# jfinkels/parseable

Added fragments of literal movement grammars for constructing permuta…

…tions of first-order propositional logic formulae with no free variables and no vacuous quantifiers, respectively.
 @@ -27,13 +27,15 @@ \documentclass[draft]{article} \usepackage{amsthm} \usepackage{amssymb} +\usepackage{amsmath} +\usepackage[small, bf]{caption} \usepackage{complexity} -\usepackage{float} +%\usepackage{float} \usepackage{syntax} \usepackage[pdftitle={Parsing first-order propositional logic}, pdfauthor={Jeffrey Finkelstein}]{hyperref} -\floatstyle{ruled} -\restylefloat{figure} +%\floatstyle{ruled} +%\restylefloat{figure} \newtheorem{theorem}{Theorem} @@ -266,6 +268,62 @@ A related problem is determining if $\lang{\pn}\in\INDEXED$. \end{openproblem} +\autoref{fig:closedlmg} is a fragment of a combinatorial, linear, non-erasing literal movement grammar for \lang{\pc}. +\autoref{fig:nvqlmg} is a fragment of a combinatorial, linear, non-erasing literal movement grammar for \lang{\pn}. + +\begin{figure} + \caption{ + \label{fig:closedlmg} + Literal movement grammar for \lang{\pc}. + $Q_i$ represents a quantification of variable $x_i$. + After quantification, this grammar allows any number of instances of the variable $x_i$ to occur in rule $V_i$. + The rules for producing the other symbols of first-order propositional logic have been omitted, as represented by the vertical ellipsis. + Note: this grammar can derive strings with vacuous quantifiers. + } + \begin{align*} + & S(\epsilon) \rightarrow \epsilon \\ + & S(X) \rightarrow Q_i(X) \\ + & Q_i(X\forall Y x_i Z) \rightarrow V_i(XYZ) \\ + & Q_i(X x_i Y \forall Z) \rightarrow V_i(XYZ) \\ + & V_i(X x_i Y) \rightarrow V_i(XY) \\ + & V_i(X) \rightarrow S(X) \\ + & \vdots + \end{align*} +\end{figure} + +\begin{figure} + \caption{ + \label{fig:nvqlmg} + Literal movement grammar for \lang{\pn}. + $V_i$ represents an instance of variable $x_i$. + After generating any number of instances of variable $x_i$, it is quantified by the rule $Q_i$. + After that, production of instances of variable $x_{i+1}$ is allowed. + Shortcuts are provided for skipping any unused variables to avoid vacuous quantification. + The rules for producing the other symbols of first-order propositional logic follow from rule $R$ and are omitted here, as represented by the second vertical ellipsis. + Note: this grammar can derive strings with free variables. + } + \begin{align*} + & S(\epsilon) \rightarrow \epsilon \\ + & S(X) \rightarrow Q_i(X) \\ + & V_1(X x_i Y) \rightarrow V_1(XY) \\ + & V_1(X) \rightarrow Q_1(X) \\ + & V_1(X) \rightarrow V_2(X) \\ + & Q_1(X\forall Y x_1 Z) \rightarrow V_2(XYZ) \\ + & Q_1(X x_1 Y \forall Z) \rightarrow V_2(XYZ) \\ + & \vdots \\ + & Q_n(X x_n Y \forall Z) \rightarrow R(XYZ) \\ + & Q_n(X \forall Y x_n Z) \rightarrow R(XYZ) \\ + & \vdots + \end{align*} +\end{figure} + +\begin{todo} + Definition of combinatorial, linear, non-erasing literal movement grammars. +\end{todo} +\begin{todo} + Determine the complexity of the literal movement grammar in \autoref{fig:closedlmg} and \autoref{fig:nvqlmg}. +\end{todo} + \bibliographystyle{plain} \bibliography{bibliography}