From de4b384e0ad70d13f2b855e36249a1df0064f669 Mon Sep 17 00:00:00 2001 From: Nathaniel Wesley Filardo Date: Wed, 26 Feb 2014 18:14:09 -0500 Subject: [PATCH 1/1] Meager beginnings --- .gitignore | 11 + COPYING | 661 +++++++++++++++++++++++++++++++++++++++ biblio.bib | 122 ++++++++ glossary.tex | 34 ++ main.tex | 252 +++++++++++++++ strautintro.tex | 147 +++++++++ strlang.tex | 55 ++++ treeautintro.tex | 115 +++++++ treelang.tex | 24 ++ zoo-str/fsm.tex | 104 ++++++ zoo-str/nested-stack.tex | 9 + zoo-str/pda.tex | 63 ++++ zoo-str/stack.tex | 4 + zoo-str/tm.tex | 35 +++ zoo-tree/awcbb.tex | 2 + zoo-tree/awedc.tex | 0 zoo-tree/ra.tex | 5 + zoo-tree/regular.tex | 27 ++ zoo-tree/tahom.tex | 5 + 19 files changed, 1675 insertions(+) create mode 100644 .gitignore create mode 100644 COPYING create mode 100644 biblio.bib create mode 100644 glossary.tex create mode 100644 main.tex create mode 100644 strautintro.tex create mode 100644 strlang.tex create mode 100644 treeautintro.tex create mode 100644 treelang.tex create mode 100644 zoo-str/fsm.tex create mode 100644 zoo-str/nested-stack.tex create mode 100644 zoo-str/pda.tex create mode 100644 zoo-str/stack.tex create mode 100644 zoo-str/tm.tex create mode 100644 zoo-tree/awcbb.tex create mode 100644 zoo-tree/awedc.tex create mode 100644 zoo-tree/ra.tex create mode 100644 zoo-tree/regular.tex create mode 100644 zoo-tree/tahom.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..dfdbca7 --- /dev/null +++ b/.gitignore @@ -0,0 +1,11 @@ +.*.swp +*.aux + +main.bbl +main.blg +main.fdb_latexmk +main.glo +main.log +main.out +main.pdf +main.xdy diff --git a/COPYING b/COPYING new file mode 100644 index 0000000..dba13ed --- /dev/null +++ b/COPYING @@ -0,0 +1,661 @@ + GNU AFFERO GENERAL PUBLIC LICENSE + Version 3, 19 November 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU Affero General Public License is a free, copyleft license for +software and other kinds of works, specifically designed to ensure +cooperation with the community in the case of network server software. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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It is safest +to attach them to the start of each source file to most effectively +state the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + + Copyright (C) + + This program is free software: you can redistribute it and/or modify + it under the terms of the GNU Affero General Public License as published by + the Free Software Foundation, either version 3 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU Affero General Public License for more details. + + You should have received a copy of the GNU Affero General Public License + along with this program. If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If your software can interact with users remotely through a computer +network, you should also make sure that it provides a way for users to +get its source. For example, if your program is a web application, its +interface could display a "Source" link that leads users to an archive +of the code. There are many ways you could offer source, and different +solutions will be better for different programs; see section 13 for the +specific requirements. + + You should also get your employer (if you work as a programmer) or school, +if any, to sign a "copyright disclaimer" for the program, if necessary. +For more information on this, and how to apply and follow the GNU AGPL, see +. diff --git a/biblio.bib b/biblio.bib new file mode 100644 index 0000000..a44602e --- /dev/null +++ b/biblio.bib @@ -0,0 +1,122 @@ +@book{sipser:theorycomp, + author={Michael Sipser}, + title={Introduction to the Theory of Computation}, + publisher={Course Technology}, + address={25 Thompson Place, Boston, MA, 02210, USA}, + year={2006}, + edition={Second Edition}, + isbn={978-0-534-95097-2}, +} + +@article{vidal:pfsm, + title = {Probabilistic Finite-State {Machines—Part} I}, + volume = {27}, + url = {http://www.inf.ed.ac.uk/teaching/courses/inf2a/readings/probabilistic-FSM-I.pdf}, + number = {7}, + journal = {{IEEE} TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE}, + author = {Enrique Vidal and Franck Thollard and Colin de la Higuera and Francisco Casacuberta and Rafael C. Carrasco}, + month = jul, + year = {2005}, + pages = {1013--1025}, +} + +@article{mohri:fstlsp, + title = {Finite-state transducers in language and speech processing}, + volume = {23}, + url = {http://dl.acm.org/citation.cfm?id=972698}, + number = {2}, + urldate = {2013-12-11}, + journal = {Computational linguistics}, + author = {Mohri, Mehryar}, + year = {1997}, + pages = {269–311}, +} + +@article{carrasco:learndetreg, + title = {Learning deterministic regular grammars from stochastic samples in polynomial time}, + volume = {33}, + issn = {0988-3754, 1290-{385X}}, + url = {http://www.rairo-ita.org/action/displayAbstract?fromPage=online&aid=8249903}, + doi = {10.1051/ita:1999102}, + number = {1}, + urldate = {2013-12-11}, + journal = {{RAIRO} - Theoretical Informatics and Applications}, + author = {Carrasco, Rafael C. and Oncina, Jose}, + month = aug, + year = {2002}, + pages = {1--19} +} + +@incollection{pereira:wfst, + title = {Speech recognition with weighted finite-state transducers}, + url = {http://link.springer.com/openurl?id=doi:10.1007/978-3-540-49127-9_28&from=SL}, + urldate = {2013-12-11}, + booktitle = {Springer Handbook of Speech Processing}, + publisher = {Springer}, + author = {Pereira, Fernando and Riley, Michael}, + year = {2008}, + pages = {559–584}, +} + +@article{jacquemard:tamodeq, + title = {Tree automata with equality constraints modulo equational theories}, + volume = {75}, + url = {http://www.sciencedirect.com/science/article/pii/S1567832607000884}, + number = {2}, + urldate = {2013-11-11}, + journal = {Journal of Logic and Algebraic Programming}, + author = {Jacquemard, Florent and Rusinowitch, Michael and Vigneron, Laurent}, + year = {2008}, + pages = {182–208}, +} + +@misc{gauwin:languageautomata, + title = {Language, Automata and Logic for Finite Trees}, + url = {http://www.labri.fr/perso/ogauwin/talks/tree-automata.pdf}, + urldate = {2013-11-05}, + author = {{Olivier Gauwin}}, + month = feb, + year = {2010}, +} + +@inproceedings{gecseg:classes, + title = {On some classes of tree automata and tree languages}, + volume = {25}, + url = {http://www.emis.ams.org/journals/AASF/Vol25/gecseg.pdf}, + urldate = {2013-11-05}, + booktitle = {{ANNALES-ACADEMIAE} {SCIENTIARUM} {FENNICAE} {SERIES} A1 {MATHEMATICA}}, + author = {Gécseg, Ferenc}, + year = {2000}, + pages = {325–336}, +} + +@article{godoy:hom, + title = {The {HOM} problem is decidable}, + abstract = {We close affirmatively a question which has been open for long time: decidability of the {HOM} +problem. The {HOM} problem consists in determining, given a tree homomorphism H and a regular tree language L represented by a tree automaton, whether H(L) is regular. In order to decide the +{HOM} problem, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and +that the equivalence and inclusion problems are decidable for images of regular languages through +tree homomorphisms. Our contributions are based on the following new constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of tree automata with arbitrary disequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of +a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping +mechanism. We combine the above constructions adequately to provide an algorithm deciding the {HOM} problem. This is the journal version of a paper presented in the 42nd {ACM} Symposium on Theory of Computing ({STOC} 2010). Here, we provide all proofs and examples. Moreover, we obtain better complexity results via the modification of some proofs and a careful complexity analysis. In particular, the obtained time complexity for the decision of {HOM} is a tower of three exponentials.}, + author = {{Guillem Godoy} and {Omer Giménez}}, +} + +@book{tata, + title = {Tree Automata Types and Applications}, + url = {https://gforge.inria.fr/frs/download.php/10994/tata.pdf}, + author = {{Hubert Comon} and {Max Dauchet} and {Remi Gilleron} and {Florent Jacquemard} and {Denis Lugiez} and {Christof Loding} and {Sophie Tison} and {Marc Tommasi}}, + month = nov, + year = {2008} +} + +@incollection{abdulla:simantichain, + title = {When simulation meets antichains}, + url = {http://link.springer.com/chapter/10.1007/978-3-642-12002-2_14}, + urldate = {2014-01-06}, + booktitle = {Tools and Algorithms for the Construction and Analysis of Systems}, + publisher = {Springer}, + author = {Abdulla, Parosh Aziz and Chen, Yu-Fang and Holík, Lukáš and Mayr, Richard and Vojnar, Tomáš}, + year = {2010}, + pages = {158–174}, +} diff --git a/glossary.tex b/glossary.tex new file mode 100644 index 0000000..0701133 --- /dev/null +++ b/glossary.tex @@ -0,0 +1,34 @@ +%\newglossaryentry{}{ +%% +% name={}, +%% +% description={} +%} + +\newglossaryentry{fulltrio}{ +% + name={full trio}, +% + description={A \gls{trio} also closed under {\em arbitrary} + homomorphism} +% +} + +\newglossaryentry{regop}{ +% + name={regular language operation}, +% + description={One of union, intersection, or Kleene star} +} + +\newglossaryentry{trio}{ +% + name={trio}, +% + description={A family of languages which is closed under + $\epsilon$-free homomorphism image, homomorphism pre-image, and + intersection with regular languges} +% +} + + diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..a8ffb8a --- /dev/null +++ b/main.tex @@ -0,0 +1,252 @@ +% Recommend building with +% latexmk -pdf main.tex + +\documentclass[letterpaper]{article} + +\DeclareSymbolFont{AMSb}{U}{msb}{m}{n} +\DeclareMathAlphabet{\mathsc}{T1}{cmr}{m}{sc} +\SetMathAlphabet{\mathsc}{bold}{T1}{cmr}{bx}{sc} % sigh + +\title{An Automata Zoo% +% +\footnote{The source to this document is available under the terms of the +GNU Affero Public License version 3 (see {\tt COPYING} beside the main file, +or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}} + +\author{Nathaniel Wesley Filardo} + +\usepackage{fullpage} +\usepackage{xcolor} +\usepackage{amsmath,amssymb,latexsym} +\usepackage{keyval} +\usepackage{graphicx} +\usepackage{xspace} +\usepackage{etoolbox} +\usepackage[color,all]{xypic} +\usepackage{calc} +\usepackage{subfig} +\usepackage{multicol} +\usepackage{multirow} +\usepackage{proof} +\usepackage{qtree} +\usepackage{listings} + \lstloadlanguages{Haskell} +\usepackage{floatrow} +\usepackage{caption} +\usepackage{comment} + % http://stackoverflow.com/questions/2193307/how-to-get-latex-to-hyphenate-a-word-that-contains-a-dash +\usepackage[shortcuts]{extdash} +\usepackage{pifont} +\usepackage{stmaryrd} +\usepackage{bold-extra} +\usepackage{xspace} +\usepackage{tabu} +\usepackage{natbib} +\usepackage{hyperref} + +\usepackage[xindy,toc]{glossaries} % AFTER: hyperref + +% Do something clever with line numbers, since we cite them all the time. +\newcommand{\lineref}[2]{\hyperref[lst:#1:#2]{Line \ref{lst:#1}:\ref{lst:#1:#2}}} + +% Figure drawing stuff +\newlength\templena +\newlength\templenb +\newsavebox{\tempboxa} +\newsavebox{\tempboxb} +\newsavebox{\tempboxc} +\newsavebox{\tempboxd} +\newsavebox{\tempboxe} + +\definecolor{lightred}{rgb}{1,.25,.25} + +% Math +\newcommand{\defeq}{\stackrel{\mbox{\tiny def}}{=}} + +%\newcounter{rllstctr} +%\newcommand{\rllstctrautorefname}{L.} +\newsavebox\rllstboxa +\newsavebox\rllstboxb +\newlength\rllstlenl +\newlength\rllstlenr +\lstnewenvironment{rllst}[1][] +{ +\lstset{#1} +\setbox\rllstboxa=\hbox\bgroup +} +{ +\egroup +% \refstepcounter{rllstctr} % OLD - SEPARATE COUNTER FOR RLL LISTINGS +\refstepcounter{equation} % NEW - JUST USE EQUATION COUNTER +\label{rll:\lstname} +% \setbox\rllstboxb=\hbox{(L. \arabic{rllstctr})} % OLD - MARK AS (L. 3) +\setbox\rllstboxb=\hbox{(\arabic{equation})} % NEW - MARK AS (3) +\trivlist\item +\setlength{\rllstlenl}{\linewidth/2-\wd\rllstboxa/2} +\setlength{\rllstlenr}{\linewidth/2-\wd\rllstboxa/2-\wd\rllstboxb} +\hspace{\rllstlenl}\usebox{\rllstboxa}\hspace{\rllstlenr}\usebox{\rllstboxb} +\relax\endtrivlist +} + +\newcommand{\eqnref}[1]{\hyperref[eqn:#1]{(\ref{eqn:#1})}} +\newcommand{\Figref}[1]{\hyperref[fig:#1]{Figure~\ref{fig:#1}}} +\newcommand{\ftnref}[1]{footnote~\hyperref[ftn:#1]{\ref{ftn:#1}}} % highlight only the footnote # +\newcommand{\rllref}[1]{\hyperref[rll:#1]{(\ref{rll:#1})}} +\newcommand{\secref}[1]{\hyperref[sec:#1]{\S~\ref{sec:#1}}} +\newcommand{\Tabref}[1]{\hyperref[tab:#1]{Table~\ref{tab:#1}}} + +\newcommand{\paren}[1]{\left(#1\right)} +\newcommand{\brak}[1]{\left[#1\right]} +\newcommand{\abs}[1]{\left\lvert#1\right\rvert} +\newcommand{\set}[1]{\left\{#1\right\}} +\newcommand{\pair}[2]{\langle#1,#2\rangle} +\newcommand{\expect}[1]{\left\langle#1\right\rangle} +\newcommand{\sem}[1]{\left\llbracket #1\right\rrbracket} + +\newcommand{\pset}[1]{\mathcal{P}\paren{#1}} + +\makeatletter +\define@key{autinfo}{member} {\def\autmember {#1}} +\define@key{autinfo}{empty} {\def\autempty {#1}} +\define@key{autinfo}{univ} {\def\autuniv {#1}} +\define@key{autinfo}{equiv} {\def\autequiv {#1}} +\define@key{autinfo}{subset} {\def\autsubset {#1}} +\define@key{autinfo}{disjoin} {\def\autdisjoin{#1}} + +\define@key{autinfo}{determinize}{\def\autmiscdet{#1}} +\define@key{autinfo}{minimize} {\def\autmiscmin{#1}} +\define@key{autinfo}{compl} {\def\autcompl {#1}} +\define@key{autinfo}{relcompl} {\def\autrelcomp{#1}} +\define@key{autinfo}{intersect} {\def\autregintr{#1}} % regular +\define@key{autinfo}{union} {\def\autregun {#1}} % regular +\define@key{autinfo}{kstar} {\def\autregkstr{#1}} % regular +\define@key{autinfo}{kplus} {\def\autregkpls{#1}} +\define@key{autinfo}{efhom} {\def\autabsehom{#1}} % trio +\define@key{autinfo}{invhom} {\def\autabsmoh {#1}} % trio +\define@key{autinfo}{reginter} {\def\autabsregi{#1}} % trio +\define@key{autinfo}{hom} {\def\autabshom {#1}} % full trio + +\define@key{autinfo}{regops} {\def\autregintr{#1} % all regops at once + \def\autregun {#1} + \def\autregkstr{#1}} + +\define@key{autinfo}{trio} {\def\autabsehom{#1} % all trio ops at once + \def\autabsmoh {#1} + \def\autabsregi{#1}} + +\define@key{autinfo}{fulltrio} {\def\autabshom {#1} % all full trio ops at once + \def\autabsmoh {#1} + \def\autabsregi{#1}} + +\def\autinfo{\@ifnextchar[\autinfo@{\autinfo@[]}} % handle opt. arg. +\def\autinfo@[#1]#2{% + \def\autmember {} + \def\autempty {} + \def\autuniv {} + \def\autequiv {} + \def\autsubset {} + \def\autdisjoin{} + \def\autmiscdet{} + \def\autmiscmin{} + \def\autcompl {} + \def\autrelcomp{} + \def\autregintr{} + \def\autregun {} + \def\autregkstr{} + \def\autregkpls{} + \def\autabsehom{} + \def\autabsmoh {} + \def\autabsregi{} + \def\autabshom {} + + \setkeys{autinfo}{#2} + {\centering + \begin{longtabu}{|X[l]|X[2,l]|} \hline + \textbf{Decision Predicates} & \\ \hline + \ifdefempty{\autmember} {}{Membership testing & \autmember \\ \hline} + \ifdefempty{\autempty} {}{Emptiness & \autempty \\ \hline} + \ifdefempty{\autuniv} {}{Universality & \autuniv \\ \hline} + \ifdefempty{\autequiv} {}{Equivalence & \autequiv \\ \hline} + \ifdefempty{\autsubset} {}{Subset & \autsubset \\ \hline} + \ifdefempty{\autdisjoin}{}{Disjointness & \autsubset \\ \hline} + \textbf{Closure Properties} & \\ \hline + \ifdefempty{\autregkstr}{}{Kleene star & \autregkstr \\ \hline} + \ifdefempty{\autregkpls}{}{Kleene plus & \autregkpls \\ \hline} + \ifdefempty{\autcompl} {}{General Complement & \autcompl \\ \hline} + \ifdefempty{\autrelcomp}{}{Relative Complement & \autrelcomp \\ \hline} + \ifdefempty{\autregintr}{}{Intersection & \autregintr \\ \hline} + \ifdefempty{\autregun }{}{Union & \autregun \\ \hline} + \ifdefempty{\autabsmoh} {}{Inverse Hom. & \autabsmoh \\ \hline} + \ifdefempty{\autabshom} {}{Arbitrary Hom. & \autabshom \\ \hline} + \ifdefempty{\autabsehom}{}{$\epsilon$-free Hom. & \autabsehom \\ \hline} + \ifdefempty{\autabsregi}{}{Intersect reg. lang. & \autabsregi \\ \hline} + \textbf{Automata Operations} & \\ \hline + \ifdefempty{\autmiscdet}{}{Determinizable & \autmiscdet \\ \hline} + \ifdefempty{\autmiscmin}{}{Minimizable & \autmiscmin \\ \hline} + \end{longtabu} + } +} +\makeatother + +\newcommand{\socalled}[1]{``#1''} +\newcommand{\latin}[1]{#1} % Style guide says not to italicize + % common latin, since it needlessly draws + % attention to unimportant text + +\newcommand{\eg}{\latin{e.g.}, } +\newcommand{\ie}{\latin{i.e.}, } +\newcommand{\etc}{\latin{etc}\.} +\newcommand{\etal}{\latin{et al}\.} +\newcommand{\apriori}{\latin{a priori}\xspace} +\newcommand{\perse}{\latin{per se}\xspace} + +\newcommand{\defn}[1]{{\boldmath\bfseries #1}} +\newcommand{\config}{{\ensuremath\mathcal{C}}} +\newcommand{\alang}[1]{\mathcal{L}\paren{#1}} +\newcommand{\sts}{\ensuremath\mathcal{Q}} +\newcommand{\stsf}{\ensuremath\mathcal{Q}_f} +\newcommand{\stsi}{\ensuremath\mathcal{Q}_i} + +\newcommand{\maininclude}[2]{\section{#1} \label{sec:#2} \input{#2}} + +\newcommand{\Note}[1]{} +\renewcommand{\Note}[1]{{\large\textcolor{red}{\textbf{[#1]}}}} +\newcommand{\NoteNWF}[1]{\Note{#1 --NWF}} + +\input{glossary} +\makeglossaries + +\begin{document} + +\maketitle + + +\part{Finite Strings} +\maininclude{Languages}{strlang} +\maininclude{Automata Overview}{strautintro} +\maininclude{Finite-state Machine}{zoo-str/fsm} +\maininclude{Push-down Automata}{zoo-str/pda} +\maininclude{Stack Automata}{zoo-str/stack} +\maininclude{Nested Stack Automata}{zoo-str/nested-stack} +\maininclude{Turing Machine}{zoo-str/tm} + +%\part{Infinite Strings} + +\part{Finite Trees} +\maininclude{Languages}{treelang} +\maininclude{Automata Overview}{treeautintro} +\maininclude{Regular Tree Automata}{zoo-tree/regular} +\maininclude{(Dis)Equality-Constrained Tree Automata}{zoo-tree/awedc} +\maininclude{Constraints Between Brothers}{zoo-tree/awcbb} +\maininclude{Reduction Automata}{zoo-tree/ra} +\maininclude{Homomorphism-Equality Automata}{zoo-tree/tahom} + +%\part{Infinite Trees} + +\appendix +\bibliographystyle{abbrvnat} +\bibliography{biblio} +%\printindex +\glsaddall +\printglossaries +\end{document} diff --git a/strautintro.tex b/strautintro.tex new file mode 100644 index 0000000..cf282c7 --- /dev/null +++ b/strautintro.tex @@ -0,0 +1,147 @@ +String automata represent a {\em mechanical} description of languages over +$\Sigma$ (\ie subsets of the strings which may be formed from the +characters of a particular alphabet, $\Sigma$). There are numerous +equivalent ways of stating this, possibly the most useful being that an +automaton can be thought of as an {\em indicator function} of type $\Sigma^* +\to 2$, signaling whether a given string is or is not part of its language. + +\subsection{Execution of Automata} + +At the core of each automaton will be a \defn{transition function}, +$\delta$, which will take the current configuration of the machine (which we +shall denote in general with $c \in \config$) and a character of $\Sigma$ to +produce a new configuration (usually very closely related to the input +configuration, having been changed through a constrained interface). Each +automaton will specify exactly one \defn{initial configuration}, $c_0 \in +\config$, and will designate some configurations as \defn{accepting +configurations}, $\config_F \subseteq \config$. + +To test whether a particular string $\vec{s} = s_1 \ldots s_n \in \Sigma^*$ +is in its language, one starts with the automaton in its initial +configuration, feeds its transition function each character of the string in +turn (\ie $s_1$, then $s_2$, and so on, up to $s_n$), and then looks to see +if the automaton is now in an accepting configuration. More formally, we +can define a \defn{run} of an automaton as a string composed of alternating +elements of $\config$ and $\Sigma$. If, after transitioning on $s_i$ the +machine is in configuration $c_i$, then $c_0 s_1 c_1 s_2 c_2 \ldots s_n c_n$ +is the run of the machine on $\vec{s} \in \Sigma^*$. A run is said to be +accepting if $c_n$ is an accepting configuration (\ie $c_n \in \config_F$).% +% +\footnote{It is equally sensible to feed strings through an automaton in +reverse and ask if, after receiving the first symbol of the (un-reversed) +string the machine is in an accepting configuration.} + +\subsection{Nondeterministic Automata} + +The above scenario described a \defn{deterministic} automaton, which always +make a particular move given a configuration and an input character. +Nondeterministic automata instead may take on {\em one of many} +configurations after receiving a character (as well as at initialization). +To model these, we define the transition function to result in a {\em set} +of configurations.% +% +\footnote{Many texts instead use a \defn{transition relation}. While +equivalent, we find it useful to distinguish the inputs and outputs of the +transition process.} +% +Now a run of our nondeterministic machine looks like $C_0 s_1 C_1 \ldots s_n +C_n$ where each $C_i \subseteq \config$ and $c' \in C_{i+1}$ iff +$\exists_{c \in C_i} . c' \in \delta\paren{c,s_{i+1}}$. We consider the +string $\vec{s}$ accepted by the automaton if {\em there exists} a run, as +defined earlier, where each $c_i \in C_i$, and $c_n \in \config_F$. + +\subsection{Transducers} + +A useful generalization of automata is from acceptors ($\Sigma^* \to 2$) to +transducers ($\Sigma^* \to \Sigma'^*$). Here, $\delta$ not only outputs a +new configuration, it additionally outputs (sometimes optionally) a +character of the \defn{output alphabet} $\Sigma'$. Nondeterministic +transducers' transition functions typically output a set of +configuration/character pairs ($(c, s') \in \config \times \Sigma'$). + +\subsection{Weighted Automata} + +Related to transducers are weighted automata, which may be thought of as +scoring or ranking input strings. That is, rather than being indicator +functions $\Sigma^* \to 2$, they capture functions from $\Sigma^*$ to a +semi-ring, $R = \paren{0_R, 1_R, +_R, \times_R}$, such as $\mathbb{N}$ or +$\mathbb{R}$. The typical definition of a weighted string automata has +$\delta$ producing an element of $R$ on each transition and adds two new +functions, which assign weights to the initial ($\config_0 \to R$) and +final ($\config_F \to R$) states.% +% +\footnote{For weighted {\em deterministic} automata, there is only one +initial configuration and so one typically elides the former function.} +% +The weight of a run is then the +$R$-product of its initial and final states' weights and the weights of all +transitions therein. The weight of a string is the $R$-sum of all accepting +runs for that string.% +% +\footnote{Equivalently, we can think of a typical weighted automaton as a +non-deterministic {\em transducer to $R^*$} followed by per-run $R$-product +and cross-run $R$-sum reductions.} + +\subsection{Descriptive Taxonomy} + +We have said that an automaton's transition function $\delta$ has type +$\config \times \Sigma \to \config$, but that the input and output +configurations are often closely related. In such a case, we may say that +$\delta$ is \defn{characterized} by a function of a different return +type. What we mean by this is that the information bottleneck represented +by this alternate return type is sufficient to describe the change between +elements of $\config$ on a transition. See the entry for +\hyperref[sec:zoo-str/pda]{Push-down Automata} for a simple example. + +\subsection{Operational Taxonomy} + +In an effort to neatly summarize many properties of string automata, entries +will typically be a brief prose description of the family followed by a +table containing common properties. Uncommon properties or note-worthy +features will be left to prose. + +Here, we use the common table to describe itself. Given a string $s \in +\Sigma^*$, automata $A, A', \ldots$ and their languages $\alang{A}, +\alang{A'}, \ldots$ ($\subseteq \Sigma^*$)... +\autinfo{ + member={$s \in \alang{A}$?}, + empty={$\alang{A} = \emptyset$?}, + univ={$\alang{A} = \Sigma^*$?}, + equiv={$\alang{A} = \alang{A'}$?}, + subset={$\alang{A} \subseteq \alang{A'}$?}, +% + kstar={Find $B$ s.t. $\alang{B} = \alang{A}^*$}, + kplus={Find $B$ s.t. $\alang{B} = \alang{A}^+$}, + compl={Find $B$ s.t. $\alang{B} = \Sigma^* \setminus \alang{A}$}, + relcompl={Find $B$ s.t. $\alang{B} = \alang{A} \setminus \alang{A'}$}, +% + intersect={Find $B$ s.t. $\alang{B} = \alang{A} \cap \alang{A'}$?}, + union={Find $B$ s.t. $\alang{B} = \alang{A} \cup \alang{A'}$?}, +% + hom={Find $B$ for hom. $h$ s.t. $\alang{B} = h\paren{\alang{A}}$}, + invhom={Find $B$ for hom. $h$ s.t. $\alang{A} = h\paren{\alang{B}}$}, + efhom={Find $B$ for $\epsilon$-free hom. $h$ s.t. $\alang{B} = + h\paren{\alang{A}}$}, +% + reginter={Find $B$ for \hyperref[sec:zoo-str/fsm]{regular language} + $L$ s.t. $\alang{B} = \alang{A} \cap L$.}, +% + determinize={Find a deterministic $B$ with language equal to + (nondet.) $A$.}, + minimize={Find a ``smallest'' $B$ with language equal to $A$.% + % + \footnote{``Smallest'' is not always straightforward to define. + The goal, loosely, is to minimize the amount of information + necessary to describe the automaton's configuration.}}, +} + +Of course, not every row in the table is independent of the others. The +following implications (at least) hold in all cases: +\begin{itemize} + \item Subset testing implies equivalence testing. + \item Relative complement closure implies general complement closure. + \item Intersection and general complement closures imply both subset testing + and relative complement closure. + \item Closure under arbitrary homomorphism implies closure under + $\epsilon$-free homomorphisms. +\end{itemize} diff --git a/strlang.tex b/strlang.tex new file mode 100644 index 0000000..f527c09 --- /dev/null +++ b/strlang.tex @@ -0,0 +1,55 @@ +A string language $L$ is a set of finite strings over some alphabet +$\Sigma$.% +% +\footnote{More formally, it is a subset of the underlying set of the free +monoid of $\Sigma$. Following the literature, we tend to suppress writing +the forgetful functor, and simply refer to the set as $\Sigma^*$.} +% +For example, if $\Sigma = \set{a,b}$, then ``the string $a$'', ``the set of +strings with an even number of $a$s and an odd number of $b$s (and no other +characters)'', and ``all strings composed of $x$ many $a$s, $y$ many $b$s, +then $x*y$ more $a$s'' are all languages over $\Sigma$. + +We may define string languages in terms of other string languages by +\begin{enumerate} + \item \label{lang:str:ops:set} Set theoretic operations: + \begin{enumerate} + \item Intersection: + $L_1 \cap L_2 \defeq \set{ s \middle\vert s \in L_1 ~\wedge~ s \in L_2 }$ + \item Union: + $L_1 \cup L_2 \defeq \set{ s \middle\vert s \in L_1 ~\vee~ s \in L_2 }$ + \item Relative complement: + $L_1 \setminus L_2 \defeq \set{s \middle\vert s \in L_1 ~\wedge~ s \not\in L_2 }$ + \item (General) complement, $L^C$: the set of strings over $\Sigma$ + which are not in $L$. We might write this as + $F_\text{mon}\Sigma \setminus L$ or $\Sigma^* \setminus L$ + (see below). + + \end{enumerate} + \item \label{lang:str:ops:kleene} Keene's operators: + \begin{enumerate} + \item Kleene plus: $L^+ \defeq \set{ s \cdot t \middle\vert s \in L ~\wedge~ t \in L^+ }$ + \item Kleene star: $L^* \defeq L^+ \cup \set{ \epsilon }$, where +$\epsilon$ represents the empty string.% +% +\footnote{The empty string is the identity of the string concatenation operator. +Some other texts use $\lambda$ for this purpose.} +% +\footnote{Many newcomers trip over the distinction between ``the empty string is +in the language'' ($\epsilon \in L$) and ``the language is empty'' ($L = +\emptyset$).} + \end{enumerate} + \item \label{lang:str:ops:hom} Homomorphic operations: + \begin{enumerate} + \item Homomorphism: $h(L) \defeq \set{ h(s) \middle\vert s \in L}$ + where $h(\epsilon) = \epsilon$ and $\forall_{s,t} . h(s \cdot t) = h(s) \cdot h(t)$. + \item $\epsilon$-free homomorphism, for $h$ which + additionally satisfies $\forall_{s \ne \epsilon} . h(s) \ne \epsilon$. + \item Inverse homomorphism: $h^{-1}(L) \defeq \set{ s \middle\vert h(s) \in L}$ + \end{enumerate} + \item Miscellaneous operations: + \begin{enumerate} + \item Concatenation: $L_1 \cdot L_2 \defeq \set{ s \cdot t \middle\vert s \in L_1 ~\wedge~ t \in L_2}$ + \item Reversal: $L^R \defeq \set{ c_1 \ldots c_n \middle\vert c_i \in \Sigma ~\wedge~ c_n \ldots c_1 \in L }$ + \end{enumerate} +\end{enumerate} diff --git a/treeautintro.tex b/treeautintro.tex new file mode 100644 index 0000000..c31e6d3 --- /dev/null +++ b/treeautintro.tex @@ -0,0 +1,115 @@ +Just as string automata describe sets of strings over some alphabet of +characters, tree automata describe sets of trees whose nodes are labeled +with elements of some set $\Sigma$. We focus initially on {\em ranked} tree +automata. + +\subsection{Notation for Trees} + +\Note{Positions} +\Note{subterm-at-position} + +$\Sigma_n \defeq \set{ \sigma \in \Sigma \middle\vert \mbox{ar}(\sigma) = n}$ +and $\Sigma_+ \defeq \Sigma\setminus\Sigma_0$. + +\subsection{Execution of Automata} + +As with the string case, there will be configurations $\config$, and a +privileged subset $\config_F$. However, transition functions are +quite different, and in fact split into two sub-classes of tree automata:% +% +\footnote{The transition functions presented here are for {\em +deterministic} automata. See \autoref{xxx} for details of non-deterministic +automata.} +% +\begin{itemize} +% + \item $\delta^{op} : \config \to \mathcal{T}(\Sigma \sqcup \config)$ defines a + \defn{top-down} automaton, which can be thought of generating a tree + root-first; the positions in the output which are configurations are + recursively expanded into trees. +% + \item $\delta : \mathcal{T}(\Sigma \sqcup \config) \to (1 + \config)$ defines a + \defn{bottom-up} automaton, which can be thought of as consuming a tree + leaves-first. Note that we allow, in general, the automata to refuse to + label a node (the $1$ return value), but certain classes of automata will + not avail themselves of this option. +% +\end{itemize} + +A \defn{run} of a tree automaton to be a tree in $\mathcal{T}\paren{\Sigma +\times (1 + \mathcal{T}(\Sigma \sqcup \config) \times \config)}$: an +augmented tree where each node is additionally, optionally, labeled by input +and output of $\delta$. For each node $\sigma \times \mbox{inr}(t \times +c)$, $\delta t = c$ (or $\delta^{op} c = t$), and all positions of $t$ which +are in $\Sigma$ must agree with the first projection of the run itself, and +all positions of $t$ which are in $\config$ must be so-labeled by the run. +\Note{That's not very clear, but I think it's right.} + +\subsection{Nondeterminism} + +\subsection{Plies} + +An important subclass of tree automata are the \defn{single-ply} automata, +where $\delta$ is restricted to manipulation of trees of the form +$\mathcal{T}(\Sigma_+ \sqcup \config)$. + +\subsection{Constraints} + +A natural novelty of tree automata is one of {\em constraints} between +positions in a (sub)tree. In general, one can think of such constraints as +including $\mathcal{T}(\Sigma)$ as a projection of $\config$, though with +restricted handling within the transition function,\eg only comparison for +(dis)equality. + +The \defn{constraint path} is the concatenation of the paths between the +{\em labeled node whose labeling introduced the constraint} and the +constrained positions. + +\subsubsection{Metaconstraints} + +In general, tree automata with arbitrary constraints have very poor closure +properties (see \autoref{xxx}). A variety of restrictions have been +developed throughout the literature and are summarized here. + +\paragraph{Brothers} The transition function may only mutually constrain +positions which differ only in the last index. + +\paragraph{Contained} All positions tested for equality must be within the +fragment of the tree being labeled (or generated); that is, constraint paths +may not involve a $\config$-labeled run node other than at their apex. + +\paragraph{Opaque} If a position is constrained by a transition, no other +constraint path may cross this position. + +\paragraph{Reduction} There exists a partial order on $\config$ such that +$\forall_{t \in \mathcal{T}(\Sigma \sqcup \config), c \in \config}$, $t$ and +$c$ being related by transition implies that $\forall_{c' \in t} . c' \le +c$, strictly if the transition between $t$ and $c$ induces any constraints. + +\subparagraph{Deep-Reduction} As above, but strictness is implied only when, +additionally, a position referenced by a constraint would be non-Contained. + +\paragraph{Stated} All positions constrained must be labeled with a +$\config$. + +\subparagraph{$f$-Same-Stated} In addition to the above, all positions +mutually constrained must be labeled and the $f$-image of these +configurations must be identical. When $f$ is the identity function or +otherwise clear from context, we may simply say ``Same-Stated''. + +\subsubsection{Interrelation of Metaconstraints} + +Single-ply automata necessarily satisfy Stated, as every node in a run is +labeled. Single-ply automata subject to Contained necessarily satisfy +Opaque; this is not true of multi-ply automata, as there may be crossing +constraints within a single multi-ply transition. + +In general, Contained and Stated together imply Opaque. + +\subsection{Transducers} + +\subsection{Weighted Automata} + +\subsection{Descriptive Taxonomy} + +\subsection{Operational Taxonomy} diff --git a/treelang.tex b/treelang.tex new file mode 100644 index 0000000..d8dfb20 --- /dev/null +++ b/treelang.tex @@ -0,0 +1,24 @@ +\subsection{Ranked Tree Languages} + +Trees open some new possibilities and so we shall need some new notation and +a new set of operations. For an excellent and thourough introduction, see +\cite[Preliminaries]{tata}. We limit ourselves here to a quick summary. + +A {\em ranked} tree language $L$ is a set of finite trees over some +\defn{signature} (also \defn{ranked alphabet}) $\Sigma$, with arity function +$\mbox{ar} : \Sigma \to \mathbb{N}$. Every node of a tree labeled with +$\sigma \in \Sigma$ has exactly $\mbox{ar}\paren{\sigma}$-many children.% +% +\footnote{More formally, a ranked tree language is a subset of the carrier +of the free algebra over $\paren{\Sigma,\mbox{ar}}$.} We use +$\mathcal{T}(\Sigma,\mbox{ar})$ for such a set of ranked trees; often +$\mbox{ar}$ will be implicit and we will just write $\mathcal{T}(\Sigma)$. +We use the notation $\mathcal{T}(\Sigma \sqcup X)$ to mean the set of trees +whose labels come either from $\Sigma$ (with the appropriate arity) or a +(disjoint) set of ``variables'' $X$, with $\mbox{ar}(x \in X) \defeq 0$. +The notation $\mathcal{T}(\Sigma \times \mathcal{Q})$ will be used for trees +whose labels are pairs of elements from $\Sigma$ and $\mathcal{Q}$; we +define $\mbox{ar}(\sigma \times q \in \Sigma \times \mathcal{Q}) \defeq +\mbox{ar}(\sigma)$. + +The set-theoretic operations carry over as might be expected. diff --git a/zoo-str/fsm.tex b/zoo-str/fsm.tex new file mode 100644 index 0000000..858dbe2 --- /dev/null +++ b/zoo-str/fsm.tex @@ -0,0 +1,104 @@ +Also called finite-state acceptor, or just state machine. Everyones' +favorite and simplest automaton family. + +Deterministic FSMs are completely characterized by a state set, typically +denoted $\mathcal{Q}$, an initial state $q_0 \in \mathcal{Q}$, and a set of +accepting states, $\mathcal{Q}_f \subseteq \mathcal{Q}$. (In our notation, +we have $\mathcal{C} = \mathcal{Q}$, $c_0 = q_0$, and $\mathcal{C}_f = +\mathcal{Q}_f$.) $\delta$ simply selects the new state of the machine. + +A deterministic FSM (``DFA'') may be rendered as a directed graph whose +vertices are the elements of $\mathcal{Q}$ and whose edges are labeled with +characters of $\Sigma$. $\delta$ may be read off from the graph by finding +the input configuration and following the (unique!) edge labeled by the +input character to the output configuration. + +Nondeterministic FSMs (``NFA''s) have a similar graphical rendering, save +that some edges may be labeled with the empty string $\epsilon$. The output +of $\delta$ given $(c_i, s)$ is the set of all nodes reachable from $c_i$ by +following a path composed of any number (possibly zero) $\epsilon$-edges, +exactly one $s$-edge, and then any number of $\epsilon$ edges.% +% +\footnote{Computationally, however, faster algorithms exist that perform +only one of the two $\epsilon$-closure stages suggested here. Typically, +one will traverse only paths labeled by $s\epsilon^*$, having initialized +with $C_0$ as the $\epsilon$ closure of $Q_0$.} For convenience, one will +often see nondeterministic FSMs with initial state {\em sets}; these are +equivalent to a designated single state and $\epsilon$-edges. + +See \href{http://wikipedia.org/Finite-state\_Machine}{Wikipedia's treatment} +or \cite[Ch. 1]{sipser:theorycomp} for thourough introductions. + +FSMs' recognition corresponds exactly with +\href{http://wikipedia.org/Regular\_language}{regular languages} and as such +they are closed under all regular operations. + +\autinfo{ + empty={Yes; $O(v \log v + e \log e)$ + to test reachability of accepting states. \cite{xxx}}, + member={Yes; $O(n)$ in length of input. \cite{xxx}}, + subset={Yes (by implication); + complementation avoidable in some cases \cite{abdulla:simantichain}}, +% + compl={Yes; take as accepting states $\sts \setminus \stsf$ \cite{xxx}}, +% + intersect={Yes; cross-product construction}, + union={Yes; cross-product construction for DFA, trivial for NFA}, +% + kstar={Yes; $O\paren{\abs{Q_f} + \abs{Q_i}}$ changes to the machine \cite{xxx}}, + kplus={Yes; $O\paren{\abs{Q_f}}$ changes to the machine \cite{xxx}}, +% + reginter={Yes, by definition}, +% + determinize={Yes; at possible exponential space cost. \cite{xxx}} +} + +\subsection{Probabilistic Finite State Automata} + +\cite{vidal:pfsm} provides an excellent overview of FSMs weighted with +probabilities. (Unless otherwise indicated, cross-references in this section +refer to that paper.) The authors distinguish several classes: +% +\begin{itemize} +% + \item ``Probabilistic Finite-State Automata'' (PFA, \S2.2) +% + \item ``Acyclic PFA`` (APFA, \S2.2) +% + \item ``$\lambda$-PFA'' (\S2.3) +% + \item ``Deterministic Probabilistic Finite-State Automata'' (DPFA, \S2.4) +% +\end{itemize} +PFA are non-deterministic but do not permit empty-label transitions. +$\lambda$-PFA permit empty-label transitions, easing certain constructions, +but are expressively equivalent to PFA (\S4.3). DPFA are less expressive +(prop. 10) but more readily analysed. + +If we ignore the probabilistic annotations, these automata are equivalent to +\hyperref[sec:zoo-str/fsm]{FSM}s. Therefore, in the tables below, we mean +to test not just the set of strings but the {\em distribution} of strings. +In the case of membership testing, we instead wish to know the +probability assigned to a string. For unions, we mean the pointwise-mean +of the two distributions (def. 10), while for intersection we mean {\em +coemmision probability} (the pointwise product of the distributions). + +For ($\lambda$-)PFA, we have: +\autinfo{ + member={Yes, $O(n\abs{\sts})$.}, + determinize={No; \S4.1; prop 10.} +} +While for DPFA, we have: +\autinfo{ + member={Yes; $O(n)$ in the length of the input.}, + union={No; \S4.2, prop 9 and appendix A.}, + intersect={Yes; \S5}, + minimize={Yes; \S4.1; see \cite{carrasco:learndetreg} and \cite{mohri:fstlsp}} +} + +\Note{These tables are quite sparsely populated; some entries are trivial, +while some I've probably overlooked in the paper.} + +\subsection{Probabilistic Finite State Transducers} + +\Note{Pull from \cite{pereira:wfst}.} diff --git a/zoo-str/nested-stack.tex b/zoo-str/nested-stack.tex new file mode 100644 index 0000000..c0d263e --- /dev/null +++ b/zoo-str/nested-stack.tex @@ -0,0 +1,9 @@ +There are several classes of automata which fall under the label ``nested +stack automata'', so we find it useful to say a few words in common, despite +the differing decision and closure properties. + +A nested stack automaton is a \hyperref[sec:zoo-str/stack]{stack automaton} +where the elements of the stack, $\Gamma$, are themselves stacks. (\ie +$\Gamma = \Gamma'^*$.) In all cases, the transition function is permitted +to push or pop from the ``outer'' stack and/or the top-most inner stack, +even if it may read more widely. diff --git a/zoo-str/pda.tex b/zoo-str/pda.tex new file mode 100644 index 0000000..19583fe --- /dev/null +++ b/zoo-str/pda.tex @@ -0,0 +1,63 @@ +A push-down automaton (PDA) is a \hyperref[sec:zoo-str/fsm]{FSM} equipped +with a stack. Let $\Gamma$ be the set of symbols on the stack and $\sts$ +the set of states of the FSM, as before; then $\config = \mathcal{Q} \times +\Gamma^*$. Initial configurations are typically defined, WOLOG, to have an +empty stack. + +Each transition of the automaton is given the top symbol of the +stack as well as the FSM's state and input character, and may manipulate the +stack by optionally popping the top symbol, and optionally pushing a new +symbol. That is, $\delta$ may be characterized by the type $\Sigma +\times \mathcal{Q} \times \paren{1 + \Gamma} \to \mathcal{Q} \times +\paren{1 + \Gamma} \times 2$, where $1 + \Gamma$ indicates an optional push +to the stack and $2$ indicates a boolean decision to pop.% +% +\footnote{Attempting to pop an emtpy stack is assumed to leave the stack +empty. We might have been more precise by forbidding popping an empty stack +and instead have said that $\delta$ was fully characterized by the type +$\paren{\Sigma \times \sts \to \sts \times \paren{1 + \Gamma}} \times +\paren{\Sigma \times \sts \times \Gamma \to \sts \times \paren{1 + \Gamma} +\times 2}$, but this more-strict view does not seem worth-while.} +% + +There are two, equivalent \cite{xxx}, definitions of accepting +configurations for PDAs: +% +\begin{itemize} +% + \item The ``accepting state'' condition requires that the final + configuration be in an accepting $\mathcal{Q}_f$, but may have arbitrary stack + contents.% + % + \footnote{That is, $\pi_1 c_n \in \mathcal{Q}_f$.} +% + \item The ``emtpy stack'' condition requires only that the stack be + empty at the end of the string, and does not care in which state of + $\mathcal{Q}$ the machine finds itself. + % + \footnote{That is, $\pi_2 c_n = \epsilon$.} +% +\end{itemize} + +\autinfo{ + empty={Yes \cite{xxx}}, +} + +\subsection{Non-deterministic PDAs} + +Non-deterministic PDAs' recognition corresponds exactly to context free languages. +Deterministic PDAs can recognize the subset of context-free languages, +appropriately called ``deterministic context free languages''. +\Note{cites and hrefs} + +See \href{http://wikipedia.org/Push-down\_automaton}{Wikipedia's treatment} +or \cite[Ch. 2]{sipser:theorycomp} for thourough introductions. + +\autinfo{ + empty={Yes \cite{xxx}}, + univ={Undecidable; proof via ``computation history'' of a + \hyperref[sec:zoo-str/tm]{TM}. \cite{xxx}}, + equiv={Undecidable; $\Sigma^*$ is recognizable, but universality is + undecidable.}, +} + diff --git a/zoo-str/stack.tex b/zoo-str/stack.tex new file mode 100644 index 0000000..b3f9e43 --- /dev/null +++ b/zoo-str/stack.tex @@ -0,0 +1,4 @@ +A stack automaton is a \hyperref[sec:zoo-str/pda]{PDA} given read access to +the entire stack. Here $\delta$ is characterized by the type $\Sigma +\times \mathcal{Q} \times \Gamma^* \to \mathcal{Q} \times +\paren{1 + \Gamma} \times 2$. diff --git a/zoo-str/tm.tex b/zoo-str/tm.tex new file mode 100644 index 0000000..3e96e62 --- /dev/null +++ b/zoo-str/tm.tex @@ -0,0 +1,35 @@ +\paragraph{Turing Machine} +A FSM augmented with an unbounded tape. A model of arbitrary computation. + +Let $\Gamma$ denote the tape alphabet (with $\Sigma \subseteq \Gamma$), then +$\config = \sts \times \mathbb{N} \times \Gamma^\mathbb{N}$, representing +the FSM state, the current position on the tape, and the tape contents.% +% +\footnote{Unlike most of the other automata in the zoo, the input to a TM as +defined here is not given character-by-character, but is encoded into the +seed machine configuration. Of course, an equivalent formulation, allowing +the machine to await the next character of its input, is possible, but note +that there remains some fundamental level of asynchrony -- the machine may +compute without bound between requests for the next character.} + +$\delta$ is characterized by the type $\paren{\sts \times \Gamma} \to +\paren{\sts \times \Gamma \times 3}$; the input $\Gamma$ is the symbol at +the current position, while the output $\Gamma$ will be written there in the +next configuration; the $3$ indicates a choice between incrementing, +decrementing (if nonzero), or not changing the current position.% +% +\footnote{Attempting to decrease while at position $0$ is the same as a +no-operation; again, we could have given a more precise characterization of +$\delta$ with the attendant loss of clarity. Note, however, that a machine +with a tape indexed by $\mathbb{Z}$ has no such corner case and is equivalent +to the presentation given here. \cite{post:1936}} + +\autinfo{ + empty={Undecidable}, + member={Semidecidable}, + intersect={Yes}, + regops={Yes}, + kplus={Yes}, + determinize={Yes, at exponential space and time cost}, + minimize={No; the set of minimal TMs is not even recognizable.} +} diff --git a/zoo-tree/awcbb.tex b/zoo-tree/awcbb.tex new file mode 100644 index 0000000..da56efc --- /dev/null +++ b/zoo-tree/awcbb.tex @@ -0,0 +1,2 @@ +\Note{AWCBB is single-ply AWEDC subject to Brother and Contained +metaconstraints (and, by implication, Opaque and Stated).} diff --git a/zoo-tree/awedc.tex b/zoo-tree/awedc.tex new file mode 100644 index 0000000..e69de29 diff --git a/zoo-tree/ra.tex b/zoo-tree/ra.tex new file mode 100644 index 0000000..c88435e --- /dev/null +++ b/zoo-tree/ra.tex @@ -0,0 +1,5 @@ +\Note{RA is single-ply AWEDC subject to Reduction metaconstraint (and, by +implication, Stated).} + +\Note{\cite[p133]{tata} discusses the relaxation of RA to permit AWCBB-style +constraints freely; what I've called ``Deep-Reduction'' above. This class has decidable emptyness testing.} diff --git a/zoo-tree/regular.tex b/zoo-tree/regular.tex new file mode 100644 index 0000000..d7bac68 --- /dev/null +++ b/zoo-tree/regular.tex @@ -0,0 +1,27 @@ +Regular tree languages are tree automata without constraints with a finite +$\config$ set, directly analagous to the finite string automata case. They +are typically presented as single-ply constructions, though this is not +essential -- any multi-ply regular automaton has a weakly equivalent +single-ply automaton, and the reverse direction is clearly trivial. + +Given a family of trees accepted by a regular tree automata, the set of +strings formed by labels along paths in this set is a regular finite string +language. The same is true of accepting runs. + +For bottom-up automata, we have: +\autinfo{ + empty={Linear in the size of the automaton; \cite[Thm 1.7.4]{tata}}, + member={ALOGTIME-complete; \cite[Thm 1.7.2]{tata}}, + univ={PTIME for det. aut., EXPTIME for nondet. \cite[Thm 1.7.7]{tata}}, + equiv={Product of sizes for det. aut., exponential for nondet. \cite[Thm 1.7.8]{tata}}, + compl={Yes, but may require determinization; \cite[Thm 1.3.1]{tata}}, + intersect={Yes, determinization not required; \cite[Thm 1.3.1]{tata}}, + union={Yes; \cite[Thm 1.3.1]{tata}}, + determinize={Yes; \cite[Thm 1.1.9]{tata}}, +} +It is additionally possible to determine if an automaton accepts only a +singleton set in PTIME. + +Non-deterministic top-down automata are equivalent to bottom-up automata +\cite[Thm 1.6.1]{tata}, but deterministic top-down automata are weaker. +\cite[Prop 1.6.2]{tata}. diff --git a/zoo-tree/tahom.tex b/zoo-tree/tahom.tex new file mode 100644 index 0000000..e81313b --- /dev/null +++ b/zoo-tree/tahom.tex @@ -0,0 +1,5 @@ +\Note{Various classes are introduced by \cite{godoy:hom}. +$TA_{\text{hom},\neq}$ is multi-ply AWEDC subject to Contained and Same-Stated +(and, by implication, Opaque) only for equality constraints; disequality +constraints are without metaconstraints. +} -- 2.50.1