# jeffreykegler/Marpa-theory

Update for changes to Leo logic

• Loading branch information...
1 parent 17206b5 commit eb6fa182ece2f321c99b0ae619cfd39f5efebdbc Jeffrey Kegler committed Jun 20, 2013
Showing with 18 additions and 24 deletions.
1. +18 −24 recce.ltx
 @@ -1,7 +1,7 @@ % Copyright 2013 Jeffrey Kegler % This document is licensed under % a Creative Commons Attribution-NoDerivs 3.0 United States License. -\documentclass[12pt,draft]{amsart} +\documentclass[12pt]{amsart} \usepackage{algorithm} \usepackage{algpseudocode} \usepackage{url} @@ -2021,12 +2021,10 @@ and a parse location \Vloc{i}, \end{theorem} \begin{proof} -In this context \Vloc{i} is to be treated -as an integer, the ordinal of Earley set \Vloc{i}. EIM's have the form $[\Vah{x}, \Vorig{x}]$. \Vorig{x} is the origin of the EIM, which in Marpa cannot be after the current -Earley set \Vloc{i}, +Earley set at \Vloc{i}, so that \begin{equation*} 0 \le \Vorig{x} \le \Vloc{i}. @@ -2038,7 +2036,7 @@ which depends on \Cg{}. Since duplicate EIM's are never added to an Earley set, the maximum size of Earley set \Vloc{i} is therefore \begin{equation*} -\Vloc{i} \times \size{\Cfa} = \order{\var{i}}.\qedhere +\Vloc{i} \times \size{\Cfa} = \order{\Vloc{i}}.\qedhere \end{equation*} \end{proof} @@ -2051,12 +2049,12 @@ Marpa's Earley sets are correct. The proof is by triple induction, that is, induction with a depth down to 3 levels. -We number the levels of induction from -outermost to innermost, -0, 1 and 2. +We number the levels of induction +0, 1 and 2, +starting with the outermost. The level 0 induction is usually called the outer induction. The level 1 induction is usually called the inner induction. -The level 2 induction is referred to by its number. +Level 2 induction is referred to by number. The outer induction is on the Earley sets. The outer induction hypothesis is that all Earley sets @@ -2335,7 +2333,7 @@ the Earley set is an ordered set, and that new Earley items are added at the end. From Theorem \ref{t:es-count}, we know that -the number of Earley items is \On{}, +the number of Earley items is finite, so a traversal of them must terminate. Consider, for the purposes of the level 2 induction, @@ -2400,6 +2398,7 @@ we see that a Earley reduction result of generation $\var{n}+1$ will be added to \EVtable{\Marpa}{i} corresponding to every Earley reduction result +in generation $\var{n}+1$ of \EVtable{\Leo}{i}, as well as one corresponding to every prediction that results from @@ -2497,10 +2496,10 @@ the previous two equalities. For the complexity proofs, we consider only Marpa grammars without nulling symbols. -When we examined correctness, -we showed that this rewrite +We showed that this rewrite is without loss of generality -in Section \ref{s:nulling}. +in Section \ref{s:nulling}, +when we examined correctness. For complexity we must also show that the rewrite and its reversal can be done in amortized \Oc{} time and space @@ -2991,9 +2990,9 @@ form where $\Vsym{rightmost} \deplus \Vsym{A}$. Such a rule is right recursive by definition. This is contrary to the assumption for the reductio. -We therefore conclude that the length of a right derivation with no -step that uses a right recursive rule, -must be less than or equal to \var{c}. +We therefore conclude that the length of a right derivation +must be less than or equal to \var{c}, +unless at least one step of that derivation uses a right recursive rule. \end{proof} \subsection{The complexity results} @@ -3145,12 +3144,7 @@ including the space for tracking links. \end{theorem} \begin{proof} -One link is required to record -each attempt to add an Earley item. -Each link requires \Oc{} space. -The bound of $\order{\var{n}^3}$ results -from these two observations, -Theorem \ref{t:O1-links-per-eim}, +By Theorem \ref{t:O1-links-per-eim}, and Theorem \ref{t:ambiguous-tries}. \end{proof} @@ -3255,11 +3249,11 @@ let it be understood that \end{equation*} We require that, for some constant \var{c}, -possibly dependant on the grammar \Cg{}, +possibly dependent on the grammar \Cg{}, that every token length be less than \var{c}, \begin{equation} \label{e:restriction1} -\forall \, \token{[\Vsym{x}, \Vloc{x}, length]}, +\forall \, \token{[\Vsym{x}, \Vloc{x}, \var{length}]}, \; \var{length} < \var{c}, \end{equation} and that

#### 0 comments on commit eb6fa18

Please sign in to comment.