diff --git a/main.tex b/main.tex index 8eb9f885902a59464a376065cb85ff43c374e3fb..e3d0a9d85e78829e959577e244c2e068bc4615ee 100644 --- a/main.tex +++ b/main.tex @@ -114,21 +114,7 @@ A non-deterministic \msomi (\nmsomi) $S$ from $\ssign$-structures to $\tsign$-st We call string-to-string (monadic) interpretations the particular case when the two signatures have unary symbols and one binary symbol which is a linear order (this can be enforced syntactically). An \expreg function is a string-to-string monadic interpretation. -We denote by \textsf{ExpF} the class of \expreg functions. -As an example, consider the following function. Let $\Sigma=\{a,b\}$ -and $\Gamma = \Sigma{\times} \{0,1\}\cup \set{\#}$. Let $u\in\Sigma^*$. -Given a -subset $U\subseteq Pos(u)$, we let $u_U\in \Gamma^*$ such that -$|u_U| = |u|$ and for all positions $p$, $u_U(p) = (u(p), p\in -U)$. Given another subset $V$, $U\leq_{lex} V$ if $U = V$ or the -smallest $x$ such that $x\in V$ iff $x\not\in U$ satisfies $x\in V$ -and $x\not\in U$. Finally, we let $subsets(u) = \prod_{U\subseteq Pos(u) \text{ in - lexicographic order}} u_U\#$. For example, -$$ -subsets(ab) = (a,0)(b,0)\#(a,0)(b,1)\#(a,1)(b,0)\#(a,1)(b,1)\# -$$ -It is easy to see that $subsets\in\textsf{ExpF}$. - +We denote by \textsf{ExpF} the class of \expreg functions. @@ -252,7 +238,7 @@ pebble transducer model. candidate: \end{remark} \begin{example} -Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \set{0,1}\cup \set{\natural})^*$. +Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup{0,1}\cup \set{\natural})^*$. 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 @@ -292,7 +278,7 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give \begin{question} -Does $\ipt=\mt\circ\msot$ hold? +Does $\ipt\subseteq \mt\circ\msot$ hold? \end{question} \section{Some remaining questions}