From Lexical Analysis to Parsing

(字句解析から構文解析へ)

6rd lecture, May 25, 2018

Language Theory and Compilers

http://www.sw.it.aoyama.ac.jp/2018/Compiler/lecture6.html

Martin J. Dürst

© 2005-18 Martin J. Dürst 青山学院大学

Today's Schedule

• Minitest
• flex homework
• Limitations of regular languages
• Differences between lexical analysis and parsing
• Context-free grammars
• Push-down automata

Minitest

flex Homework: Lexical Analysis for C

Deadline: May 31, 2017 (Thursday), 19:00, box in front of room O-529 (building O, 5th floor)

Hints for flex Homework (Leftovers)

Additional Hints for Homework

• Start simple and proceed in small steps
• Command line for all steps:
  flex xml.l && gcc lex.yy.c && ./a <flex_in.txt | diff - flex_out.txt
• Example input, output

Compilation Stages

1. Lexical analysis
2. Parsing (syntax analysis)
3. Semantic analysis
4. Optimization (or 5)
5. Code generation (or 4)

Table of Formal Language Types

Limitations of Regular Languages/Grammars and FSAs

Can the following languages be represented with a regular expression?

• The language where all words consist of n 0es followed by n 1s
• The language where all words consist of symbols a, b, and c so that words are palindromes (e.g. acbca)
• The language where all words consist of the symbols ( and ), properly nested as in a formula

All these languages cannot be accepted by FSAs because they have limited (finite) memory.

Comparing Lexical Analysis and Parsing

Regular Grammars and Context-Free Grammars

Regular grammar:

• Right linear grammar or left linear grammar

Context free grammar:

• The left hand side of all rewriting rules is a single non-terminal symbol
• The right hand side is not restricted (any number of terminal and non-terminals possible)
• Examples: A → cBd, B → ccB, S → cBcAd
• Meaning of free: Not dependent, not influenced by
• For programming languages, the correctness of syntax can be decided locally, and does not depend on context
  (however, there may be semantic constraints, e.g. whether a variable is defined or not, and what type it has)

Example of Context-Free Grammar

S → aSa | bSb | c

Examples of generated words: c, aca, bcb, abaabcbaaba

Language being generated: A single c in the middle, surrounded by 0 or more a and b so that the resulting word is a palindrome

This language cannot be accepted by an automaton with finite memory (e.g. FSA)

We need to extend FSAs to create more powerful automata

We will add a push-down stack

Push-Down Stack and Push-Down Symbols

• A push-down stack stores push-down symbols
• Push-down symbols are different from input symbols
• Only the symbol on the top of the stack can be seen/checked
• Symbols can be added to or removed from the stack one at a time
• There is a special symbol at the bottom of the stack, the bottom marker
• The bottom marker cannot be removed

Example of Push-Down Automaton

• Σ = {a, b, c}
• The stack symbols are A, B, and Z (bottom marker)
• The stack is written with the top on the left
• A/ε means that if the top of the stack is A, then remove A
• A/A means that if the top of the stack is A, then leave it as is
• A/BA means that if the top of the stack is A, then replace that with BA (i.e. put B on top)

How a Push-Down Automaton Works

• In many ways the same as for FSAs (states, start state, final states, transitions,...)
• At the start, the bottom marker is visible (i.e. the stack is empty)
• The transition is selected based on:
  • The next input symbol (examples: a, b), and
  • The stack symbol visible on the top of the stack (examples: B, Z)
• A transition is one of:
  
  1. Remove one symbol from the top of the stack (example: b, B/ε)
  2. Keep the stack as is (example: c, A/A)
  3. Put a new symbol on top of the stack (example: a, B/AB)
• The input is accepted if (variants can be shown to be equivalent):
  1. The bottom marker is visible
  2. The automaton is in a final state at the end of the input
  3. Both conditions (1. and 2.) are met

Deterministic and Nondeterministic Push-Down Automata

• The grammar S → aSa | bSb | c can be accepted by a deterministic push-down automaton
• The grammar S → aSa | bSb | ε cannot be accepted by a deterministic push-down automaton
  Why: There is no marker in the middle of the word
  
• The power of deterministic and nondeterministic push-down automata is different
• For efficient parsing, it is important to use a grammar with a deterministic pushdown automaton
• Fortunately, this is easy for programming languages

(there are other aspects of context-free languages/grammars that affect parsing speed)

Homework

(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.

Glossary

nested

入れ子 (になっている)

palindrome

回文、左右対称な語

push-down symbol

プッシュダウン記号

bottom marker

ボトムマーカ

deterministic push-down automaton

決定性プッシュダウンオートマトン

nonteterministic push-down automaton

非決定性プッシュダウンオートマトン