Parsing Context-Free Languages

(文脈自由言語の構文解析)

7th lecture, May 31, 2019

Language Theory and Compilers

http://www.sw.it.aoyama.ac.jp/2019/Compiler/lecture7.html

Martin J. Dürst

© 2005-19 Martin J. Dürst 青山学院大学

Today's Schedule

• Leftovers and summary from last lecture
• flex homework
• Notations for grammars
• Constructing a grammar
• Parse tree and abstract syntax tree
• Difficulty of parsing
• Top-down and bottom-up parsing

Collection of Homework from Last Lecture

(bring to next lecture, will be collected)

For a programming language that you know (e.g. C, Java, Ruby,...), search for a grammar on the Web, print it out, and carefully study it.

Leftovers from Last Lecture

Summary of Last Lecture

• Regular expressions and related tools are limited in their descriptive power
  (example: properly nested parentheses,...)
• Regular expressions,... cannot be used for parsing, but are very useful for lexical analysis
• Separating lexical analysis and parsing leads to a fast compiler with a clean structure
• Grammars for (programming) languages can be written as context-free grammars
• Context-free grammars can be recognized with push-down automata
• Nondeterministic push-down automata are more powerful than deterministic ones, but their implementation is more complex and slower

Homework Due Yesterday

Example solution: xml.l

A Difficult Regular Expression: C Comment

For better readability, we will match /x x/, and use spaces

Reference: Mastering Regular Expressions, Jeffrey E.F. Friedl, pp. 168,...

Homework from Last Lecture

都合により削除

Compound Statement for C

compound_statement
        : '{' '}'
        | '{' statement_list '}'
        | '{' declaration_list '}'
        | '{' declaration_list statement_list '}'
        ;

declaration_list
        : declaration
        | declaration_list declaration
        ;

statement_list
        : statement
        | statement_list statement
        ;

Block for Java

Be careful to distinguish e.g. literal {} (appearing as such in Java) and grammar-level {} (indicating 0 or more repetitions)

Block:
     { BlockStatements }

BlockStatements:
     { BlockStatement }

BlockStatement:
     LocalVariableDeclarationStatement
     ClassOrInterfaceDeclaration
     [ Identifier : ] Statement

Different Ways to Express a Grammar

Simple Grammar

• Example: C grammar
• Metasymbols: only →/: and |
• Rewriting rules have the same form as in language theory

BNF

• Example: Java grammar
• Metasymbols: →/:, | (or line break), {}, [], (),...
  (the Java grammar uses fonts to distinguish metasymbols ({}) and terminal symbols ({}))
• The right side of a rewriting rule is similar to a regular expression

Formal Rewriting Rules and BNF

There are many different ways to write a grammar:

1. Simplest: Only a list of rewriting rules
   
2. Connect all the right hand sides that have the same left hand side with |
   ⇒ No fundamental change from 1 (syntactic sugar)
3. Add equivalent of ? in regular expressions (present/absent, often written with [...])
   ⇒ Can be rewritten as two different rules
4. Add equivalent of * in regular expressions (0 or more repetitions, often written {...})
   ⇒ Can be rewritten with [] and list

Writing grammars as above is often called BNF (Backus-Naur Form), EBNF (Extended...), or ABNF (Augmented...),...

Rewriting Connected Right Hand Sides

A → B C | D E

⇒

A → B C
A → D E

Rewriting Presence/Absence

A → B [ C ] D

⇒

A → B C D
A → B D

Rewriting BNF Repetition

A → B { C } D

⇒

A → B [ CList ] D
CList → C | CList C

How to Create a Grammar

1. Write down a simple example word of the language
2. Convert the example to tokens types (result of lexical analysis)
3. Give names to the various phenomena in the example (e.g.: ...expression, ...statement, etc.)
4. Create draft rewriting rules
5. Repeat 1.-4. with more difficult examples, check, and fix

Example of Grammar Creation

• Goal: A grammar for formulæ such as 5 + 3

Goal of Parsing

• Acceptance/rejection of input
• Processing of input:
  • Direct computation of result (for uses such as "pocket calculator")
  • Creation of abstract syntax tree
• Easy to understand error messages

Result of Parsing: Parse Tree and Abstract Syntax Tree

Parse tree (concrete syntax tree):

• Leaves are non-terminals (during analysis) and terminals
• Internal nodes are non-terminals
• Purpose: Research/analysis of grammars and parsing methods

Abstract syntax tree:

• Leaves are subset of terminals (identifiers, literals,...)
• Nodes correspond to some non-terminals, but are often labeled with terminals (operators, keywords,...)
• Some terminals are ignored (parentheses,...)

Examples of Parse Tree and Abstract Syntax Tree

Parser Implementation: Top-Down or Bottom-Up

Top-down parsing:

Build the parse tree from the top (root, start symbol)

Bottom-up parsing:

Build the parse tree from the bottom (terminal symbols)

During parsing, there may be several (small) parse trees

Difficulty of Parsing

• Deterministic and non-deterministic push-down automata differ in recognition power
• A general algorithm to recognize context-free languages is O(n³)
• A O(n) algorithm is highly desirable, but this means that the grammar has to be somewhat limited
• In research on parsing, many different methods have been studied:
  • Methods easy to understand for humans
  • Methods with few limitations on grammars
  • Grammars for which a parser can be implemented by hand
  • Grammars for which a parser can be implemented automatically

Very General Parsing Method

(Cocke–Younger–Kasami (CYK) algorithm)

• Convert grammar to Chomsky normal form
  (only rules of the forms A → BC, D → d, and S → ε are allowed)
• Consider subsequences of the input of length 1, 2, ...
  
  • (for subsequences of length ≥3) consider many different ways of splitting
    (Dymanic Programming, same as matrix multiplication problem in Data Structures and Algorithms)
  • Check and store all applicable rewriting rules
• Computational complexity is O(n³ · g) (where g is the size of the grammar)

Homework

Deadline: June 6, 2019 (Thursday), 19:00

Where to submit: Box in front of room O-529 (building O, 5th floor)

Format: A4 single page (using both sides is okay; NO cover page, staple in top left corner if more than one page is necessary), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

In the problems below, n, +, -, *, and / are terminal symbols. Any other letters are non-terminal symbols. n denotes an arbitrary number, and the other symbols denote the four basic arithmetic operations.

1. For the three grammars below, construct all the possible parse trees for words of length 5. Find the grammar that allows all and only those parse trees that produce correct results.
   1. E → n | E - E
   2. E → n | n - E
   3. E → n | E - n
2. Same as in problem 1 for the four grammars below:
   1. E → n | E + E | E * E
   2. E → n | E + n | E * n
   3. E → T | E + T; T → n | T * n
   4. E → T | E * T; T → n | T + n
3. (bonus problem) Based on what you learned from problems 1 and 2, create a grammar that allows to correctly calculate expressions with the four arithmetic operations (without parentheses). Check this grammar with expressions of length 5.
4. Bring your notebook computer to the next lecture

Glossary

parse tree (concrete syntax tree)

解析木

abstract syntax tree

構文木 (抽象構文木)

syntactic sugar

糖衣構文

top-down parsing

下向き構文解析

bottom-up parsing

上向き構文解析

pocket calculator

電卓

Chomsky normal form

Chomsky 標準形

four (basic) arithmethic operations

四則演算