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z_suftHjoa-&Ga80#Rg+VTMh*V@uFC0*Oj^V2QqH*Ri8j8 z4Xvt=c^04jf;= + +\title{Minimization of Symbolic Automata} +\author{Jonathan Homburg} +\date{} + +\usepackage{algorithm2e} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{mathtools} +\usepackage{centernot} +\usepackage{changepage} +\usepackage{framed} +\usepackage{graphicx} +\usepackage{latexsym} +\usepackage{enumerate} +\usepackage{tikz} +\usepackage{pgfplots} +\usepackage{verbatim} +\usepackage{stmaryrd} + +\long\def\/*#1*/{} + +\DeclarePairedDelimiter\abs{\lvert}{\rvert} +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} + +\makeatletter +\let\oldabs\abs +\def\abs{\@ifstar{\oldabs}{\oldabs*}} + +\newcommand{\suchthat}{\hphantom l | \hphantom l} +\newcommand{\denotation}[1]{\llbracket #1 \rrbracket} + +\setbeamersize{text margin left = 8mm, text margin right = 8mm} + +\begin{document} +\begin{frame}[plain] +\titlepage + +Based on "Minimization of Symbolic Automata" by Loris D'Antoni \& Margus Veanes +\end{frame} + +\begin{frame} +\frametitle{Intuition} +\begin{itemize} +\item A deterministic finite automaton (DFA) is defined over a finite alphabet. Characters from this alphabet are given to the DFA as input and also label the state transitions. +\item A symbolic finite automaton (SFA) has inputs defined over a (possibly infinite) domain set and transitions labeled by a set of predicates over that domain. +\begin{itemize} +\item e.g. an SFA might have $\mathbb{Z}$ as a domain and might have predicates $\text{odd}(x)$ and $\text{even}(x)$ for transitions. +\end{itemize} +\end{itemize} +\end{frame} + +\begin{frame} +\begin{definition} +An \alert{effective Boolean algebra} is a tuple $A=(D, \Psi, \denotation{\cdot}, \perp, \top, \wedge, \vee, \neg)$ such that +\begin{itemize} +\item $D$ is a recursively enumerable set of domain elements. +\item $\Psi$ is a recursively enumerable set of predicates closed under $\wedge, \vee, \neg$ and with $\bot, \top \in \Psi$ +\item $\denotation{\cdot} : \Psi \to 2^D$ (called the denotation function) is recursively enumerable and has: +\begin{itemize} +\item $\denotation{\bot} = \varnothing$ and $\denotation{\top} = D$ +\item $\forall \hphantom l \varphi, \psi \in \Psi$, $\denotation{\phi \wedge \psi}= \denotation{\varphi}\cap\denotation{\psi}$, $\denotation{\phi \vee \psi} = \denotation{\varphi}\cup\denotation{\psi}$, and $\denotation{\neg \phi} = D \setminus \denotation{\varphi}$ +\end{itemize} +\end{itemize} +\end{definition} +For $\varphi \in \Psi$, if $\denotation{\varphi} \neq \varnothing$ then $\varphi$ is said to be satisfiable and we write $IsSat(\varphi)$. It is generally required that satisfiability checking is decidable. +\end{frame} + +\begin{frame} +\begin{definition} +A \alert{symbolic finite automaton} is a tuple $M=(Q, A, q^0, \Delta, F)$ such that +\begin{itemize} +\item $Q$ is a finite set of states +\item $A$ is an effective Boolean algebra, called the alphabet +\item $q^0 \in Q$ is the initial state +\item $\Delta \subseteq Q \times \Psi_A \times Q$ is a finite set of transitions +\item $F \subseteq Q$ is the set of final states +\end{itemize} +\end{definition} +Most notation used for SFAs is directly analogous to those used for DFAs. e.g. $L(M),\delta(p,a), \overset{\leftarrow}{\Delta}(q)$ +\end{frame} + +\begin{frame} +\frametitle{Properties of a Symbolic Automaton} +Let $M=(Q, A, q^0, \Delta, F)$ be an SFA +\begin{itemize} +\item $M$ is \alert{determinstic} iff $\forall (p, \varphi, p'), (p, \varphi', q') \in \Delta$ we have $IsSat(\varphi \wedge \varphi')$ implies $q = q'$ +\item $M$ is \alert{complete} iff $\forall a \in D_A, p \in Q$ $\exists q \in Q, \varphi \in \Psi_A$ such that $(p, \varphi, q) \in \Delta$ and $a \in \denotation{\varphi}$ +\item $M$ is \alert{clean} iff $\forall (p, \varphi, q) \in \Delta$, $p$ is reachable from $q^0$ and $IsSat(\varphi)$ +\item $M$ is \alert{normalized} iff $\forall p,q \in Q$ there is at most one transition from $p$ to $q$ +\end{itemize} +It is nearly always assumed that an SFA is clean and normalized. + +Four our purposes, it will generally be assumed that every SFA is deterministic and complete. +\end{frame} + +\begin{frame} +\frametitle{Minimality of SFAs} +\begin{definition} +An SFA M is \alert{minimal} if it is deterministic, complete, clean normalized, and for all $p,q \in Q$, $p=q$ if and only if $L_p(M) = L_q (M)$. +\end{definition} + +\begin{theorem} +If $M$ and $N$ are minimal SFAs such that $L(M) = L(N)$ then $M$ and $N$ are equivalent automata up to the relabeling of states and equivalence of predicates. +\end{theorem} + +\end{frame} + +\begin{frame} +\frametitle{$M$-equivalence} +\begin{definition} +Two states $p,q \in Q$ are \alert{$M$-equivalent}, denoted $p \equiv_M q$, if $L_p(M) = L_q(M)$. +\end{definition} + +\begin{theorem} +If $M$ is a clean, complete and deterministic SFA then $M_{/\equiv_M}$ is minimal and $L(M) = L(M_{/\equiv_M})$. +\end{theorem} +\end{frame} + +\begin{frame} +\frametitle{Moore's Algorithm (DFA)} +Let $M = (Q, \Sigma, q^0, \delta, F)$ be a DFA.\\ +Define $\equiv_0$ on $Q$ such that $p \equiv_0 q \iff p,q \in F$ or $p,q \not\in F$ +Recursively define $\equiv_i$ on $Q$ such that +$$p \equiv_i q \iff p \equiv_{i-1} q \text{ and } \forall a \in \Sigma, \delta(p,a) \equiv_{i-1} \delta(q,a)$$ +Continue until $\equiv_k$ and $\equiv_{k-1}$ are equivalent relations.\\ +$M_{/\equiv_k}$ is minimal. +\end{frame} + +\begin{frame} +\frametitle{Moore's Algorithm (SFA)} +Let $M = (Q, A, q^0, \Delta, F)$ be an SFA.\\ +Define $\equiv_0$ on $Q$ s.t. $p \equiv_0 q \iff (p,q) \in (F^c \times F) \cup (F \times F^c)$ +Recursively define $\equiv_i$ on $Q$ such that +\begin{align*} +p \equiv_{i} q &\iff p \equiv_{i-1} q \text{ or } \\ +&\exists (p,\varphi,p'), (q, \psi, q') \in \Delta \text{ s.t. } p' \equiv_{i-1} q' \text{ and IsSat($\varphi \wedge \psi$)} +\end{align*} +Continue until $\equiv_k$ and $\equiv_{k-1}$ are equivalent relations\\ +Define $\equiv_M$ on $Q$ such that $p \equiv_M q \iff p \centernot\equiv_k q$\\ +$M_{/\equiv_M}$ is minimal +\end{frame} + +\begin{frame} +\frametitle{Hopcroft's Algorithm (DFA)} +\begin{algorithm}[H] +\SetKwInOut{Input}{input} +\Input{DFA $M = (Q, \Sigma, q^0, \delta, F)$} +$P = \{ F, Q \setminus F \}$ \tcp{$Q$ is partitioned} +$W = \{F\} $ \tcp{Working set} +\While{$W \neq \varnothing$} +{ + remove any $R$ from $W$\\ + \For{$a \in \Sigma$} + { + $S = \delta^{-1} (R, a)$ \tcp{states leading into $R$ on $a$} + \For{$T \in P$} + { + \If{$T \cap S \neq \varnothing$ and $T \setminus S \neq \varnothing$} + { + Split($T$, $T \cap S$, $T \setminus S$) \tcp{refines $P,W$. Adds smaller of $\{T \cap S, T \setminus S\}$ to $W$} + } + } + } +} +Return $M_{/\equiv_P}$ +\end{algorithm} + +\end{frame} + +\begin{frame} +\frametitle{Hopcroft's Algorithm (SFA)} +\begin{algorithm}[H] +\SetKwInOut{Input}{input} +\Input{SFA $M = (Q, A, q^0, \Delta, F)$} +$P = \{ F, Q \setminus F \}$ \tcp{$Q$ is partitioned} +$W = \{F\} $ \tcp{Working set} +\While{$W \neq \varnothing$} +{ + remove any $R$ from $W$\\ + \For{$\varphi \in$ Minterms($M$)} + { + $S = \delta^{-1} (R, \varphi)$ \tcp{states leading into $R$ on $a$} + \For{$T \in P$} + { + \If{$T \cap S \neq \varnothing$ and $T \setminus S \neq \varnothing$} + { + Split($T$, $T \cap S$, $T \setminus S$) \tcp{refines $P,W$. Adds smaller of $\{T \cap S, T \setminus S\}$ to $W$} + } + } + } +} +Return $M_{/\equiv_P}$ +\end{algorithm} + +\end{frame} + +\begin{frame} +\frametitle{Modified Hopcroft's Algorithm (SFA)} +\begin{algorithm}[H] +\SetKwInOut{Input}{input} +\Input{SFA $M = (Q, A, q^0, \Delta, F)$} +$P = \{ F, Q \setminus F \}$ \\ +$W = \{F\} $\\ +\While{$W \neq \varnothing$} +{ + remove any $R$ from $W$\\ + $S = \overset{\leftarrow}{\Delta}(R)$ \tcp{set of all states going into $R$} + $\varGamma = \{S \ni p \mapsto \bigvee_{(p,\varphi,q) \in \Delta, q \in R}\varphi\}$ \tcp{$\varGamma(p)$ is disjunction of all predicates going into $R$} + \While{$\exists T \in P$ with $T \cap S \neq \varnothing$ and $\exists p_1, p_2 \in T$ s.t. $\text{IsSat}(\neg (\varGamma(p_1) \iff \varGamma(p_2)))$} + { + Let $a \in \denotation{\neg(\varGamma(p_1) \iff \varGamma(p_2))}$\\ + $T_1 = \{ p \in T \suchthat a \in \varGamma(p)\}$\\ + Split($T$, $T_1$, $T \setminus T_1$) + } + +} +Return $M_{/\equiv_P}$ (Note: optional optimization steps are skipped) +\end{algorithm} +\end{frame} + +\end{document} \ No newline at end of file