diff --git a/main.tex b/main.tex index 390f482a3c90c722340a6e373266d759ad9f34d8..7970e99ad8eb003dd862d890279ea0d826657531 100644 --- a/main.tex +++ b/main.tex @@ -32,10 +32,16 @@ \newcommand{\mtt}{\textsf{MTT}\xspace} \newcommand{\rev}{\textsf{rev}\xspace} +\newcommand{\rtm}{\textsf{RTM}\xspace} +\newcommand{\drtm}{\textsf{DRTM}\xspace} +\newcommand{\reg}{\textsf{Reg}} -\newcommand{\dlin}{\textsc{DLinSpace}\xspace} +\newcommand{\dlinspace}{\textsc{DLinSpace}\xspace} +\newcommand{\dlogspace}{\textsc{DLogSpace}\xspace} +\newcommand{\dptime}{\textsc{DPTime}\xspace} +\newcommand{\dexptime}{\textsc{DExpTime}\xspace} \newcommand{\ssign}{\mathfrak{S}} \newcommand{\tsign}{\mathfrak{T}} @@ -96,7 +102,7 @@ \paragraph{Signature} A \emph{signature} $\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow \nat$. Abusing notations, we will often write $R\in \ssign$ instead of $R\in S$. -\paragraph{Structures} A \emph{structure} $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)-\subseteq \dom(A)^r$, with $r=\ar(R)$. +\paragraph{Structures} A \emph{structure} (sometimes \emph{model}) $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)\subseteq \dom(A)^r$, with $r=\ar(R)$. \paragraph{Operations} Any structure over a signature can be seen as a structure over a larger signature where additional symbols are interpreted as empty sets. @@ -107,9 +113,9 @@ The \emph{product} of two structures $A,B$ over a signature $\ssign$ is a struct The \emph{ordered coproduct} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigoplus_{i\leq n} A_i$ over the signature $\ssign+\set{\sqsubseteq}$, where $\sqsubseteq$ has arity $2$. The domain of $\bigoplus_{i\leq n} A_i$ is $\bigoplus_{i\leq n} \dom_{A_i}=\bigcup_{i\leq n} \dom_{A_i}{\times}\set{i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n} \inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$. -The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^1 + \ssign^{>1}\times\set{\sqsubseteq}$. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$. -The interpretation of a symbol $R$ is $\bigoplus_{i\leq n} \inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$. - +The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^{1} + \ssign^{>1}\times\set{\sqsubseteq}$, where $\ssign^{1}$, $\ssign^{>1}$ denote the symbols of arity one and at least two, respectively. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$. +The interpretation of a symbol $R$ of arity $1$ is $\bigotimes_{i\leq n} \inter_{A_i}(R)$. The interpretation of $(R,\sumorder)$ is: $$\bigcup_{j\leq n}\quad \bigotimes_{i<j}\Delta_{i,k} \times\inter_{A_j}\times\bigotimes_{j<i\leq n}\dom_{A_i}$$ +where $\Delta_{i,k}=\set{(a,\ldots,a)\in \dom_{A_i}^k\mid\ a\in \dom_{A_i}}$ is the $k$-fold diagonal of $\dom_{A_i}$. The \emph{powerset structure} of a structure $A$ over signature $\ssign$ is a structure $\pow A$ over signature $\ssign\uplus \set{\subseteq}$, where $\subseteq$ is a binary symbol. The domain of $\pow A$ is $\pow{\dom_A}$. The interpretation of a symbol $R$ is the set $\set{(\set{x_1},\dots,\set{x_k})\mid\ (x_1,\ldots,x_n)\in \inter_A(R)}$. The interpretation of $\subseteq$ is the actual set inclusion $\set{(x,y)\mid\ x\subseteq y\subseteq \dom_A}$. \paragraph{Word models} @@ -234,26 +240,6 @@ An \msomi of dimension $d$ has growth $O((2^n)^d)=O(2^{dn})=2^{O(n)}$. -\section{The case of the successor} - -\begin{theorem} -\msomi with successor is equivalent to \dlin reductions. -\end{theorem} - -\begin{proof} -Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton. -From a linear bounded automaton, the next configuration relation can be defined in \mso. -\end{proof} - -\begin{question} - - -Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ \cdots \circ f_{a_n}(\epsilon)$ compute \dlin ? - -\end{question} - - - \section{Nested marble transducer} We define a model of transducers based on marble transducers. @@ -299,7 +285,7 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give \end{remark} \begin{example} -Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup{0,1}\cup \set{\natural})^*$. +Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup\set{0,1,\natural})^*$. What the function $f$ does is list all subsets of positions (without $\sharp$) and output the word labelled with the corresponding subset. The labelled words are output in a specific order, and separated by $\natural$. Let us decribe the order in which these subsets of positions are listed. Over a word $u$ without $\sharp$, $f$ outputs the list of all annotations, in the lexicographic order over the subset of positions of $u$, \eg @@ -352,8 +338,35 @@ Does $\ipt\subseteq \mt\circ\msot$ hold? \end{question} +\section{Rational Turing Machine} +A \emph{Rational Turing Machine} is a Turing Machine with a reachability relation over configurations that is rational. + + +\begin{theorem} +A function is \expreg if and only if it is definable by a \drtm. +\end{theorem} + +\begin{theorem} +For rational turing machines, the following hold: +\begin{enumerate} +\item \drtm = \dlinspace +\item \dlogspace= \dptime +\end{enumerate} +\end{theorem} + + +\section{The case of the successor} + +\begin{theorem} +\msomi with successor is equivalent to \dlinspace reductions. +\end{theorem} + +\begin{proof} +Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton. +From a linear bounded automaton, the next configuration relation can be defined in \mso. +\end{proof} \section{Some remaining questions} @@ -386,6 +399,8 @@ Given a (fixed) \msomi realizing $f$, can one compute the image of a word $u$ in \item recursive programming language corresponding to or being captured by \msomi ? \item Krohn-Rhodes like decomposition \eg $\msoi\circ \mathsf{exp} \circ \msot$, with $\mathsf{exp}$ being some simple class of exponential growth function. + + \item Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ \cdots \circ f_{a_n}(\epsilon)$, with sequential functions compute \dlinspace ? \end{itemize}