Definitions, operations, and properties for languages
Automata, grammars, and derivation
Example Answers for Homework
Course Contens
Theory
Compilers
Other applications
Front end
language theory, automata
lexical analysis, parsing
regular expressions, text/data formats
Back end
optimization, code generation
Importance of Formal Language Theory
Model for data formats and programming languages
Model for computation and recognition
Terms used for Natural Languages and Formal Languages
Basic Terms
Terms for formal languages:
A word is composed of symbols following some rules
A word is a string/sequence of symbols
Example: Words such as a, abc, aaabbb,
and abcba can be created using symbols a,
b, and c
The empty word (ε) is also a word
Definition of Word
A word or a language are defined using a finite set of symbols (or
letters) Σ
Σ is called alphabet (example: Σ = {a,
b, c})
A word over Σ is a sequence of 0 or more symbols from
Σ
The number of symbols in a word is called the length of a word
The length of a word w is written |w|
Example: |abcaba| = 6; |ε| = 0
Symbols are also words of length 1 (example: b)
Concatenation Operation for Words
A new word can be created by lining up two words after each other
This is called concatenation operation for (on) words
The concatenation operation is represented without an explicit symbol
(similar to multiplication in high school)
Example: The concatenation of words w and v is
written wv
Application example: With w = abc and v
= cba, wv = abccba
The concatenation of a word (or symbol) with itself is written using an
exponent: w2 = ww =
abcabc, a5 = aaaaa,...
Properties of Concatenation
Associativity: For any words w, v, and
u: (wv)u =
w(vu)
The length of a concatenation is the sum of the lengths of its operands:
|wv| = |w| + |v|
Definition of Language
A language over Σ is a set of words over Σ
Examples for lanuages over Σ
={a,b,c}:
Empty set: {}
Set containing only the empty word: {ε}
Σ (set of words of length 1 over Σ):
{a,b,c}
Set of words of length 3 (over Σ) (size of set: 27)
Set of all words over Σ
Set of (all) words (over Σ) starting with a
Set of words representing the weather at each of the prefectural
governments for each of the days of next week, if a stands for
sunny, b stands for cloudy, and c stands for
rainy/snowy (size of each word: 7; number of
words: 47)
More Examples of Languages
Σ = {a,..., z}, set of keywords of the programming language
C
Σ = {0,..., 9, a,..., f, A,..., F, x,...}, set of integer
literals of C
Σ = {characters from ASCII or Unicode}, (set of)
grammatically correct C programs
Σ = {a,..., z, (, ), ¬, ∧, ∨}, set of all well-formed
formulæ of predicate logic
Σ = {Latin letters,...}, set of all English words
Σ = {Latin letters,...}, set of all French words
Σ = {Kanji, Kana,...}, set of all Japanese words
Σ = {Kanji, Kana,...}, set of all grammatically correct
Japanese sentences
Operations on Languages
Operations on languages are combinations of operations on sets and
operations on words.
Set union of languages
Set intersection of languages
Set difference of langugages
Concatenation operation for languages: For languages A and
B, their concatenation AB is the set {
wv | w∈A,
v∈B }
As for words, we write L2 for
LL,...
Kleene closure: Concatenating the same language 0 or more times
written L*; L* =
⋃∞i=0Li
Example: L = {a, b} =>
L* = {ε, a, b,
aa, ab, ba, bb, aaa,
...}
Main Problems in Formal Language Theory
How can languages be defined in a way that is simple and easy to
understand?
How can words be produced from definitions of languages?
How to decide whether some sequence of symbols is a word in some
language?
How can such decisions be implemented easily and executed quickly?
How can we associate syntax with sematics?
Languages and Automata and Grammars
An automaton is a model for a machine that
accepts/recognizes/distinguishes words in a given language
A grammar is a set of rules to create a language
There are many different kinds of automata and grammars
These different kinds have different ranges of languages that can be
accepted/generated
Language theory distinguishes mainly four types of languages
There is an ordered subset relationship between these four types
For each type of language, there is a corresponding type of automaton and
a corresponding type of grammar
Table of Formal Language Types
(Chomsky hierarchy)
文法
grammar
Type
Lanugage type
automaton
句構造文法
phrase structure grammar (psg)
0
phrase structure language
Turing machine
文脈依存文法
context-sensitive grammar (csg)
1
context-sensitive language
linear-bounded automaton
文脈自由文法
context-free grammar (cfg)
2
context-free language
push-down automaton
正規文法
regular grammar (rg)
3
regular language
finite state automaton
The Turing machine is a model for computation in general
Context-free languages are used for parsing
Regular languages are used for lexical analysis
Types of Automata
Automata types are distinguised by the restrictions on their "external
memory":
0. The external memory is a tape of unlimited length: Turing machine
1. The external memory is a tape of limited length: linear-bounded
automaton
2. The external memory is a stack where only the top can be accessed:
push-down automaton
3. There is no external memory: finite state automaton
Example of a Grammar for a Formal Language
S, A: nonterminal symbols (upper case)
a, o, y: terminal symbols (lower case)
rewriting rules:
S → aSo
S → Aw
A → ya
S: initial symbol (start symbol)
Example of derivation of a word from the grammar:
S → aSo →
aaSoo → aaAoo → aayaoo
S ⇒ aayaoo
Definition of Grammar
A finite set of nonterminal symbols N (usually upper case)
A finite set of terminal symbols Σ (usually lower case,
N ∩ Σ ＝ {})
A finite set of rewriting rules P (also called production
rules)
A start symbol S (S ∈ N, the symbol on
the left side of the first rewriting rule if not explicitly specified)
A grammar is defined as a quadruple (N, Σ,
P, S)
書換規則
(rewriting rule)
書換規則一つは α → β と書く
α は左辺 (left-hand side)、β は右辺
(right-hand side)
α とβ は
0以上の非終端記号と終端記号の列
左辺には非終端記号が最低一つ
例: aD → aDDb, EF → abc,
F → Fb, D → ε
反例: bc → Dc, ε → b
導出
(derivation)
文法から語を作るプロセス
初期記号から開始
一回の導出では一つの書き換え規則を一回適用:
現在ある (非)終端記号の中に
書換規則の左辺と同じ部分列を特定
この部分列を書換規則の右辺と置換
複数の導出が可能な場合、任意に選択
(選択により、別の語を作成)
結果が終端記号だけの列の時点で導出が終了
→結果が (文法が定義する) 言語の一つの語
適用可能な書換規則がない場合、導出が失敗
文法と導出の例
文法: S → aba (1), S → aDTa (2),
T → CDTa (3), T → CDa
(4), DC → QC (5), QC → QD (6),
QD → CD (7), aC → aa (8), Da → ba (9),
Db → bb (10)
(数値は書き換え規則の番号、下線は書換規則の適応範囲、普通
(宿題を含め) 省略)
導出の一例: S →2aDTa→4aDCDaa→5aQCDaa→9aQCbaa→6aQDbaa→7aCDbaa→8aaDbaa→10aabbaa
