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Constructing a Parse Tree Initial Conditions Before we begin we need to have two things: 1. A Grammar 2. A string we wish to parse So then what is the procedure: 1. Use the grammar to produce a derivation resulting in the given string 2. Use

  1. Constructing a Parse Tree

  2. Initial Conditions Before we begin we need to have two things: 1. A Grammar 2. A string we wish to parse So then what is the procedure: 1. Use the grammar to produce a derivation resulting in the given string 2. Use the derivation to produce the parse tree

  3. Example 1: We have the following grammar: <S> ::= <round> <square> | <outer> <round> ::= ( <round> ) | ( ) <square> ::= [ <square> ] | [ ] <outer> ::= ( <outer> ] | ( <inner> ] <inner> ::= ) <inner> [ | ) [ We will derive the following string: (())[[]]

  4. Example 1: Producing the derivation A derivation is produce using the following steps: 1. Start with the start rule and select one of its options 2. We then continue to replace each rule until we reach a terminal working from left to right. 3. We repeat step 2 until all non-terminals are replaced by terminals, and we have produced the target string. Goal: Derive ( ( ) ) [ [ ] ] <S> => <round> <square> => ( <round> ) <square> => ( ( ) ) <square> => ( ( ) ) [ <square> ] => ( ( ) ) [ [ ] ] --> Finished

  5. Example 1: Producing the parse tree <S> => <round> <square> <S> / \ <round> <square> => ( <round> ) <square> / | \ / | \ ( <round> ) / | \ => ( ( ) ) <square> / \ | | \ ( ) | | \ => ( ( ) ) [ <square> ] [ <square> ] => ( ( ) ) [ [ ] ] / \ [ ]

  6. Example 2 Grammar: <S1> ::= <S1> + <S2> | <S2> <S2> ::= <S2> * <S3> | <S3> <S3> ::= ( <S1> ) | a | b | c String: a + b * c

  7. Example 2: Derivation Goal: Derive a + b * c <S1> => <S1> + <S2> => <S2> + <S2> => <S3> + <S2> => a + <S2> => a + <S2> * <S3> => a + <S3> * <S3> => a + b * <S3> => a + b * c

  8. Example 2: Parse tree <S1> => <S1> + <S2> <S1> / | \ <S1> + <S2> => <S2> + <S2> | / | \ <S2> / | \ => <S3> + <S2> | / | \ <S3> | | | => a + <S2> | | | | a | | | => a + <S2> * <S3> <S2> * <S3> => a + <S3> * <S3> | | <S3> | => a + b * <S3> | | b | => a + b * c c

  9. We can also derive a string from the bottom up Goal: Use Example 2 grammar to derive a + b * c In This case we start with the rightmost terminal and continue to replace with non-terminals until we reach the start rule. => a + b * c => a + b * <S3> => a + <S3> * <S3> => a + <S2> * <S3> => a + <S2> => <S3> + <S2> => <S2> + <S2> => <S1> + <S2> <S1> => <S1> + <S2> We then build the parse tree starting from the bottom

  10. Constructing Abstract Syntax Trees

  11. Parse Trees to ASTs An Abstract Syntax Tree (AST) is a simplified form of a Parse Tree which is useful for interpreting/converting code from one language to another. Thus, it is useful for the compiling process. An AST typically is a Binary Tree and requires that there are no non-terminal symbols left in the tree, and this then requires that we have the following: ◮ A Parse Tree ◮ A notion of traversing the tree (we will assume an In Order traversal)

  12. The Conversion Process <S1> / | \ <S1> + <S2> | / | \ <S2> / | \ | / | \ <S3> | | | | | | | a | | | <S2> * <S3> | | <S3> | | | b | c

  13. Example <S1> <S1> / | \ / | \ <S1> + <S2> a + <S2> | / | \ / | \ <S2> / | \ / | \ | / | \ / | \ <S3> | | | | | | | | | | | | | a | | | | | | <S2> * <S3> <S2> * <S3> | | | | <S3> | <S3> | | | | | b | b | c c

  14. Example <S1> <S1> / | \ / | \ <S1> + <S2> a + <S2> | / | \ / | \ <S2> / | \ b | \ | / | \ | \ <S3> | | | | | | | | | | | a | | | | | <S2> * <S3> * <S3> | | | <S3> | | | | | b | | c c

  15. Example <S1> <S1> / | \ / | \ <S1> + <S2> a + <S2> | / | \ / | \ <S2> / | \ b * \ | / | \ \ <S3> | | | | | | | | | a | | | | <S2> * <S3> <S3> | | | <S3> | | | | | b | | c c

  16. Example <S1> <S1> / | \ / | \ <S1> + <S2> a + <S2> | / | \ / | \ <S2> / | \ b * c | / | \ <S3> | | | | | | | a | | | <S2> * <S3> | | <S3> | | | b | c

  17. Example <S1> <S1> / | \ / | \ <S1> + <S2> a + * | / | \ / \ <S2> / | \ b c | / | \ <S3> | | | | | | | a | | | <S2> * <S3> | | <S3> | | | b | c

  18. Example <S1> + / | \ / \ <S1> + <S2> a * | / | \ / \ <S2> / | \ b c | / | \ <S3> | | | | | | | a | | | <S2> * <S3> | | <S3> | | | b | c

  19. Example <S> / \ <round> <square> / | \ / | \ ( <round> ) / | \ / \ | | \ ( ) | | \ [ <square> ] / \ [ ]

  20. Example <S> <S> / \ / \ <round> <square> <round> <square> / | \ / | \ / | \ / | \ ( <round> ) / | \ ( <round> ) / | \ / \ | | \ / \ | | \ ( ) | | \ ( ) | | \ [ <square> ] [ <square> ] / \ / \ [ ] [ ]

  21. Example <S> <S> / \ / \ <round> <square> <round> <square> / | \ / | \ / | \ / | \ ( <round> ) / | \ ( ) ) / | \ / \ | | \ / | | \ ( ) | | \ ( | | \ [ <square> ] [ <square> ] / \ / \ [ ] [ ]

  22. Example <S> <S> / \ / \ <round> <square> ) <square> / | \ / | \ / | \ / | \ ( <round> ) / | \ ( ( ) / | \ / \ | | \ | | \ ( ) | | \ | | \ [ <square> ] [ <square> ] / \ / \ [ ] [ ]

  23. Example <S> ) / \ / \ <round> <square> ( <square> / | \ / | \ / \ / | \ ( <round> ) / | \ ( ) / | \ / \ | | \ | | \ ( ) | | \ | | \ [ <square> ] [ <square> ] / \ / \ [ ] [ ]

  24. Example <S> ) / \ / \ <round> <square> ( <square> / | \ / | \ / \ / | \ ( <round> ) / | \ ( ) / | \ / \ | | \ | | | ( ) | | \ | | | [ <square> ] [ ] ] / \ / [ ] [

  25. Example <S> ) / \ / \ <round> <square> ( <square> / | \ / | \ / \ / | \ ( <round> ) / | \ ( ) [ ] ] / \ | | \ / ( ) | | \ [ [ <square> ] / \ [ ]

  26. Example <S> ) / \ / \ <round> <square> ( ] / | \ / | \ / \ / | \ ( <round> ) / | \ ( ) [ [ ] / \ | | \ ( ) | | \ [ <square> ] / \ [ ]

  27. Example <S> ) / \ / \ <round> <square> ( ] / | \ / | \ / \ / \ ( <round> ) / | \ ( ) [ ] / \ | | \ / ( ) | | \ [ [ <square> ] / \ [ ]

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