection} \end{lem} \begin{proof} - \textcolor{red}{TODO} + More induction on $t$! + \begin{itemize} + \item[(Leaf)] + \item[(Node)] + \end{itemize} \end{proof} \begin{lem} @@ -331,7 +347,45 @@ \section{Projection} \end{lem} \begin{proof} - \textcolor{red}{TODO} + Even more induction on $t$! + \begin{itemize} + \item[(Leaf)] For $t = \ast$, let $q = \Leaf(p)$ and $pt = r$. + By \Cref{eq:pieo3} in \Cref{dfn:pieo}. + \begin{align*} + \proj_f(\push(q,\pkt, d, pt)) &= \Leaf(\proj(\push(p, \pkt, d, r), f)) \\ + &= + \begin{cases} + \Leaf(\push(\proj(p, f), \pkt, r)) & f(d) \text{ holds true}\\ + \Leaf(\proj(p, f)) & \text{otherwise} + \end{cases}\\ + &= + \begin{cases} + \push(\proj_f(q), \pkt, pt) & f(d) \text{ holds true}\\ + \proj_f(q) & \text{otherwise} + \end{cases} + \end{align*} + \item[(Node)] For $t = \Node(ts)$, let $q = \Internal(qs, p)$ and $pt = (i, r) :: pt^\prime$. + Define + \begin{align*} + p^\prime = \push(p, i, d, r) && + q^\prime = \push(qs[i] \pkt, d, pt^\prime) && + qs^\prime &= qs[q^\prime/i] + \end{align*} + By \Cref{eq:pieo3} in \Cref{dfn:pieo} and the inductive hypothesis, + \begin{align*} + \proj(p^\prime, f) &= + \begin{cases} + \push(\proj(p, f), i, r) & f(d) \text{ holds true}\\ + \proj(p, f) & \text{otherwise} + \end{cases}\\ + \proj_f(qs^\prime, \pkt, d, pt^\prime)) &= + \begin{cases} + \push(\proj_f(qs[i]), \pkt, pt^\prime) & f(d) \text{ holds true}\\ + \proj_f(qs[i]) & \text{otherwise} + \end{cases} + \end{align*} + Since $\push(q, \pkt, d, pt)) = \Internal(qs^\prime, p^\prime)$, the desired result follows. + \end{itemize} \end{proof} @@ -448,7 +502,6 @@ \section{Embedding \& Simulation} \end{proof} \begin{thm} - \label{thm:profit} Let $t_1, t_2 \in \Topo$. If $f$ embeds $t_1$ into $t_2$, then $$R = \{(q, \overline{f}(q)) \mid q \in \PIEOTree(t_1)\}$$ is a simulation. 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$\vdash_{f^\prime} q$. \end{dfn} +\newpage + \section{Projection} \begin{dfn} @@ -465,7 +469,6 @@ \section{Projection} \end{itemize} \end{proof} - \section{Embedding \& Simulation} \begin{dfn} @@ -598,33 +601,79 @@ \section{Embedding \& Simulation} $$\pop(q, g) = (\pkt, q^\prime) \implies \pop(\overline{f}(q), g) = (\pkt, \overline{f}(q^\prime))$$ \end{lem} +\textcolor{red}{Almost a clone of the proof \cite[Lemma ~5.7]{OG}.} + \begin{proof} - Let $q_2 = \overline{f}(q_1)$ and suppose $\pop(q_1, g)$ is defined. - By \Cref{lem:pop_undef} and \cite[Lemma ~5.6]{OG}, $\pop(\widehat{f}(\proj_g(q_1)))$ is defined. - Hence, $\pop(\proj_g(q_2))$ is defined via \Cref{thm:cd}. - By \Cref{lem:pop_undef}, $\pop(q_2, g)$ is defined. - Let's say - \begin{align*} - \pop(q_1, g) = (\pkt_1, q_1^\prime) && \pop(q_2, g) = (\pkt_2, q_2^\prime) - \end{align*} - By \Cref{lem:pop}, + We'll proceed by induction on $t_1$. + Suppose $t_1 = \ast$. + By \cite[Lemma ~5.3]{OG}, $t_2 = \ast$ as well. + By \Cref{dfn:f_bar}, $\overline{f}$ is the identity. + Hence, + $$\pop(q, g) = (\pkt, q^\prime) \implies \pop(\overline{f}(q), g) = \pop(q, g) = (\pkt, q^\prime) = (\pkt, \overline{f}(q^\prime))$$ + Suppose $t_1 = \Node(ts)$. + Let \begin{align*} - \pop(\proj_g(q_1)) = (\pkt_1, \proj_g(q_1^\prime)) - && - \pop(\proj_g(q_2)) = (\pkt_2, \proj_g(q_2^\prime)) + q = \Internal(qs, p) && + q^\prime = \Internal(qs^\prime, p^\prime) && + \pop(p, g) = (j, p^\prime) \end{align*} - By \cite[Lemma ~5.7]{OG}, - $$\pop(\widehat{f}(\proj_g(q_1))) = (\pkt_1, \widehat{f}(\proj_g(q_1^\prime)))$$ - By \Cref{thm:cd}, + For any prefix $\alpha$ of some $f(i)$ (where $i \in [1, |ts|]$), we'll show \begin{align*} - \pkt_1 &= \pkt_2\\ - \widehat{f}(\proj_g(q_1^\prime)) &= \proj_g(q_2^\prime) \label{eq:popsim} \tag{$\dagger$} + \pop(\overline{f}(q)_\alpha, g) &= (\pkt, \overline{f}(q^\prime)_\alpha) &\text{ if $\alpha$ is a prefix of $f(j)$} \tag{$\dagger$} \label{eq:popinduc}\\ + \overline{f}(q)_\alpha &= \overline{f}(q^\prime)_\alpha &\text{ otherwise} \end{align*} - Since our choice of $g$ was arbitrary, notice \Cref{eq:popsim} holds for all $g \in \mathcal F$ \textcolor{red}{(this is not true!)}. - Hence, using \Cref{thm:cd}, - $$\proj_{g^\prime}(\overline{f}(q_1^\prime)) = \widehat{f}(\proj_{g^\prime}(q_1^\prime)) = \proj_{g^\prime}(q_2^\prime)$$ - for all $g^\prime \in \mathcal F$. - At last, \Cref{lem:pieotree_eq} yields $\overline{f}(q_1^\prime) = q_2^\prime$. + by inverse induction on $\alpha$. Instantiating \Cref{eq:popinduc} with $\alpha = \epsilon$ yields the desired result. + \begin{itemize} + \item Suppose $\alpha = f(i)$. + If $\alpha$ is a prefix of $f(j)$, $i = j$ by injectivity and \cite[Definition ~5.2, Equation (3)]{OG}. + Recall $\pop(qs[j], g) = (\pkt, qs^\prime[j])$. + Hence, by the outer inductive hypothesis, + $$ + \pop(\overline{f}(q)_\alpha, g) = + \pop(\overline{f_j}(qs[j]), g) = + (\pkt, \overline{f_j}(qs^\prime[j])) = + (\pkt, \overline{f}(q^\prime)_\alpha) + $$ + \item Once more, suppose $\alpha = f(i)$. + If $\alpha$ is not a prefix of $f(j)$, then $i \neq j$. + Since $qs[i] = qs^\prime[i]$, + $$ + \overline{f}(q)_\alpha = + \overline{f_i}(qs[i]) = + \overline{f_i}(qs^\prime[i]) = + \overline{f}(q^\prime)_\alpha + $$ + \item Suppose $\alpha$ is some strict prefix of $f(j)$, pointing to a node with $m$ children. + Let + \begin{align*} + \overline{f}(q)_\alpha = \Internal(qs_\alpha, p_\alpha) && + \overline{f}(q^\prime)_\alpha = \Internal(qs^{\prime}_\alpha, p^{\prime}_\alpha) + \end{align*} + There exists $k \in [1, m]$ such that $\alpha \cdot k$ is a prefix of $f(j)$. + By the inner inductive hypothesis, + \begin{align*} + qs_\alpha[i] &= \overline{f}(q)_{\alpha \cdot i} = \overline{f}(q^\prime)_{\alpha \cdot i} = qs_\alpha[i] \text{ for $i \in [1,m]$ with $i \neq k$} \label{eq:empty1} \tag{!}\\ + \pop(qs_\alpha[k], g) &= \pop(\overline{f}(q)_{\alpha \cdot k}, g) = (\pkt, \overline{f}(q^\prime)_{\alpha \cdot k}) = (\pkt, qs^\prime[k]) + \end{align*} + Via the construction in \Cref{dfn:f_bar} and since $\pop(p, g) = p^\prime$, $\pop(p_\alpha, g) = (k, p_\alpha^\prime)$. + Putting everything together, $\pop(\overline{f}(q)_\alpha, g) = (\pkt, \overline{f}(q^\prime)_\alpha)$, as desired. + + \item Suppose $\alpha$ is some strict prefix of some $f(i)$ but not $f(j)$, pointing to a node with $m$ children. + Let + \begin{align*} + \overline{f}(q)_\alpha = \Internal(qs_\alpha, p_\alpha) && + \overline{f}(q^\prime)_\alpha = \Internal(qs^{\prime}_\alpha, p^{\prime}_\alpha) + \end{align*} + Since $p$ and $p^\prime$ agree on all indices $i$ such that $f(i)$ is a child of $\alpha$, $p_\alpha = p^\prime_\alpha$. + For $i \in [1,m]$, since $\alpha \cdot i$ is not a prefix of $f(j)$, the inner inductive hypothesis yields + \begin{align*} + qs_\alpha[i] = \overline{f}(q)_{\alpha\cdot i} = \overline{f}(q^\prime)_{\alpha\cdot i} = qs^\prime_\alpha[i] \label{eq:empty2} \tag{!!} + \end{align*} + Putting everything together, $\overline{f}(q)_\alpha = \overline{f}(q^\prime)_\alpha$, as desired. + \end{itemize} + \textbf{NOTE:} + even when $\alpha \cdot i$ is not a prefix of any $f(i)$, \Cref{eq:empty1} and \Cref{eq:empty2} hold! + Both $\overline{f}(q)_{\alpha \cdot i}$ and $\overline{f}(q^\prime)_{\alpha \cdot i}$ would be PIEO trees with empty PIEOs on all leaf and internal nodes. \end{proof} \leqnomode @@ -678,12 +727,7 @@ \section{Embedding \& Simulation} \end{thm} \begin{proof} - We'll show the conditions from \Cref{dfn:sim} hold. - \begin{enumerate} - \item See \Cref{lem:sim1} - \item See \Cref{lem:sim2} - \item See \Cref{lem:sim3} - \end{enumerate} + By \Cref{lem:sim1}, \Cref{lem:sim2}, and \Cref{lem:sim3}, the conditions from \Cref{dfn:sim} hold. \end{proof} \newpage From d20033376f5433eea7f966b154cee28007ed59c4 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Tue, 23 Jul 2024 12:27:00 -0400 Subject: [PATCH 17/31] Add well-formedness scaffolding --- pieo-trees/notes.pdf | Bin 152124 -> 153055 bytes pieo-trees/notes.tex | 23 ++++++++++++++++++----- 2 files changed, 18 insertions(+), 5 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index c4a8982a5e230b9c39294811e54827317c248e29..92230d4ec74aeb238c9c9c3685d648fb963a2ea4 100644 GIT binary patch delta 38487 zcmV)7K*ztlqzT`v36LcMGcc1tv?!5J3V%Vrq_~cV;6CK?LsGI*zMW{R@*pc!G&!ps zn@dvU&>Q>Lr!m0XJqM}PVLy>WVla(9ziu>!&R);x?4MV(_4$vRtG_+hV#Xv9QuEo( 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zVc>MK)N~6Vy`*`FnLUlyv z!l_o_7OzS&N1`72@C_nK^)Rk!Hdmx8r&NuEok*8No~eLRH|gw6hoV&h17(xcHMQ8^ z9`IApSoMr=46@5508*SdgNaU8o`YvvGkRjSqL8zfYw8*_znoAunO@g~Yl24OwV6>I z5|90g`DyIwQs6DF`zmPz@xn`hGa3@*Dsa@QdXz# 0$, then $\pop(q, f)$ is well-defined and $\vdash q^\prime$, where $\pop(q, f) = (\pkt, q^\prime)$. + \end{enumerate} +\end{thm} + +\begin{proof} + \textcolor{red}{TBD} +\end{proof} + \newpage \section{Projection} From f91d0cc44c20286baec956bd6f6f4d41bcf372de Mon Sep 17 00:00:00 2001 From: polybeandip Date: Wed, 24 Jul 2024 13:18:01 -0400 Subject: [PATCH 18/31] Typos: missing _f --- pieo-trees/notes.pdf | Bin 153055 -> 153065 bytes pieo-trees/notes.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index 92230d4ec74aeb238c9c9c3685d648fb963a2ea4..7ee10696e0644b2118c64e92176722dc287f9834 100644 GIT binary patch delta 2472 zcmV;Z30L;ts|o3=39xGq0yZ*}VOc1D-C4nkR=A)x8>zC>^ihONLx;5Z2Z 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Define $|\cdot|_f : \PIEOTree(t) \to \mathbb N$ by \begin{align*} - |\Leaf(p)| = |p|_f && |\Internal(qs, p)| = \sum_{i=1}^{|qs|} |qs[i]|_f + |\Leaf(p)|_f = |p|_f && |\Internal(qs, p)|_f = \sum_{i=1}^{|qs|} |qs[i]|_f \end{align*} We say that $q \in \PIEOTree(t)$ is \emph{well-formed} w.r.t $f$, denoted $\vdash_f q$, if it adheres to the following rules. 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+\begin{document} + +\section{Relevant Definitions and Lemmas} + +\subsection{Definition 1} + +\noindent \textbf{\textit{Defining $|\cdot|$ on PIFOs With Regards to Pushes}} + +\hfill \break + +\noindent \textbf{Definition 1A}: \textit{Pushing an element to a PIFO increments how many times it appears} + $$\frac{p \in \text{PIFO} \hspace{0.7cm} |p|_i = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{PUSH}(p, i, r) = p' \hspace{0.7cm} n' = n+1}{|p'|_i = n'}$$ + +\noindent \textbf{Definition 1B:} \textit{Pushing an element to a PIFO does not change how much others appear} +$$\frac{p \in \text{PIFO} \hspace{0.7cm} |p|_i = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{PUSH}(p, j, r) = p' \hspace{0.7cm} i \neq j}{|p'|_i = n}$$ + +\noindent \textbf{Definition 1C:} \textit{Pushing an element to a PIFO increments its size} +$$\frac{p \in \text{PIFO} \hspace{0.7cm} |p| = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{PUSH}(p, i, r) = p' \hspace{0.7cm} n' = n+1}{|p'| = n'}$$\newline + +\subsection{Definition 2} +\noindent \textbf{\textit{Defining $|\cdot|$ on PIFOs With Regards to Pops}} + +\hfill\break + +\noindent \textbf{Definition 2A}: \textit{Popping an element from a PIFO decrements how many times it appears} + $$\frac{p \in \text{PIFO} \hspace{0.7cm} |p|_i = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{POP}(p) = (i, p') \hspace{0.7cm} n' = n-1}{|p'|_i = n'}$$ + +\noindent \textbf{Definition 2B:} \textit{Popping an element from a PIFO does not change how much others appear} +$$\frac{p \in \text{PIFO} \hspace{0.7cm} |p|_i = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{POP}(p) = (j, p') \hspace{0.7cm} i \neq j}{|p'|_i = n}$$ + +\noindent \textbf{Definition 2C:} \textit{Popping an element from a PIFO decrements its size} +$$\frac{p \in \text{PIFO} \hspace{0.7cm} |p| = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{POP}(p) = (i, p') \hspace{0.7cm} n' = n-1}{|p'| = n'}$$\newpage + +\section{Lemmas 1 and 2} +\noindent \textbf{\textit{Defining $|\cdot|$ on PIFO Trees With Regards to Pushes and Pops}}\newline + +\subsection{Lemma 1:} + +\noindent \textbf{\textit{Pushing to a PIFO Tree increments its size}} + +$$\frac{q \in \text{PIFOTree} \hspace{0.7cm} |q| = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} \text{push}(q, pkt, p) = q' \hspace{0.7cm} n' = n+1}{|q'| = n'}$$ + +\noindent \textbf{Proof}. We proceed upon this proof by structural induction over the definition of $|\cdot|$ as follows.\newline + +\noindent \textbf{Base Case}: $q = \text{Leaf}(p)$ for some packet $p$.\newline + +\noindent By definition 3.8, we have that $|\text{Leaf}(p)|_i = |p|_i$, where $|p|_i$ is the number of occurrences of $i$ in PIFO $p$. The definition of $push$ in 3.6 gives the following: + +$$\frac{\text{PUSH}(p, pkt, r) = p'}{push(\text{Leaf}(p), pkt, r) = \text{Leaf}(p')}$$\\[-10pt] + +\noindent Where $\text{Leaf}(p')$ is the result of $push$ing into $\text{Leaf}(p)$.\newline + +\noindent By definition 1A, we now know that $|p'|_i = |p|_i + 1$.\newline + +\noindent By definition 3.8, we know that $|\text{Leaf}(p')|_i = |p'|_i$ and $|\text{Leaf}(p)|_i = |p|_i$.\newline + +\noindent Combining these together, we obtain the following: + +$$|\text{Leaf}(p')| = |\text{Leaf}(p)| + 1$$ + +\noindent Thus, we have proven our base case.\\[10pt] + +\noindent \textbf{Inductive Case}:\newline + +\noindent Assume an arbitrary PIFO $p$, Node $q = \text{Internal}(qs, p)$, packet $pkt$, path $pt$, index $i$ and rank $r$.\newline + +\noindent \textbf{Inductive Hypothesis}: $|qs[i]| = n \implies |push(qs[i], pkt, pt)| = n+1$\newline + +\noindent \textbf{Show}: $|\text{Internal}(qs, p)| = m \implies |\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt)| = m+1$\newline + +\noindent Note that $push$ is defined for internal nodes as follows: + +$$\frac{\text{push}(qs[i], pkt, pt) = q' \hspace{1cm} \text{PUSH}(p, i, r) = p'}{\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt) = \text{Internal}(qs[q'/i], p')}$$\\[-10pt] + +\noindent Since $q$ is a valid PIFO, we know from Definition 3.8 the following: + +$$m = |q| = \sum_{j=1}^{|qs|} |qs[j]|$$ + +\noindent Thus, we can further conclude the following: + +$$|\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt)| = \sum_{j=1}^{|qs|} |qs[q'/i][j]|$$ + +\noindent From our Inductive Hypothesis, we know that $|q'| = |push(qs[i], pkt, pt| = n+1$.\newline + +\noindent From the definition of $push$, pushing changes only $qs[i]$, while the remaining subtrees remain the same. Subsequently, we can assert that $\forall +1 \leq j \leq |qs|, i \neq j \implies |qs[j]| = |qs[q'/i][j]$. With this in mind, we make the following claim:\\[-15pt] + +$$|\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt)| = \sum_{j=1}^{|qs|} |qs[j]| - |qs[i]| + |qs[q'/i][i]|$$ + +\noindent Substituting what we know, this gives us the following: + +$$|\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt)| = m - n + n + 1$$ + +$$\implies |\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt)| = m + 1$$\\[-15pt] + +\noindent With this, we have shown the desired equality, and proven our Inductive case.\newline + +\noindent Thus, it follows that Lemma 1 holds.\newline + +\section{Proofs For Lemma 3.9 (Well-Formedness)} + +\subsection{Proof of Lemma 3.9.1} +\noindent \textit{\textbf{Well-Formedness Is Preserved in PIFO Trees Upon Pushes}}\newline + +\noindent We proceed with this proof by inducting upon the definition of $push$ for PIFO Trees.\newline + +\noindent \textbf{Base Case : $Leaf(p)$}\newline + +\noindent By definition 3.6 of $push$, we have the following for Leaf nodes: + +$$\frac{\text{PUSH}(p, pkt, r) = p'}{push(\text{Leaf}(p), pkt, r) = \text{Leaf}(p')}$$\\[-10pt] + +\noindent It follows that after pushing, the resultant tree is expressed as $\text{Leaf}(p')$ for some packet $p'$ By definition 3.8 (cited above), we know the following holds for any arbitrary PIFO $p$:\\[-10pt] + +$$\frac{}{\vdash \text{Leaf}(p)}$$ + +\noindent We now have: $\forall p, pkt, r, \vdash push(\text{Leaf}(p), pkt, r)$\newline + +\noindent With this, our Base Case is proven.\\[10pt] + +\noindent \textbf{Inductive Case:} $q = \text{Internal}(qs, p), \vdash q$\newline + +\noindent \textbf{Inductive Hypothesis}: $\forall 1 \leq i \leq |qs|. \vdash push(qs[i], pkt, pt)$\newline + +\noindent Recall Definition 3.8 for $\vdash$ on Internal Nodes: + +$$\frac{\forall 1 \leq i \leq |qs|. \vdash qs[i] \land |p|_i = |qs[i]|}{\vdash \text{Internal}(\text{qs, p})}$$\newline + +\noindent Recall Definition 3.6 for $push$ on Internal Nodes: + +$$\frac{\text{push}(qs[i], pkt, pt) = q' \hspace{1cm} \text{PUSH}(p, i, r) = p'}{\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt) = \text{Internal}(qs[q'/i], p')}$$\newline + +\noindent Now, let $q' = push(q, pkt, (i, r) :: pt)$, and let $p' = \text{PUSH}(p, i, r)$.\newline + +\noindent \textbf{Show}: $\vdash q \implies \vdash \text{Internal}(qs[q'/i], p')$.\newline + +\noindent Using definition 3.6 and our Inductive Hypothesis, we can conclude that $\vdash q'$. Furthermore, note that the only subtree to be modified in $qs$ is in $qs[i]$, per the definition of $push$.\newline + +\noindent \textbf{Now we make the following claim:} + +\begin{alignat*}{5} +&(1) \vdash \text{Internal}(qs, p) && \hspace{0.3cm} \textit{By definition}\\[10pt] +&(2) \implies \forall 1 \leq j \leq |qs|. \hspace{0.2cm} \vdash qs[j] \land |p|_j = |qs[j]|&& \hspace{0.3cm} \textit{Inversion Lemma (Definition 3.8)}\\[5pt] +&(3) \implies |p|_i = |qs[i]|&& \hspace{0.3cm} \textit{Instance of universal quantifier (2)}\\[5pt] +&(4) \indent |p'|_i = |p|_i + 1&& \hspace{0.3cm} \textit{Definition 1A}\\[5pt] +&(5) \indent |q'| = |qs[i]| + 1&& \hspace{0.3cm} \textit{Lemma 1}\\[5pt] +&(6) \implies |p|_i + 1 = |qs[i]| + 1&& \hspace{0.3cm} \textit{Addition Property of Equality}\\[5pt] +&(7) \implies |p'|_i = |q'|&& \hspace{0.3cm} \textit{Substitution (4, 5, 6)}\\[5pt] +&(8) \indent \forall 1 \leq j \leq |p'|, i \neq j \implies |p'|_j = |p|_j&& \hspace{0.3cm} \textit{Definition 1B}\\[5pt] +&(9) \implies \forall 1 \leq j \leq |p'|, i \neq j \implies |p'|_j = |qs[j]|&& \hspace{0.3cm} \textit{By (2) and (8)}\\[5pt] +&(10) \implies \forall 1 \leq j \leq |p'|, |p'|_j = |qs[q' / i][j]|&& \hspace{0.3cm} \textit{By (7) and (9)}\\[5pt] +&(11) \indent \forall 1 \leq j \leq |qs|. \vdash qs[q'/i][j]&& \hspace{0.3cm} \textit{Inductive Hypothesis}\\[5pt] +&(12) \implies \forall 1 \leq j \leq |qs|. \vdash qs[q'/i][j] \land |p'|_j = |qs[q'/i][j]|&&\hspace{0.3cm} \textit{By (10) and (11)}\\[5pt] +&(13) \implies \vdash \text{Internal}(qs[q'/i], p') && \hspace{0.3cm} \textit{Definition of } \vdash\\[-10pt] +\end{alignat*} + +\noindent With this, we have proven our Inductive Statement and completed the proof. We have shown that pushing to an arbitrary PIFO Tree preserves its well-formedness.\\[-10pt] + +\subsection{Proof of Lemma 3.9.2} +\noindent \textit{\textbf{Well-Formedness Is Preserved in PIFO Trees Upon Pops}}\newline + +\noindent We proceed with this proof by inducting upon the definition of $pop$ for PIFO Trees.\newline + +\noindent \textbf{Base Case : $Leaf(p)$}\newline + +\noindent By definition 3.4 of $pop$, the following holds for Leaf nodes: + +$$\frac{\text{POP}(p) = (pkt, p')}{pop(\text{Leaf}(p)) = (pkt, \text{Leaf}(p'))}$$ + +\noindent It follows that after popping, the resultant tree is of form $\text{Leaf}(p')$ for some packet $p'$. By definition 3.8 (cited above), we know the following holds for any arbitrary PIFO $p$:\\[-10pt] + +$$\frac{}{\vdash \text{Leaf}(p)}$$ + +\noindent We now have: $\forall p, \vdash p'$ where $(pkt, p') = pop(\text{Leaf}(p))$.\newline + +\noindent With this, our Base Case is proven. + +\end{document} + diff --git a/pieo-trees/well-formedness/refs.bib b/pieo-trees/well-formedness/refs.bib new file mode 100644 index 0000000..e69de29 From fb378e7d8befb7c7e3ce3212b242c9ae2f14fc5a Mon Sep 17 00:00:00 2001 From: Kabir Samsi Date: Sun, 28 Jul 2024 15:30:05 -0400 Subject: [PATCH 20/31] Lemma 2 for |.| on PIFO Trees --- .../well-formedness/pifo_wellformed.pdf | Bin 154490 -> 179771 bytes .../well-formedness/pifo_wellformed.tex | 68 ++++++++++++++++++ 2 files changed, 68 insertions(+) diff --git a/pieo-trees/well-formedness/pifo_wellformed.pdf b/pieo-trees/well-formedness/pifo_wellformed.pdf index a09f6c7dd61718c3e983d962305274f8b7a91708..612f6514ae5b412c6c39c855e4f0eb918d6f4e11 100644 GIT binary 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Thus, it follows that Lemma 1 holds.\newline +\subsection{Lemma 2:} + +\noindent \textbf{\textit{Popping from a PIFO Tree decrements its size}} + +$$\frac{q \in \text{PIFOTree} \hspace{0.7cm} |q| = n \hspace{0.7cm} n \in \mathbb{N} \hspace{0.7cm} (pkt, q') = \text{pop}(q) \hspace{0.7cm} n' = n-1}{|q'| = n'}$$ + +\noindent \textbf{Proof}. We proceed upon this proof by structural induction over the definition of $|\cdot|$ as follows.\newline + +\noindent \textbf{Base Case}: $q = \text{Leaf}(p)$ for some packet $p$.\newline + +\noindent By definition 3.8, we have that $|\text{Leaf}(p)|_i = |p|_i$, where $|p|_i$ is the number of occurrences of $i$ in PIFO $p$. The definition of $pop$ in 3.6 gives the following: + +$$\frac{\text{POP}(p) = p'}{pop(\text{Leaf}(p)) = \text{Leaf}(pkt, p')}$$\\[-10pt] + +\noindent Where $\text{Leaf}(p')$ is the result of $pop$ping from $\text{Leaf}(p)$.\newline + +\noindent By definition 2A, we now know that $|p'|_i = |p|_i - 1$.\newline + +\noindent By definition 3.8, we know that $|\text{Leaf}(p')|_i = |p'|_i$ and $|\text{Leaf}(p)|_i = |p|_i$.\newline + +\noindent Combining these together, we obtain the following: + +$$|\text{Leaf}(p')| = |\text{Leaf}(p)| - 1$$ + +\noindent Thus, we have proven our base case.\\[10pt] + +\noindent \textbf{Inductive Case}:\newline + +\noindent Assume an arbitrary PIFO $p$, Node $q = \text{Internal}(qs, p)$ and index $i$.\newline + +\noindent \textbf{Inductive Hypothesis}: $|qs[i]| = n \implies |q'| = n-1$ where $(pkt, q') = pop(qs[i])$.\newline + +\noindent \textbf{Show}: $|\text{Internal}(qs, p)| = m \implies |q''| = m-1$ where $(pkt, q'') =\text{pop}(\text{Internal}(qs, p)$.\newline + +\noindent Note that $pop$ is defined for internal nodes as follows: + +$$\frac{\text{POP(p) = (i, p') \hspace{1cm} \text{pop}(qs[i]) = (pkt, q')}}{{\text{pop}(\text{Internal}(qs, p)) = (pkt, \text{Internal}(qs[q'/i], p'))}}$$\\[-10pt] + +\noindent Since $q$ is a valid PIFO, we know from Definition 3.8 the following: + +$$m = |q| = \sum_{j=1}^{|qs|} |qs[j]|$$ + +\noindent Thus, we can further conclude the following: + +$$(pkt, q'') = \text{pop}(\text{Internal}(qs, p) \implies |q''| = \sum_{j=1}^{|qs|} |qs[q'/i][j]|$$ + +\noindent From our Inductive Hypothesis, we know that $|q'| = n-1$ where $pkt, q' = \text{pop}(qs[i])$.\newline + +\noindent From the definition of $pop$, popping changes only $qs[i]$, while the remaining subtrees remain the same. Subsequently, we can assert that $\forall +1 \leq j \leq |qs|, i \neq j \implies |qs[j]| = |qs[q'/i][j]$. With this in mind, we make the following claim:\\[-15pt] + +$$(pkt, q'') = \text{pop}(\text{Internal}(qs, p)) \implies |q''| = \sum_{j=1}^{|qs|} |qs[j]| - |qs[i]| + |qs[q'/i][i]|$$ + +\noindent Substituting what we know, this gives us the following: + +$$(pkt, qs'') = \text{pop}(\text{Internal}(qs, p)) \implies |q''| = m - n + n - 1$$ + +$$\implies (pkt, qs'') = \text{pop}(\text{Internal}(qs, p)) \implies |qs''| = m - 1$$\\[-15pt] + +\noindent With this, we have shown the desired equality, and proven our Inductive case.\newline + +\noindent Thus, it follows that Lemma 2 holds.\newline + + \section{Proofs For Lemma 3.9 (Well-Formedness)} \subsection{Proof of Lemma 3.9.1} From abfdea6170c64e6bbfbbbba5a87a69bed3dfe4bc Mon Sep 17 00:00:00 2001 From: Kabir Samsi Date: Sun, 28 Jul 2024 15:39:51 -0400 Subject: [PATCH 21/31] Format and font consistency fixes --- ...ellformed.pdf => pifo_tree_wellformed.pdf} | Bin 179771 -> 179958 bytes ...ellformed.tex => pifo_tree_wellformed.tex} | 62 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zOZX@s@~TYB7jJZRepYW@_HZeL1*&(+++VOKJ#EPtkk%1xHE8=@SCSOCGdo>Lb}}xw z!l!6SzVE?gu_ft`7P$P}Ebt|Z-A3~(BKD^^FVY5FSeNkFk37M8zJroF`W;8fVK zUi*L0X^#&FQf^A|zQ;vP$-!23uqZSz)~(g`G%G{Udq@@>br-vnv6+hI%co)(6mL;5 z15CY6o-Z0@?ojUV-BAW@0&!rN`w0f?!hF5Xv0*n%R?~cSk|$ud*?$7pJOna1F)^34yaFf$ zGC46Zm*Bht!~!-pmyufn7ni-g0!M!^F)?E}WMpAEVPi2dF*Y_eH!@;lVqrKrFlAw3 zW;kR%JYqF6IX5slF*h_}Ghs6|WMg77GC5*7WHLEqH#lZ9IX+z=K0XR_baG{3Z3=jt zV`2aSCPpT|KMYI^42;Dfl6w`1WB{{yrXtu}8xd^oqhNLkh-9$`k Date: Mon, 29 Jul 2024 01:04:11 -0400 Subject: [PATCH 22/31] Proof for Lemma 3.9.2, fixes from earlier --- .../well-formedness/pifo_tree_wellformed.pdf | Bin 179958 -> 162547 bytes .../well-formedness/pifo_tree_wellformed.tex | 65 +++++++++++++++--- 2 files changed, 54 insertions(+), 11 deletions(-) diff --git a/pieo-trees/well-formedness/pifo_tree_wellformed.pdf b/pieo-trees/well-formedness/pifo_tree_wellformed.pdf index 5fc7520ecb6ba323812f35e0d92db9a4d59f41c4..d2d9cfc9908b4cdd0965a317571ed73284585ded 100644 GIT binary patch delta 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b/pieo-trees/well-formedness/pifo_tree_wellformed.tex index 0d4ee59..b77e121 100644 --- a/pieo-trees/well-formedness/pifo_tree_wellformed.tex +++ b/pieo-trees/well-formedness/pifo_tree_wellformed.tex @@ -11,9 +11,13 @@ \begin{document} -\title{Preservation Of Well-Formedness In PIFO Trees} +\section{Preservation of Well-Formedness In PIFO Trees} -\maketitle +\noindent This set of definitions and proofs completes the proof sketch outlined in Section 3.3, Definition 3.8 of Formal Abstractions In Packet Scheduling. In particular, it fleshes out the relationships between the functions push, pop and $|\cdot|$, in order to formally define (and by extension, prove) the preservation of well-formedness in PIFO Trees.\newline + +\noindent This defines how pushing and popping modifies length in PIFOs, and supplies lemmas (with proof) clarifying how length changes with regards to pushing and popping in PIFO Trees. This then proves preservation of well-formedness in PIFO Trees following push and pop operations.\newline + +\noindent This also lays some of the foundation for proving for well-formedness in PIEO Trees. Proofs of pushing translate almost one-to-one for PIEO Trees, while popping follows as an extension of this system, but with an inculsion of a totally ordered predicate. \section{Relevant Definitions and Lemmas} @@ -87,7 +91,7 @@ \subsection{Lemma 1: Pushing to a PIFO Tree increments its size} $$\frac{\text{push}(qs[i], pkt, pt) = q' \hspace{1cm} \text{PUSH}(p, i, r) = p'}{\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt) = \text{Internal}(qs[q'/i], p')}$$\\[-10pt] -\noindent Since $q$ is a valid PIFO, we know from Definition 3.8 the following: +\noindent Since $q$ is a PIFO Tree, we know from Definition 3.8 the following: $$m = |q| = \sum_{j=1}^{|qs|} |qs[j]|$$ @@ -150,7 +154,7 @@ \subsection{Lemma 2: Popping from a PIFO Tree decrements its size} $$\frac{\text{POP(p) = (i, p') \hspace{1cm} \text{pop}(qs[i]) = (pkt, q')}}{{\text{pop}(\text{Internal}(qs, p)) = (pkt, \text{Internal}(qs[q'/i], p'))}}$$\\[-10pt] -\noindent Since $q$ is a valid PIFO, we know from Definition 3.8 the following: +\noindent Since $q$ is a PIFO Tree, we know from Definition 3.8 the following: $$m = |q| = \sum_{j=1}^{|qs|} |qs[j]|$$ @@ -209,13 +213,13 @@ \subsection{Proof of Lemma 3.9.1} $$\frac{\text{push}(qs[i], pkt, pt) = q' \hspace{1cm} \text{PUSH}(p, i, r) = p'}{\text{push}(\text{Internal}(qs, p), pkt, (i, r) :: pt) = \text{Internal}(qs[q'/i], p')}$$\newline -\noindent Now, let $q' = \text{push}(q, pkt, (i, r) :: pt)$, and let $p' = \text{PUSH}(p, i, r)$.\newline +\noindent Now, let $q' = \text{push}(qs[i], pkt, pt)$, and let $p' = \text{PUSH}(p, i, r)$.\newline \noindent \textbf{Show}: $\vdash q \implies \vdash \text{Internal}(qs[q'/i], p')$.\newline \noindent Using definition 3.6 and our Inductive Hypothesis, we can conclude that $\vdash q'$. Furthermore, note that the only subtree to be modified in $qs$ is in $qs[i]$, per the definition of $\text{push}$.\newline -\noindent \textbf{Now we make the following claim:} +\noindent \textbf{Now we make the following claim:}\\[-20pt] \begin{alignat*}{5} &(1) \vdash \text{Internal}(qs, p) && \hspace{0.3cm} \textit{By definition}\\[10pt] @@ -236,7 +240,7 @@ \subsection{Proof of Lemma 3.9.1} \noindent With this, we have proven our Inductive Statement and completed the proof. We have shown that pushing to an arbitrary PIFO Tree preserves its well-formedness.\\[-10pt] \subsection{Proof of Lemma 3.9.2} -\noindent \textit{\textbf{Well-Formedness Is Preserved in PIFO Trees Upon Pops}}\\[10pt] +\noindent \textit{\textbf{Well-Formedness Is Preserved in PIFO Trees Upon Pops}}\\[5pt] \noindent We proceed with this proof by inducting upon the definition of $\text{pop}$ for PIFO Trees.\newline @@ -244,15 +248,54 @@ \subsection{Proof of Lemma 3.9.2} \noindent By definition 3.4 of $\text{pop}$, the following holds for Leaf nodes: -$$\frac{\text{POP}(p) = (pkt, p')}{\text{pop}(\text{Leaf}(p)) = (pkt, \text{Leaf}(p'))}$$ +$$\frac{\text{POP}(p) = (pkt, p')}{\text{pop}(\text{Leaf}(p)) = (pkt, \text{Leaf}(p'))}$$\\[-15pt] -\noindent It follows that after popping, the resultant tree is of form $\text{Leaf}(p')$ for some packet $p'$. By definition 3.8 (cited above), we know the following holds for any arbitrary PIFO $p$:\\[-10pt] +\noindent It follows that after popping, the resultant tree is of form $\text{Leaf}(p')$ for some packet $p'$. By definition 3.8 (cited above), we know the following holds for any arbitrary PIFO $p$:\\[-20pt] -$$\frac{}{\vdash \text{Leaf}(p)}$$ +$$\frac{}{\vdash \text{Leaf}(p)}$$\\[-10pt] \noindent We now have: $\forall p, \vdash p'$ where $(pkt, p') = \text{pop}(\text{Leaf}(p))$.\newline -\noindent With this, our Base Case is proven. +\noindent With this, our Base Case is proven.\newline + + +\noindent \textbf{Inductive Case:} $q = \text{Internal}(qs, p), \vdash q$\newline + +\noindent \textbf{Inductive Hypothesis}: $\forall 1 \leq j \leq |qs|. \vdash qs[j]',$ where $(pkt[j], qs[j]') = \text{pop}(qs[j])$\newline + +\noindent Recall Definition 3.8 for $\vdash$ on Internal Nodes: + +$$\frac{\forall 1 \leq i \leq |qs|. \vdash qs[i] \land |p|_i = |qs[i]|}{\vdash \text{Internal}(\text{qs, p})}$$\\[-10pt] + +\noindent Recall Definition 3.6 for $\text{pop}$ on Internal Nodes: + +$$\frac{\text{POP(p) = (i, p') \hspace{1cm} \text{pop}(qs[i]) = (pkt, q')}}{{\text{pop}(\text{Internal}(qs, p)) = (pkt, \text{Internal}(qs[q'/i], p'))}}$$\\[-10pt] + +\noindent Now, let $(pkt, q') = \text{pop}(qs[i])$, and let $(i, p') = \text{POP}(p)$.\newline + +\noindent \textbf{Show}: $\vdash q \implies \vdash \text{Internal}(qs[q'/i], p')$.\newline + +\noindent Using definition 3.6 and our Inductive Hypothesis, we can conclude that $\vdash q'$. Furthermore, note that the only subtree to be modified in $qs$ is in $qs[i]$, per the definition of $\text{pop}$.\newline + +\noindent \textbf{Now we make the following claim:}\\[-20pt] + +\begin{alignat*}{5} +&(1) \vdash \text{Internal}(qs, p) && \hspace{0.3cm} \textit{By definition}\\[10pt] +&(2) \implies \forall 1 \leq j \leq |qs|. \hspace{0.2cm} \vdash qs[j] \land |p|_j = |qs[j]|&& \hspace{0.3cm} \textit{Inversion Lemma (Definition 3.8)}\\[5pt] +&(3) \implies |p|_i = |qs[i]|&& \hspace{0.3cm} \textit{Instance of universal quantifier (2)}\\[5pt] +&(4) \indent |p'|_i = |p|_i - 1&& \hspace{0.3cm} \textit{Definition 1A}\\[5pt] +&(5) \indent |q'| = |qs[i]| - 1&& \hspace{0.3cm} \textit{Lemma 1}\\[5pt] +&(6) \implies |p|_i - 1 = |qs[i]| - 1&& \hspace{0.3cm} \textit{Subtraction Property of Equality}\\[5pt] +&(7) \implies |p'|_i = |q'|&& \hspace{0.3cm} \textit{Substitution (4, 5, 6)}\\[5pt] +&(8) \indent \forall 1 \leq j \leq |p'|, i \neq j \implies |p'|_j = |p|_j&& \hspace{0.3cm} \textit{Definition 1B}\\[5pt] +&(9) \implies \forall 1 \leq j \leq |p'|, i \neq j \implies |p'|_j = |qs[j]|&& \hspace{0.3cm} \textit{By (2) and (8)}\\[5pt] +&(10) \implies \forall 1 \leq j \leq |p'|, |p'|_j = |qs[q' / i][j]|&& \hspace{0.3cm} \textit{By (7) and (9)}\\[5pt] +&(11) \indent \forall 1 \leq j \leq |qs|. \vdash qs[q'/i][j]&& \hspace{0.3cm} \textit{Inductive Hypothesis}\\[5pt] +&(12) \implies \forall 1 \leq j \leq |qs|. \vdash qs[q'/i][j] \land |p'|_j = |qs[q'/i][j]|&&\hspace{0.3cm} \textit{By (10) and (11)}\\[5pt] +&(13) \implies \vdash \text{Internal}(qs[q'/i], p') && \hspace{0.3cm} \textit{Definition of } \vdash\\[-10pt] +\end{alignat*} + +\noindent With this, we have proven our Inductive Statement and completed the proof. We have shown that popping from an arbitrary PIFO Tree preserves its well-formedness.\\[-10pt] \end{document} From bba1bf255aa8a2cd0083e22ec1569158caaabbf5 Mon Sep 17 00:00:00 2001 From: polybeandip Date: Mon, 29 Jul 2024 12:13:40 -0400 Subject: [PATCH 23/31] Fix well-formedness (again!) --- pieo-trees/notes.pdf | Bin 153065 -> 153098 bytes pieo-trees/notes.tex | 10 +++++----- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index 7ee10696e0644b2118c64e92176722dc287f9834..b597b0e5adcfcbcdd5a160ae34d105c96b9bb8d8 100644 GIT binary patch delta 3637 zcmV-54$ATAs|kv%39xGq0y#C4u|6n&-C0|28dng0=T~gs7Rh^h?w4BbLzK8mRJCb~ zluEX%(y+v&7;Ml8ipu5`Q7a5_j#cs> zK{4m?HKnY0_p$tmjx=#b*wQP1#yf>bCj)!KTN6%Y)DrrpRMmE{BHGlRdHed+kCUryjJ&pI-QRbBl>*61!1-}?;cU7h=lup>~})oNh}S( z2k?VD4b+e{D1|PNv%I^Q=^hr`DL*W{lJ33IQRW6}947*<-ID4HM{UjPRhDj6(`8RI z1XQ-DEqj21bn$kaSP`F(6Q(s9ARgq;=S0XpJwmo!^Nd)^A|?n|df$mu+4fy+yhXGb 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a/pieo-trees/notes.tex +++ b/pieo-trees/notes.tex @@ -231,7 +231,7 @@ \section{Well-Formedness} |\Leaf(p)|_f = |p|_f && |\Internal(qs, p)|_f = \sum_{i=1}^{|qs|} |qs[i]|_f \end{align*} - We say that $q \in \PIEOTree(t)$ is \emph{well-formed} w.r.t $f$, denoted $\vdash_f q$, if it adheres to the following rules. + We say $q \in \PIEOTree(t)$ is \emph{well-formed} w.r.t. $f$, denoted $\models_f q$, if it adheres to the following rules. \begin{align*} \axiom{} {\vdash_f \Leaf(p)} @@ -245,14 +245,14 @@ \section{Well-Formedness} } \end{align*} - We say $q$ is well-formed, denoted $\vdash q$, if there exists $f \in \mathcal F$ such that, for all $f^\prime \geq f$, $\vdash_{f^\prime} q$. + We say $q$ is $f$-well-formed, denoted $\vdash_f q$, if for all $f^\prime \geq f$, $\models_{f^\prime} q$. \end{dfn} \begin{thm} - Let $t \in \Topo$, $\pkt \in \Pkt$, $d \in \Data$, $f \in \mathcal F$, and $q \in \PIEOTree(t)$ such that $\vdash q$. + Let $t \in \Topo$, $\pkt \in \Pkt$, $d \in \Data$, $f,f^\prime \in \mathcal F$, and $q \in \PIEOTree(t)$ such that $\vdash_f q$. \begin{enumerate} - \item If $pt \in \Path(t)$, then $\push(q, \pkt, d, pt)$ is well-defined and $\vdash \push(q, \pkt, d, pt)$. - \item If $|q|_f > 0$, then $\pop(q, f)$ is well-defined and $\vdash q^\prime$, where $\pop(q, f) = (\pkt, q^\prime)$. + \item If $pt \in \Path(t)$, then $\push(q, \pkt, d, pt)$ is well-defined and $\vdash_f \push(q, \pkt, d, pt)$. + \item If $|q|_{f^\prime} > 0$ and $f^\prime \geq f$, then $\pop(q, f^{\prime})$ is well-defined and $\vdash_{f^\prime} q^\prime$, where $\pop(q, f^{\prime}) = (\pkt, q^\prime)$. \end{enumerate} \end{thm} From d880ade19282b35f371246c31b7b319e96ea9d05 Mon Sep 17 00:00:00 2001 From: Kabir Samsi Date: Mon, 29 Jul 2024 12:19:51 -0400 Subject: [PATCH 24/31] Recompiled pdfs --- pieo-trees/notes.pdf | Bin 153065 -> 145324 bytes .../well-formedness/pifo_tree_wellformed.pdf | Bin 162547 -> 160744 bytes 2 files changed, 0 insertions(+), 0 deletions(-) diff --git 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zA2~%_U4)vA(G{*YVuU6&FrfLD7$p{7gxFMh8V%g{C})yCEbq=gocH1U)*(#jgmSd8 z)h4lUCozt5nfWO3Y?X`;1fEZW@v;3Sbss9|v5e9~Grf7}X+o!d#i&ry7m7*e|BeKu z;1a~)^0LwHg+08Fdu+8G;$muq=ppy@PFTl10Iom9hJC2~$lh`N=|^&w-k+V(_Iaem z_e<@M>lN+ND3i$=sX6>mqUBbc@=|*HT;_hT>hQ!9j4&dZO`uI^yK^*D70p&$D9smFU diff --git a/pieo-trees/well-formedness/pieo_tree_wellformed.tex b/pieo-trees/well-formedness/pieo_tree_wellformed.tex index e1ca551..0eb39c4 100644 --- a/pieo-trees/well-formedness/pieo_tree_wellformed.tex +++ b/pieo-trees/well-formedness/pieo_tree_wellformed.tex @@ -1,7 +1,33 @@ -\documentclass{article} -\usepackage{graphicx} % Required for inserting images -\usepackage{amsmath} -\usepackage{amssymb} +\documentclass{amsart} + +% font +\usepackage{cmbright} + +% margin +\usepackage[margin=1in]{geometry} + +% references +\usepackage[colorlinks]{hyperref} +\PassOptionsToPackage{colorlinks}{hyperref} +\hypersetup{urlcolor = RedViolet, linkcolor = RoyalBlue, citecolor = NavyBlue} + +% basics +\usepackage[leqno]{amsmath} +\usepackage{amssymb, amsthm} +\usepackage[svgnames, dvipsnames]{xcolor} +\usepackage{mhsetup, mathtools} +\usepackage[capitalise]{cleveref} + +% commutative diagram +\usepackage{tikz-cd} + +% hats +\usepackage{yhmath} + +% PL macros +\usepackage{mathpartir} +\usepackage{stmaryrd} + \usepackage{geometry} \geometry{ a4paper, @@ -12,11 +38,37 @@ % no more indent \setlength{\parindent}{0pt} +% right-justified sections hack +\usepackage{titlesec} +\newcommand{\marginsecnumber}[1]{% + \makebox[0pt][r]{#1\hspace{6pt}}% +} +\titleformat{\section} + {\normalfont\Large\bfseries} + {\marginsecnumber\thesection} + {0pt} + {} +\titleformat{\subsection} + {\normalfont\large\bfseries} + {\marginsecnumber\thesubsection} + {0pt} + {} +\titleformat{\subsubsection} + {\normalfont\normalsize\bfseries} + {\marginsecnumber\thesubsubsection} + {0pt} + {} + \begin{document} +\pagestyle{empty} + +{\LARGE \textbf{Preservation of Well-Formedness in PIEO Trees}} + +\hrulefill\\ + \section{Relevant Definitions} -\subsection{Definition 1} \textbf{\textit{Defining $| \cdot |$ on PIEOs With Regards to Pushes}}\newline @@ -35,6 +87,7 @@ \subsection{Definition 1} \section{Relevant Lemmas} \subsection{Lemma 1} + \text{Given $f \in \mathcal{F}, pkt \in \text{Pkt}, d \in \mathcal{D}, pt \in \text{Path}, t \in \text{Topo}$ and $q \in \text{PIEOTree}(t)$:} $$\begin{cases} @@ -136,14 +189,20 @@ \subsection{Lemma 1} \newpage -\section{Proofs For Well-Formedness} +\section{Proof Of Theorem 2.2} + +Let $t \in \text{Topo}$, $\text{pkt} \in \text{Pkt}$, $d \in \mathcal{D}$, $f,f' \in \mathcal{F}$, and $q \in \text{PIEOTree}(t)$ such that $\vdash_f q$.\newline + +1. If $pt \in \text{Path}(t)$, then $\text{push}(q, pkt, d, pt)$ is well-defined and $\vdash_f \text{push}(q, pkt, d, pt)$.\newline + +2. If $|q|_{f'} > 0$ and $f' \geq f$, then $\text{pop}(q, f')$ is well-defined and $\vdash_{f'} q'$, where $\text{pop}(q, f') = (pkt, q')$. -\subsection{Proof of Lemma 2.2.1} +\subsection{Proof of Theorem 2.2.1} \textit{\textbf{Well-Formedness Is Preserved in PIEO Trees Upon Pushes}}\\[10pt] We proceed with this proof by inducting upon the definition of $\text{push}$ for PIEO Trees.\newline -\textbf{Base Case : $Leaf(p)$}\newline +\textbf{Base Case : $\text{Leaf}(p)$}\newline By definition 1.4 of $\text{push}$ for PIEOs, we have the following for Leaf nodes: @@ -194,7 +253,7 @@ \subsection{Proof of Lemma 2.2.1} &(10) \implies \forall 1 \leq j \leq |p'|, |p'|_{j, f} = |qs[q' / i][j]|_f&& \hspace{0.3cm} \textit{By (7) and (9)}\\[5pt] &(11) \indent \forall 1 \leq j \leq |qs|. \vdash_f qs[q'/i][j] && \hspace{0.3cm} \textit{Inductive Hypothesis}\\[5pt] &(12) \implies \forall 1 \leq j \leq |qs|. \vdash_f qs[q'/i][j] \land |p'|_{j, f} = |qs[q'/i][j]|_f&&\hspace{0.3cm} \textit{By (10) and (11)}\\[5pt] -&(13) \implies \vdash_f \text{Internal}(qs[q'/i], p') && \hspace{0.3cm} \textit{Definition of } \vdash \textit{given } f\\[-10pt] +&(13) \implies \vdash_f \text{Internal}(qs[q'/i], p') && \hspace{0.3cm} \textit{Definition of } \vdash_f\\[-10pt] \end{alignat*} With this, we have proven our Inductive Statement and completed the proof. We have shown that pushing to an arbitrary PIEO Tree preserves its well-formedness.\\[-10pt] From 4ffd1f30e266b653f99b40dc0fa8cf1562308b9b Mon Sep 17 00:00:00 2001 From: polybeandip Date: Thu, 1 Aug 2024 19:56:32 -0400 Subject: [PATCH 28/31] Recompile notes.pdf --- pieo-trees/notes.pdf | Bin 147254 -> 153098 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index befbce3c5d8219b2da7b37b8e7f41d0b371730c9..ddec001518e5f7278a04b41eabc2f69931dadbbd 100644 GIT binary patch delta 51857 zcmZsiLvSt(&}L)Xwr$(ijcwf|H+J&Iwr$(CZQHh;`DXsv&Fp&JRsB@g>C-JgP!Uzo z$mB|567@ehWOT%k0EbQ^fXrQc2M2Y*sl<|sCfL#OO-lDdcuswmJ2T_#FcSxbfY9CBXiEn^~rZc1K<1aXAvph-gZ|TU#SF>wyteG*tQo;yPqw9 zpzYu=z_AlS)R*hZw@A)>wbrhu_J~vnlimO1^A}_dXF=A+fPa5PEO*2R;5Y@B;HL{d3-;k^PO(~%lXXjrWz0kk&*d*Bz@PThX2uQRV@}<118NIZFlDeUYC{M1nz$FD%!8 zrbSpqp%#PjO`(kHxsNpB#2ZOSU|CHsEMdxEVcseF{yNP@_NCT+nIzyB>Z)Bc7(o>t z>oF?DXB^y8j)4(5oJ0W6#(mNoZK$yVz~&+6!`}r;nY2nP%}&6h;-O$sJG3=kp$%sz 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31/31] Bug fixes + recompilation - use macros for push, pop, PIEO, PIEOTree, etc - change iff to implies in Defn 1.1, eq (2) --- pieo-trees/notes.pdf | Bin 145383 -> 153263 bytes pieo-trees/notes.tex | 14 ++++++-------- 2 files changed, 6 insertions(+), 8 deletions(-) diff --git a/pieo-trees/notes.pdf b/pieo-trees/notes.pdf index 81050afc572c1a4cd9c90e16513b55047b0bc77e..5d9b33aea266320f31cac3cd483add31c5b35942 100644 GIT binary patch delta 52939 zcmZUYLwKMKux4Z1w#|;!v2EM#7+-8#9d~Tow$riMv7PzvGc&8Zd$+Zz_nfMeYna+< zSX4?SF$qRiCQdlYxuxO1aBQ53D459yowSJxFu;zXNN;glT*R*6;hm^TgG$eDtI9hP zdUXz-O2TYemBTsw_tjF?zD)t>@?~S8L{Sr#SixQgj<6Yn+0%Ln@r;rC1VS&e`}Mk$cFy6pN~JK_XGeX^BKKddpiM$;Gj&dD_GjtwTRz4&DZ zpXuY?H#f=~^MBubg0SC^Jbm&)m;AO!;U5R2AMU-I5?B`wgUq+MCOryWOD)!dz$!K3 zc~A?>e#%6%d4CFdlC%~orQTQ_#fr}SPi0;pkx-%X;+IsA!w2Q*L^K4<_`-7idq$LX z4Ei4k{wcIkeb13byhLLO2^{O`g(Ykm9PArK|8LjX=)Uy257Q*VB0cp>CS#bQBYkG2 z#H{^m>M;ls=i?~Q*@SmS<8^g*0Ae0`KJrztltsI=()<`QCJ_c6y+cRy8P;faV$5Xd 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b/pieo-trees/notes.tex @@ -122,7 +122,7 @@ \section{Structure \& Semantics} These three maps play nicely together: \begin{align} \pop(p, f) \text{ is undefined} &\iff \pop(\proj(p, f)) \text{ is undefined} \label{eq:pieo1}\\ - \pop(p, f) = (\pkt, p^\prime) &\iff \pop(\proj(p, f)) = (\pkt, \proj(p^\prime, f)) \label{eq:pieo2}\\ + \pop(p, f) = (\pkt, p^\prime) &\implies \pop(\proj(p, f)) = (\pkt, \proj(p^\prime, f)) \label{eq:pieo2}\\ \proj(\push(p, s, d, r), f) &= \begin{cases} \push(\proj(p, f), s, r) & f(d) \text{ holds true}\\ @@ -163,7 +163,7 @@ \section{Structure \& Semantics} \forall i \in [1, n]. \; qs[i] \in \mathbf{PIEOTree}(ts[i]) } { - \Internal(qs, p) \in \mathbf{PIEOTree}(\Node(ts)) + \Internal(qs, p) \in \PIEOTree(\Node(ts)) } \end{align*} \end{dfn} @@ -174,7 +174,7 @@ \section{Structure \& Semantics} \begin{align*} \inference{} { - \text{POP}(p, f) = (\pkt, p^\prime) + \pop(p, f) = (\pkt, p^\prime) } { \pop(\Leaf(p), f) = (\pkt, \Leaf(p^\prime)) @@ -182,7 +182,7 @@ \section{Structure \& Semantics} && \inference{} { - \text{POP}(p, f) = (i, p^\prime)\\ + \pop(p, f) = (i, p^\prime)\\ \pop(qs[i], f) = (\pkt, q^\prime) } { @@ -197,7 +197,7 @@ \section{Structure \& Semantics} \begin{align*} \inference{} { - \text{PUSH}(p, \pkt, d, r) = p^\prime + \push(p, \pkt, d, r) = p^\prime } { \push(\Leaf(p), \pkt, d, r) = \Leaf(p^\prime) @@ -205,7 +205,7 @@ \section{Structure \& Semantics} && \inference{} { - \text{PUSH}(p, i, d, r) = p^\prime\\ + \push(p, i, d, r) = p^\prime\\ \push(qs[i], \pkt, d, pt) = q^\prime } { @@ -260,8 +260,6 @@ \section{Well-Formedness} \textcolor{red}{TBD} \end{proof} -\newpage - \section{Projection} \begin{dfn}