State Illinois University School of Information Technology ITK 328 Introduction to The Theory of Computation Spring, 2005
February 21, 2005
1
Announcement and Agenda
Questions From Last Lecture? Second Test on Wednesday 02/23/05 Homework #3 Solution Context Free Language Summary After Test #1
2
Chapter 2 Context-Free Languages
What is context-free grammar?
Context free grammars were first used in the study of human languages. One way of understanding the relationship of terms such as noun, verb, and preposition and their respective phrases leads to a natural recursion because noun phrases may appear inside verb phrases and vice versa. Most compilers and interpreters contain a component called a parser uses context-free grammar to extract the meaning of a program prior to generating the compiled code or performing the interpreted execution.
Including XML (needs parser).
3
Context-Free Grammars
An Example of a Context-Free Grammar A ! 0A1 A!B B!#
1. 2. 3. 4. 5. A collection of Substitution rules (productions). Symbols in grammar are called variables. One variable is designed the start variable. The sequence of substitution is called a derivation. Strings consist of variables and can not make any derivation are called terminals.
In the above grammar: 0,1,and # are terminals, A is the start variable, A derivation of string 000#111 in the above grammar is: A 0A1 00A11 000A111 000B111 000#111
4
Context-Free Grammars
The Parse Tree of G1
A A A L(G1) is {0n#1n | n 0}
B 0 0 0 # 1 1 1
5
Formal Definition of a Context-Free Grammars
A context-free grammar is a 4-tuple (V, , R, S), where
V is a finite set called the variables, is a finite set, disjoint from V, called the terminals, R is a finite set of rules, with each rule being a variable and a string of variables and terminals, and S V is the start variable.
If u, v, and w are strings of variables and terminals, and A ! w is a rule of the grammar, we say that uAv yields uwv, written uAv uwv. * Write u v if u = v or if a sequence u1, . u2, . . ,uk exists for k 0 and u u1 u2 . . . uk v. * The language of the grammar is {w * | S w }.
6
Page 93 and 94
<SENTENCE> ! <NOUN-PHASE> <VERB-PHASE> <NOUN-PHASE> ! <CMPLX-NOUN> | <CMPLX-NOUN> <PREP-PHASE> <VERB-PHASE> !<CMPLX-VERB> | <CMPLX-VERB> <PREP-PHASE> <PREP-PHASE> ! <PREP> <CMPLX-NOUN> <CMPLEX-NOUN> ! <ARTICLE> <NOUN> <CMPLX-VERB> ! <VERB> | <VERB> <NOUN-PHASE> <ARTICLE> ! a | the <NOUN> ! boy| girl | flower <VERB> ! touches | likes | sees <PREP> ! with
a boy sees
<SENTENCE> " <NOUN-PHASE> <VERB-PHASE> " <CMPLX-NOUN> <VERB-PHASE> " <ARTICLE> <NOUN> <VERB-PHASE> " a <NOUN> <VERB-PHASE> " a boy <VERB-PHASE> " a boy <CMPLX-VERB> " a boy <VERB> ! a boy sees
7
Examples of Context-Free Grammars
Grammar G3 = ({S}, {a.b}, R, S). The set of rules, R, is
S ! aSb | SS |
This grammar generates strings such as abab, aaabbb, and aababb. If we substitute `a' by `(' and `b' by `)', then we can view L(G3) is the language of all strings of properly nested parentheses.
Grammar G4 = (V, , R, EXPR) and is {a, +, , (, )}, The rules are
EXPR ! EXPR + TERM | TERM TERM ! TERM FACTOR | FACTOR FACTOR ! (EXPR) | a The two strings a + a a and (a + a) a can be generated with grammar G4 The parse trees are:
8
EXPR EXPR TERM TERM EXPR TERM FACTOR FACTOR TERM TERM FACTOR FACTOR FACTOR EXPR TERM TERM FACTOR FACTOR a + a a ( a + a ) a EXPR
9
What Have We Learned after Test 1
Finite Automata DFA and NFA
Definition and Design of FA Equivalence of DFA and NFA (Convert NFA to DFA) Closure Properties Union, Concatenation, and Star Epsilon () move.
Regular Expression
Definition and few examples Equivalence with FA
Non-regular Languages
Property of regular language Pumping Lemma (only to proof a language is not regular)
10