• English only

# Differences

This shows you the differences between two versions of the page.

regular_expressions_for_automata_with_parallel_inputs [2009/04/28 20:26]
vkuncak
regular_expressions_for_automata_with_parallel_inputs [2015/04/21 17:32] (current)
Line 2: Line 2:

Given an alphabet $\Sigma$, we consider a larger (but still finite) alphabet $\Sigma^n$ for some $n$.  Keep in mind that $(a_1,​\ldots,​a_n) \in \Sigma_n$ is just one symbol; we often write it as Given an alphabet $\Sigma$, we consider a larger (but still finite) alphabet $\Sigma^n$ for some $n$.  Keep in mind that $(a_1,​\ldots,​a_n) \in \Sigma_n$ is just one symbol; we often write it as
-$\left(+\begin{equation*}\left( ​\begin{array}{l} ​\begin{array}{l} a_1 \\ a_1 \\ Line 9: Line 9: ​\end{array} ​\end{array} ​\right) ​\right) -$+\end{equation*}

We can consider We can consider
Line 15: Line 15:
* automata   * automata
on such alphabets. on such alphabets.
+

====== Using Propositional Formulas to Denote Finite Sets of Symbols ====== ====== Using Propositional Formulas to Denote Finite Sets of Symbols ======
Line 25: Line 26:

Language representing that the third coordinate is the logical **and** of the first two is: Language representing that the third coordinate is the logical **and** of the first two is:
-$+\begin{equation*} \left( \left( \left( \left( Line 59: Line 60: ​\right) ​\right) ​\right)^* ​\right)^* -$+\end{equation*}

Instead of considering $\Sigma^3$, we can consider $\{x,y,z\} \to \Sigma$ where $x,y,z$ are three names of variables. Instead of considering $\Sigma^3$, we can consider $\{x,y,z\} \to \Sigma$ where $x,y,z$ are three names of variables.

We then use propositional formulas to denote possible values of bits. For example, $[x \land y]$ denotes the regular expression ​ We then use propositional formulas to denote possible values of bits. For example, $[x \land y]$ denotes the regular expression ​
-$+\begin{equation*} \left( \left( ​\begin{array}{l} ​\begin{array}{l} Line 80: Line 81: ​\end{array} ​\end{array} ​\right) ​\right) -$+\end{equation*}
The bitwise **and** relation, shown above, is given by The bitwise **and** relation, shown above, is given by
-$+\begin{equation*} [z \leftrightarrow (x \land y)]^* [z \leftrightarrow (x \land y)]^* -$+\end{equation*}

In general In general
Line 109: Line 110:
where $p(v_1,​\ldots,​v_n)$ is a propositional formula and $(a_{i1},​\ldots,​a_{in})$ for $1 \leq i \leq k$ are all tuples of values of propositional variables for which $p(v_1,​\ldots,​v_n)$ is true. where $p(v_1,​\ldots,​v_n)$ is a propositional formula and $(a_{i1},​\ldots,​a_{in})$ for $1 \leq i \leq k$ are all tuples of values of propositional variables for which $p(v_1,​\ldots,​v_n)$ is true.

-Notational advantage: the set of variables ​can be larger, the expression ​is still the same.+Notational advantage: ​even if we increase ​the number ​of components by adding new variables, the expression ​remains ​the same.