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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.
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1 parent b890bce commit 18ea02407b739f437193421f565b4ba5301ba1d0 @jfinkels committed Feb 6, 2012
Showing with 61 additions and 3 deletions.
  1. +61 −3 parseable.tex
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64 parseable.tex
@@ -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}

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