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| | #REDIRECT [[ist:Introduction to Syntax]] |
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| = What is a Grammar =
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| = FIRST and FOLLOW sets =
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| == Computing the FIRST Set ==
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| The FIRST set for a given string or symbol can be computed as follows:
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| # If '''a''' is a terminal symbol, then FIRST('''a''') = {'''a'''} | |
| # If X is a non-terminal symbol and X -> ε is a production then add ε to FIRST(X)
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| # If X is a non-terminal symbol and X -> Y<sub>1</sub>...Y<sub>n</sub> is a production, then
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| #:a ∈ FIRST(X) if a ∈ FIRST(Y<sub>i</sub>) and ε ∈ FIRST(Y<sub>j</sub>), i>j (i.e., Y<sub>j</sub> <amsmath>\overset{*}{\Rightarrow}</amsmath> ε)
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| As an example, consider production X -> Y<sub>1</sub>...Y<sub>n</sub>
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| * If Y<sub>1</sub> <amsmath>\overset{*}{\nRightarrow}</amsmath> ε then FIRST(X) = FIRST(Y<sub>1</sub>)
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| * If Y<sub>1</sub> <amsmath>\overset{*}{\Rightarrow}</amsmath> ε and Y<sub>2</sub> <amsmath>\overset{*}{\nRightarrow}</amsmath> ε then FIRST(X) = FIRST(Y<sub>1</sub>) \ {ε} ∪ FIRST(Y<sub>2</sub>)
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| * If Y<sub>i</sub> <amsmath>\overset{*}{\Rightarrow}</amsmath> ε (∀i) then FIRST(X) = ∪<sub>i</sub>(FIRST(Y<sub>i</sub>)\{ε}) ∪ {ε}
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| The FIRST set can also be computed for a string Y<sub>1</sub>...Y<sub>n</sub> much in the same way as in case 3 above.
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| == Computing the FOLLOW Set ==
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| The FOLLOW set is computed for non-terminals and indicates the set of terminal symbols that are possible after a given non-terminal. The special symbol $ is used to represent the end of phrase (end of input).
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| # If X is the grammar's initial symbol then {$} ⊆ FOLLOW(X)
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| # If A -> αXβ is a production, then FIRST(β)\{ε} ⊆ FOLLOW(X)
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| # If A -> αX or A -> αXβ (β <amsmath>\overset{*}{\Rightarrow}</amsmath> ε), then FOLLOW(A) ⊆ FOLLOW(X)
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| The algorithm should be repeated until the FOLLOW set remains unchanged.
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| = Exercises =
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| * [[Introduction to Syntax/Exercise 1|Exercise 1]]
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| [[category:Teaching]]
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| [[category:Compilers]]
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