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regular_expressions_for_automata_with_parallel_inputs [2009/04/28 20:24] vkuncak |
regular_expressions_for_automata_with_parallel_inputs [2009/04/28 20:27] vkuncak |
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| on such alphabets. | on such alphabets. | ||
| - | Example: $\Sigma = \{0,1\}$. Let $n=3$. | + | |
| + | ====== Using Propositional Formulas to Denote Finite Sets of Symbols ====== | ||
| + | |||
| + | Suppose $\Sigma = \{0,1\}$. | ||
| + | |||
| + | Let $n=3$. | ||
| $\Sigma^3 = \{ (0,0,0),(0,0,1),\ldots,(1,1,1) \}$. | $\Sigma^3 = \{ (0,0,0),(0,0,1),\ldots,(1,1,1) \}$. | ||
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| \right) | \right) | ||
| \] | \] | ||
| - | The bitwise and relation is given by | + | The bitwise **and** relation, shown above, is given by |
| \[ | \[ | ||
| [z \leftrightarrow (x \land y)]^* | [z \leftrightarrow (x \land y)]^* | ||
| Line 105: | 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. |