Context-Free Grammars. 2IT70 Finite Automata and Process Theory
|
|
- Morgan Jones
- 5 years ago
- Views:
Transcription
1 Context-Free Grammars 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven May 18, 2016
2 Generating strings language L 1 = {a n b n n > 0} ab L 1 if w L 1 then awb L 1 production rules ab and ab 2 IT70 (2016) Context-Free Grammars 2/ 41
3 Generating strings language L 1 = {a n b n n > 0} language L 2 = (01) ab L 1 if w L 1 then awb L 1 ε L 2 if w L 2 then 01w L 2 production rules ab and ab production rules ε and 01 2 IT70 (2016) Context-Free Grammars 2/ 41
4 Variables, terminals, production rules, start symbol palindromes over {a, b} ε a b aa bb binary integer expressions E I E N I a I I0 E E +E E E E E (E) I I1 N 1 N N0 N N1 2 IT70 (2016) Context-Free Grammars 3/ 41
5 Variables, terminals, production rules, start symbol palindromes over {a, b} ε a b aa bb alternative notation ε a b aa bb binary integer expressions E I E N I a I I0 E E +E E E E E (E) I I1 N 1 N N0 N N1 alternative notation E I N E +E E E (E) I a I0 I1 N 1 N0 N1 2 IT70 (2016) Context-Free Grammars 3/ 41
6 Clicker questions L81 Consider again the grammar given by E I N E +E E E (E) I a I0 I1 N 1 N0 N1 How many of the strings aa1, a01, 011, 11a, a01+a01, (a11 a10), +101, (110) cannot be generated by the grammar, you expect? A. Two strings B. Three strings C. Four strings D. ix strings E. Can t tell 2 IT70 (2016) Context-Free Grammars 4/ 41
7 Language of a CFG context-free grammar G = (V, T, R, ) V variables and T terminals R V (V T) production rules A α V start symbol productions G (V T) (V T) γ G γ if γ = β 1 Aβ 2, A α rule of G, γ = β 1 αβ 2 production sequences γ 0 G γ 1 G G γ n 2 IT70 (2016) Context-Free Grammars 5/ 41
8 Language of a CFG context-free grammar G = (V, T, R, ) V variables and T terminals R V (V T) production rules A α V start symbol productions G (V T) (V T) γ G γ if γ = β 1 Aβ 2, A α rule of G, γ = β 1 αβ 2 production sequences γ 0 G γ 1 G G γ n language of a variable L G (A) = {w T A G w } language of the grammar L(G) = L G () = {w T G w } 2 IT70 (2016) Context-Free Grammars 5/ 41
9 More examples expression ::= term expression + term term ::= factor term factor factor ::= identifier ( expression ) identifier ::= a b c... 2 IT70 (2016) Context-Free Grammars 6/ 41
10 More examples expression ::= term expression + term term ::= factor term factor factor ::= identifier ( expression ) identifier ::= a b c... char ::= a... z A... Z... text ::= ε char text doc ::= ε element doc element ::= text <EM> doc </EM> <P> doc <OL> list </OL> listitem ::= <LI> doc list ::= ε listitem doc 2 IT70 (2016) Context-Free Grammars 6/ 41
11 Combining and splitting productions lemma CFG G = (V, T, R, ) if X 1 n 1 G γ 1,...,X k n k G γ k then X 1 X k n G γ 1 γ k where n = n 1 + +n k if X 1 X k n G γ then X 1 n 1 G γ 1,...,X k n k G γ k where n = n 1 + +n k and γ = γ 1...γ k X 1,...,X k (V T), γ 1,...,γ k (V T) 2 IT70 (2016) Context-Free Grammars 7/ 41
12 The parentheses language L () CFG ε () several production sequences for string ()(()) G G () G (()) G (()) G ()(()) G ()(()) G G () G () G ()() G ()(()) G ()(()) G G () G ()() G ()() G ()(()) G ()(()) leftmost, rightmost, mixed production sequence 2 IT70 (2016) Context-Free Grammars 8/ 41
13 Clicker question L82 Given the CFG () (). How many production sequences are there for the string (())((()))? A. (())((())) has 5 possible production sequences B. (())((())) has 6 possible production sequences C. (())((())) has 10 possible production sequences D. (())((())) has 12 possible production sequences E. Can t tell 2 IT70 (2016) Context-Free Grammars 9/ 41
14 Proving a grammar correct CFG G with production rules ab and ab for L = {a n b n n 1} it holds that L(G) = L proof induction on n: if n G w then w L, thus L(G) L induction on n: if w = a n b n then w L(G), thus L L(G) 2 IT70 (2016) Context-Free Grammars 10/ 41
15 Avoiding the inductive proofs lemma CFGs G 1 = (V 1, T 1, R 1, 1 ) and G 2 = (V 2, T 2, R 2, 2 ) moreover V 1 and V 2 disjoint define CFG G = ({} V 1 V 2, T 1 T 2, R, ) if R = { 1 2 } R 1 R 2 then L(G) = L(G 1 ) L(G 2 ) if R = { 1 2 } R 1 R 2 then L(G) = L(G 1 ) L(G 2 ) if R = { ε 1 } R 1 then L(G) = L(G 1 ) 2 IT70 (2016) Context-Free Grammars 11/ 41
16 Avoiding the inductive proofs (cont.) CFG G with production rules ab B ε bb 2 ba A ε aa then L(G) = {ab n, ba m n,m 0} proof use the lemma L G (A) = {a m m 0} and L G (B) = {b n n 0} L G ( 1 ) = {a} {b n n 0} and L G ( 2 ) = {b} {a m m 0} L(G) = {ab n n 0} {ba m m 0} 2 IT70 (2016) Context-Free Grammars 12/ 41
17 Context-free languages language L is context-free if L = L(G) for CFG G {a n b n n 0} and {ww R w {0,1} } are context-free 2 IT70 (2016) Context-Free Grammars 13/ 41
18 Context-free languages language L is context-free if L = L(G) for CFG G {a n b n n 0} and {ww R w {0,1} } are context-free theorem if L is regular then L is context-free proof for DFA D = (Q, Σ, δ, q 0, F ) put G = (Q, Σ, R, q 0 ) where R = {q aq δ(q,a) = q } {q ε q F } then L = L(G) 2 IT70 (2016) Context-Free Grammars 13/ 41
19 Chomsky Normal Form 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven May 18, 2016
20 Useless symbols CFG G = (V,T,R,) symbol X V T generating if X G w T symbol X V T reachable if α,β: G αxβ symbol X V T is useful if both generating and reachable 2IT70 (2016) Chomsky Normal Form 15/41
21 Clicker question L83 Consider the grammar AB c A a C c 2IT70 (2016) Chomsky Normal Form 16/41
22 Clicker question L83 Consider the grammar AB c A a C c Which of the following statements about variables holds true? A. 2 variables generating, 2 reachable, 1 useful B. 2 variables generating, 2 reachable, 2 useful C. 3 variables generating, 3 reachable, 2 useful D. 3 variables generating, 3 reachable, 3 useful E. Can t tell 2IT70 (2016) Chomsky Normal Form 16/41
23 Finding of generating variables CFG G = (V,T,R,) with L(G) symbol X V T is generating if X G w T and Gen(G) = {X X generating} lemma put Gen 0 = T Gen i+1 = Gen i {A A G α, α Gen i } Gen = i=0 Gen i then Gen(G) = Gen theorem CFG G = (V,T,R,) with V = V Gen(G) and R = {A α R A Gen, α Gen } then L(G) = L(G ) and all symbols of G generating 2IT70 (2016) Chomsky Normal Form 17/41
24 Finding of reachable variables CFG G = (V,T,R,) with L(G) symbol X V T is reachable if G Reach(G) = {X X reachable} αxβ and lemma put Reach 0 = {} Reach i+1 = Reach i {X A G γ, A Reach i, γαxβ } Reach = i=0 Reach i then Reach(G) = Reach theorem CFG G = (V,T,R,) with V = V Reach(G) and R = {A α R A Reach } then L(G) = L(G ) and all variables of G reachable 2IT70 (2016) Chomsky Normal Form 18/41
25 Parse Trees 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven May 18, 2016
26 Identifying production sequences parentheses grammar ε () several production sequences for string ()(()) G G () G () G ()() G ()(()) G ()(()) G G () G ()() G ()() G ()(()) G ()(()) 2IT70 (2016) Parse Trees 20/41
27 Identifying production sequences parentheses grammar ε () several production sequences for string ()(()) G G () G () G ()() G ()(()) G ()(()) G G () G ()() G ()() G ()(()) G ()(()) swapping independent productions 2IT70 (2016) Parse Trees 20/41
28 Identifying production sequences (cont.) 2IT70 (2016) Parse Trees 21/41
29 Identifying production sequences (cont.) 2IT70 (2016) Parse Trees 21/41
30 Identifying production sequences (cont.) ( ) 2IT70 (2016) Parse Trees 21/41
31 Identifying production sequences (cont.) ( ) ε 2IT70 (2016) Parse Trees 21/41
32 Identifying production sequences (cont.) ( ) ( ) ε 2IT70 (2016) Parse Trees 21/41
33 Identifying production sequences (cont.) ( ) ( ) ε ( ) 2IT70 (2016) Parse Trees 21/41
34 Identifying production sequences (cont.) ( ) ( ) ε ( ) ε 2IT70 (2016) Parse Trees 21/41
35 Identifying production sequences (once more) 2IT70 (2016) Parse Trees 22/41
36 Identifying production sequences (once more) 2IT70 (2016) Parse Trees 22/41
37 Identifying production sequences (once more) ( ) 2IT70 (2016) Parse Trees 22/41
38 Identifying production sequences (once more) ( ) ( ) 2IT70 (2016) Parse Trees 22/41
39 Identifying production sequences (once more) ( ) ( ) ε 2IT70 (2016) Parse Trees 22/41
40 Identifying production sequences (once more) ( ) ( ) ε ( ) 2IT70 (2016) Parse Trees 22/41
41 Identifying production sequences (once more) ( ) ( ) ε ( ) ε 2IT70 (2016) Parse Trees 22/41
42 Yield of a parse tree CFG G = (V, T, R, ) set PT G of all parse trees of G [X] single node tree, X V T [A ε] two node tree, root A, leaf ε for rule A ε R [A PT 1,PT 2,...,PT k ] rule A X 1 X k R parse trees PT i with root X i 2IT70 (2016) Parse Trees 23/41
43 Yield of a parse tree CFG G = (V, T, R, ) set PT G of all parse trees of G [X] single node tree, X V T [A ε] two node tree, root A, leaf ε for rule A ε R [A PT 1,PT 2,...,PT k ] rule A X 1 X k R parse trees PT i with root X i yield function yield : PT G (V T) yield([x]) = X yield([a ε]) = ε yield([a PT 1,...,PT k ]) = yield(pt 1 )... yield(pt k ) parse tree PT is complete if yield(pt) T 2IT70 (2016) Parse Trees 23/41
44 A parse tree with yield ()(()) ( ) ( ) ε ( ) ε 2IT70 (2016) Parse Trees 24/41
45 A parse tree with yield ()(()) ( ) ( ) ε ( ) ε 2IT70 (2016) Parse Trees 24/41
46 Another parse tree CFG AB A ε aaa B ε Bb A B a a A B b ε ε parse tree with yield aab 2IT70 (2016) Parse Trees 25/41
47 Another parse tree CFG AB A ε aaa B ε Bb A B a a A B b ε ε parse tree with yield aab 2IT70 (2016) Parse Trees 25/41
48 Parsing CFG G with rules ε ab ba aabb L(G)? w {a,b} 2IT70 (2016) Parse Trees 26/41
49 Parsing CFG G with rules ε ab ba aabb L(G)? ε a b b a ε awb bwa w 1 w 2 2IT70 (2016) Parse Trees 26/41
50 Parsing CFG G with rules ε ab ba aabb L(G)? ε a b b a ε awb bwa w 1 w 2 2IT70 (2016) Parse Trees 26/41
51 Parsing CFG G with rules ε ab ba aabb L(G)? a b a b a b a b ε a b b a 2IT70 (2016) Parse Trees 27/41
52 Parsing CFG G with rules ε ab ba aabb L(G)? a b a b a b a b ε a b b a ab aawbb abwab aw 1 w 2 b 2IT70 (2016) Parse Trees 27/41
53 Parsing CFG G with rules ε ab ba aabb L(G)? a b a b a b a b ε a b a b a b a b ε a b b a 2IT70 (2016) Parse Trees 28/41
54 Parsing CFG G with rules ε ab ba aabb L(G)? a b a b a b a b ε a b a b a b a b ε a b b a aabb aaawbbb aabwabb aaw 1 w 2 bb 2IT70 (2016) Parse Trees 28/41
55 Parsing CFG G with rules ε ab ba aabb L(G)? a b a b a b a b ε a b a b a b a b ε a b b a aabb aaawbbb aabwabb aaw 1 w 2 bb Thus aabb L(G) 2IT70 (2016) Parse Trees 28/41
56 Clicker question L91 parsing takes at most 2 w 1 rounds if no summand ε summands have at least one terminal or at least two variables 2IT70 (2016) Parse Trees 29/41
57 Clicker question L91 parsing takes at most 2 w 1 rounds if no summand ε summands have at least one terminal or at least two variables With the parsing procedure and restrictions above, we have that A. Parsing is linear in the length of the string B. Parsing is quadratic in the length of the string C. Parsing is exponential in the length of the string D. Can t tell 2IT70 (2016) Parse Trees 29/41
58 generated strings of terminals vs. yields of parse trees theorem CFG G = (V, T, R, ) A G w implies w = yield(pt) for parse tree PT with root A proof by induction on n: A n G w implies PT PT G(A): w = yield(pt) for all A V and w T 2IT70 (2016) Parse Trees 30/41
59 generated strings of terminals vs. yields of parse trees theorem CFG G = (V, T, R, ) A G w implies w = yield(pt) for parse tree PT with root A proof by induction on n: A n G w implies PT PT G(A): w = yield(pt) for all A V and w T thus L(G)={w T G w } {yield(pt) PT complete parse tree of G, root } 2IT70 (2016) Parse Trees 30/41
60 Clicker question L92 uppose X l G β for a CFG G. 2IT70 (2016) Parse Trees 31/41
61 Clicker question L92 uppose X l G β for a CFG G. Then it holds that A. αxγ l G αβγ for all α T and γ T B. αxγ l G αβγ for all α T and γ (V T) C. αxγ l G αβγ for all α (V T) and γ T D. αxγ l G αβγ for all α (V T) and γ (V T) E. Can t tell 2IT70 (2016) Parse Trees 31/41
62 From parse tree to leftmost production sequence theorem CFG G for parse tree PT, root A and yield w: A l G w proof induction on the height of the parse tree PT thus {yield(pt) PT complete parse tree of G, root } {w T G w }=L(G) 2IT70 (2016) Parse Trees 32/41
63 Different parse trees (harmless) a b a ε b ε aabb ε a b a ε b ε aabb ambiguous grammar ε ab 2IT70 (2016) Parse Trees 33/41
64 Different parse trees (harmful) E I E + E E * E ( E ) I a b c E E E E E + E I E + E E E E a I I I I c b c a b a*b+c a*b+c ambigious grammar 2IT70 (2016) Parse Trees 34/41
65 Different parse trees (harmful, cont.) E I E + E E * E ( E ) I a b c a c b c a b 2*3+4? wrong 2*3+4? right ambigious grammar 2IT70 (2016) Parse Trees 35/41
66 Disambiguation E T E + T T F T * F F I ( E ) I a b c syntactic categories: expression, term, factor, identifier 2IT70 (2016) Parse Trees 36/41
67 Disambiguation E E + T E T E + T T F T * F F I ( E ) I a b c T F T F I F I c I b a a*b+c syntactic categories: expression, term, factor, identifier 2IT70 (2016) Parse Trees 36/41
Context-Free Grammars. 2IT70 Finite Automata and Process Theory
Context-Free Grammars 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven Quartile2, 2014-2015 Generating strings language L 1 èa n b n Ë n 0 í ab L 1 if w L 1 then awb L 1 2 IT70
More informationParsing. Context-Free Grammars (CFG) Laura Kallmeyer. Winter 2017/18. Heinrich-Heine-Universität Düsseldorf 1 / 26
Parsing Context-Free Grammars (CFG) Laura Kallmeyer Heinrich-Heine-Universität Düsseldorf Winter 2017/18 1 / 26 Table of contents 1 Context-Free Grammars 2 Simplifying CFGs Removing useless symbols Eliminating
More informationPlan for 2 nd half. Just when you thought it was safe. Just when you thought it was safe. Theory Hall of Fame. Chomsky Normal Form
Plan for 2 nd half Pumping Lemma for CFLs The Return of the Pumping Lemma Just when you thought it was safe Return of the Pumping Lemma Recall: With Regular Languages The Pumping Lemma showed that if a
More information60-354, Theory of Computation Fall Asish Mukhopadhyay School of Computer Science University of Windsor
60-354, Theory of Computation Fall 2013 Asish Mukhopadhyay School of Computer Science University of Windsor Pushdown Automata (PDA) PDA = ε-nfa + stack Acceptance ε-nfa enters a final state or Stack is
More informationContext-Free Grammars and Languages
Context-Free Grammars and Languages Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
More informationDefinition: A grammar G = (V, T, P,S) is a context free grammar (cfg) if all productions in P have the form A x where
Recitation 11 Notes Context Free Grammars Definition: A grammar G = (V, T, P,S) is a context free grammar (cfg) if all productions in P have the form A x A V, and x (V T)*. Examples Problem 1. Given the
More informationContext-Free Grammar
Context-Free Grammar CFGs are more powerful than regular expressions. They are more powerful in the sense that whatever can be expressed using regular expressions can be expressed using context-free grammars,
More informationLecture 11 Context-Free Languages
Lecture 11 Context-Free Languages COT 4420 Theory of Computation Chapter 5 Context-Free Languages n { a b : n n { ww } 0} R Regular Languages a *b* ( a + b) * Example 1 G = ({S}, {a, b}, S, P) Derivations:
More informationAutomata Theory CS F-08 Context-Free Grammars
Automata Theory CS411-2015F-08 Context-Free Grammars David Galles Department of Computer Science University of San Francisco 08-0: Context-Free Grammars Set of Terminals (Σ) Set of Non-Terminals Set of
More informationCS5371 Theory of Computation. Lecture 7: Automata Theory V (CFG, CFL, CNF)
CS5371 Theory of Computation Lecture 7: Automata Theory V (CFG, CFL, CNF) Announcement Homework 2 will be given soon (before Tue) Due date: Oct 31 (Tue), before class Midterm: Nov 3, (Fri), first hour
More information5/10/16. Grammar. Automata and Languages. Today s Topics. Grammars Definition A grammar G is defined as G = (V, T, P, S) where:
Grammar Automata and Languages Grammar Prof. Mohamed Hamada oftware Engineering Lab. The University of Aizu Japan Regular Grammar Context-free Grammar Context-sensitive Grammar Left-linear Grammar right-linear
More informationAC68 FINITE AUTOMATA & FORMULA LANGUAGES DEC 2013
Q.2 a. Prove by mathematical induction n 4 4n 2 is divisible by 3 for n 0. Basic step: For n = 0, n 3 n = 0 which is divisible by 3. Induction hypothesis: Let p(n) = n 3 n is divisible by 3. Induction
More informationCPS 220 Theory of Computation
CPS 22 Theory of Computation Review - Regular Languages RL - a simple class of languages that can be represented in two ways: 1 Machine description: Finite Automata are machines with a finite number of
More informationContext-Free Grammars and Languages. Reading: Chapter 5
Context-Free Grammars and Languages Reading: Chapter 5 1 Context-Free Languages The class of context-free languages generalizes the class of regular languages, i.e., every regular language is a context-free
More informationAutomata Theory Final Exam Solution 08:10-10:00 am Friday, June 13, 2008
Automata Theory Final Exam Solution 08:10-10:00 am Friday, June 13, 2008 Name: ID #: This is a Close Book examination. Only an A4 cheating sheet belonging to you is acceptable. You can write your answers
More informationEinführung in die Computerlinguistik
Einführung in die Computerlinguistik Context-Free Grammars (CFG) Laura Kallmeyer Heinrich-Heine-Universität Düsseldorf Summer 2016 1 / 22 CFG (1) Example: Grammar G telescope : Productions: S NP VP NP
More informationTAFL 1 (ECS-403) Unit- III. 3.1 Definition of CFG (Context Free Grammar) and problems. 3.2 Derivation. 3.3 Ambiguity in Grammar
TAFL 1 (ECS-403) Unit- III 3.1 Definition of CFG (Context Free Grammar) and problems 3.2 Derivation 3.3 Ambiguity in Grammar 3.3.1 Inherent Ambiguity 3.3.2 Ambiguous to Unambiguous CFG 3.4 Simplification
More informationPush-Down Automata and Context-Free Languages
Chapter 3 Push-Down Automata and Context-Free Languages In the previous chapter, we studied finite automata, modeling computers without memory. In the next chapter, we study a general model of computers
More informationContext Free Grammars
Automata and Formal Languages Context Free Grammars Sipser pages 101-111 Lecture 11 Tim Sheard 1 Formal Languages 1. Context free languages provide a convenient notation for recursive description of languages.
More informationProperties of Context-free Languages. Reading: Chapter 7
Properties of Context-free Languages Reading: Chapter 7 1 Topics 1) Simplifying CFGs, Normal forms 2) Pumping lemma for CFLs 3) Closure and decision properties of CFLs 2 How to simplify CFGs? 3 Three ways
More informationGrammars and Context Free Languages
Grammars and Context Free Languages H. Geuvers and A. Kissinger Institute for Computing and Information Sciences Version: fall 2015 H. Geuvers & A. Kissinger Version: fall 2015 Talen en Automaten 1 / 23
More informationContext-free Grammars and Languages
Context-free Grammars and Languages COMP 455 002, Spring 2019 Jim Anderson (modified by Nathan Otterness) 1 Context-free Grammars Context-free grammars provide another way to specify languages. Example:
More informationSt.MARTIN S ENGINEERING COLLEGE Dhulapally, Secunderabad
St.MARTIN S ENGINEERING COLLEGE Dhulapally, Secunderabad-500 014 Subject: FORMAL LANGUAGES AND AUTOMATA THEORY Class : CSE II PART A (SHORT ANSWER QUESTIONS) UNIT- I 1 Explain transition diagram, transition
More informationComputational Models - Lecture 4 1
Computational Models - Lecture 4 1 Handout Mode Iftach Haitner. Tel Aviv University. November 21, 2016 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice Herlihy, Brown University.
More information5 Context-Free Languages
CA320: COMPUTABILITY AND COMPLEXITY 1 5 Context-Free Languages 5.1 Context-Free Grammars Context-Free Grammars Context-free languages are specified with a context-free grammar (CFG). Formally, a CFG G
More informationFormal Languages, Grammars and Automata Lecture 5
Formal Languages, Grammars and Automata Lecture 5 Helle Hvid Hansen helle@cs.ru.nl http://www.cs.ru.nl/~helle/ Foundations Group Intelligent Systems Section Institute for Computing and Information Sciences
More informationLanguages. Languages. An Example Grammar. Grammars. Suppose we have an alphabet V. Then we can write:
Languages A language is a set (usually infinite) of strings, also known as sentences Each string consists of a sequence of symbols taken from some alphabet An alphabet, V, is a finite set of symbols, e.g.
More informationGrammars and Context Free Languages
Grammars and Context Free Languages H. Geuvers and J. Rot Institute for Computing and Information Sciences Version: fall 2016 H. Geuvers & J. Rot Version: fall 2016 Talen en Automaten 1 / 24 Outline Grammars
More informationCS 373: Theory of Computation. Fall 2010
CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1 1 Normal Forms for CFG Normal Forms for Grammars It is typically easier to work with a context free language if given a CFG in a normal
More information1. (a) Explain the procedure to convert Context Free Grammar to Push Down Automata.
Code No: R09220504 R09 Set No. 2 II B.Tech II Semester Examinations,December-January, 2011-2012 FORMAL LANGUAGES AND AUTOMATA THEORY Computer Science And Engineering Time: 3 hours Max Marks: 75 Answer
More informationHarvard CS 121 and CSCI E-207 Lecture 9: Regular Languages Wrap-Up, Context-Free Grammars
Harvard CS 121 and CSCI E-207 Lecture 9: Regular Languages Wrap-Up, Context-Free Grammars Salil Vadhan October 2, 2012 Reading: Sipser, 2.1 (except Chomsky Normal Form). Algorithmic questions about regular
More informationSimplification of CFG and Normal Forms. Wen-Guey Tzeng Computer Science Department National Chiao Tung University
Simplification of CFG and Normal Forms Wen-Guey Tzeng Computer Science Department National Chiao Tung University Normal Forms We want a cfg with either Chomsky or Greibach normal form Chomsky normal form
More informationSimplification of CFG and Normal Forms. Wen-Guey Tzeng Computer Science Department National Chiao Tung University
Simplification of CFG and Normal Forms Wen-Guey Tzeng Computer Science Department National Chiao Tung University Normal Forms We want a cfg with either Chomsky or Greibach normal form Chomsky normal form
More informationFinite Automata and Formal Languages TMV026/DIT321 LP Useful, Useless, Generating and Reachable Symbols
Finite Automata and Formal Languages TMV026/DIT321 LP4 2012 Lecture 13 Ana Bove May 7th 2012 Overview of today s lecture: Normal Forms for Context-Free Languages Pumping Lemma for Context-Free Languages
More informationSuppose h maps number and variables to ɛ, and opening parenthesis to 0 and closing parenthesis
1 Introduction Parenthesis Matching Problem Describe the set of arithmetic expressions with correctly matched parenthesis. Arithmetic expressions with correctly matched parenthesis cannot be described
More informationContext Free Languages (CFL) Language Recognizer A device that accepts valid strings. The FA are formalized types of language recognizer.
Context Free Languages (CFL) Language Recognizer A device that accepts valid strings. The FA are formalized types of language recognizer. Language Generator: Context free grammars are language generators,
More informationChapter 5: Context-Free Languages
Chapter 5: Context-Free Languages Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu Please read the corresponding chapter
More informationPushdown automata. Twan van Laarhoven. Institute for Computing and Information Sciences Intelligent Systems Radboud University Nijmegen
Pushdown automata Twan van Laarhoven Institute for Computing and Information Sciences Intelligent Systems Version: fall 2014 T. van Laarhoven Version: fall 2014 Formal Languages, Grammars and Automata
More informationThis lecture covers Chapter 5 of HMU: Context-free Grammars
This lecture covers Chapter 5 of HMU: Context-free rammars (Context-free) rammars (Leftmost and Rightmost) Derivations Parse Trees An quivalence between Derivations and Parse Trees Ambiguity in rammars
More informationConcordia University Department of Computer Science & Software Engineering
Concordia University Department of Computer Science & Software Engineering COMP 335/4 Theoretical Computer Science Winter 2015 Assignment 3 1. In each case, what language is generated by CFG s below. Justify
More informationHarvard CS 121 and CSCI E-207 Lecture 10: Ambiguity, Pushdown Automata
Harvard CS 121 and CSCI E-207 Lecture 10: Ambiguity, Pushdown Automata Salil Vadhan October 4, 2012 Reading: Sipser, 2.2. Another example of a CFG (with proof) L = {x {a, b} : x has the same # of a s and
More informationIntroduction and Motivation. Introduction and Motivation. Introduction to Computability. Introduction and Motivation. Theory. Lecture5: Context Free
ntroduction to Computability Theory Lecture5: Context Free Languages Prof. Amos sraeli 1 ntroduction and Motivation n our study of RL-s we Covered: 1. Motivation and definition of regular languages. 2.
More informationThe Pumping Lemma for Context Free Grammars
The Pumping Lemma for Context Free Grammars Chomsky Normal Form Chomsky Normal Form (CNF) is a simple and useful form of a CFG Every rule of a CNF grammar is in the form A BC A a Where a is any terminal
More information(b) If G=({S}, {a}, {S SS}, S) find the language generated by G. [8+8] 2. Convert the following grammar to Greibach Normal Form G = ({A1, A2, A3},
Code No: 07A50501 R07 Set No. 2 III B.Tech I Semester Examinations,MAY 2011 FORMAL LANGUAGES AND AUTOMATA THEORY Computer Science And Engineering Time: 3 hours Max Marks: 80 Answer any FIVE Questions All
More informationComputational Models - Lecture 4 1
Computational Models - Lecture 4 1 Handout Mode Iftach Haitner and Yishay Mansour. Tel Aviv University. April 3/8, 2013 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationComputational Models - Lecture 5 1
Computational Models - Lecture 5 1 Handout Mode Iftach Haitner. Tel Aviv University. November 28, 2016 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice Herlihy, Brown University.
More informationBefore We Start. The Pumping Lemma. Languages. Context Free Languages. Plan for today. Now our picture looks like. Any questions?
Before We Start The Pumping Lemma Any questions? The Lemma & Decision/ Languages Future Exam Question What is a language? What is a class of languages? Context Free Languages Context Free Languages(CFL)
More informationEinführung in die Computerlinguistik
Einführung in die Computerlinguistik Context-Free Grammars formal properties Laura Kallmeyer Heinrich-Heine-Universität Düsseldorf Summer 2018 1 / 20 Normal forms (1) Hopcroft and Ullman (1979) A normal
More informationFLAC Context-Free Grammars
FLAC Context-Free Grammars Klaus Sutner Carnegie Mellon Universality Fall 2017 1 Generating Languages Properties of CFLs Generation vs. Recognition 3 Turing machines can be used to check membership in
More informationCPSC 313 Introduction to Computability
CPSC 313 Introduction to Computability Grammars in Chomsky Normal Form (Cont d) (Sipser, pages 109-111 (3 rd ed) and 107-109 (2 nd ed)) Renate Scheidler Fall 2018 Chomsky Normal Form A context-free grammar
More informationSolutions to Problem Set 3
V22.0453-001 Theory of Computation October 8, 2003 TA: Nelly Fazio Solutions to Problem Set 3 Problem 1 We have seen that a grammar where all productions are of the form: A ab, A c (where A, B non-terminals,
More informationPUSHDOWN AUTOMATA (PDA)
PUSHDOWN AUTOMATA (PDA) FINITE STATE CONTROL INPUT STACK (Last in, first out) input pop push ε,ε $ 0,ε 0 1,0 ε ε,$ ε 1,0 ε PDA that recognizes L = { 0 n 1 n n 0 } Definition: A (non-deterministic) PDA
More informationComputational Models - Lecture 4
Computational Models - Lecture 4 Regular languages: The Myhill-Nerode Theorem Context-free Grammars Chomsky Normal Form Pumping Lemma for context free languages Non context-free languages: Examples Push
More informationContext-Free Grammars and Languages. We have seen that many languages cannot be regular. Thus we need to consider larger classes of langs.
Context-Free Grammars and Languages We have seen that many languages cannot be regular. Thus we need to consider larger classes of langs. Contex-Free Languages (CFL s) played a central role natural languages
More informationCS5371 Theory of Computation. Lecture 9: Automata Theory VII (Pumping Lemma, Non-CFL)
CS5371 Theory of Computation Lecture 9: Automata Theory VII (Pumping Lemma, Non-CFL) Objectives Introduce Pumping Lemma for CFL Apply Pumping Lemma to show that some languages are non-cfl Pumping Lemma
More informationPushdown Automata. Reading: Chapter 6
Pushdown Automata Reading: Chapter 6 1 Pushdown Automata (PDA) Informally: A PDA is an NFA-ε with a infinite stack. Transitions are modified to accommodate stack operations. Questions: What is a stack?
More informationIntroduction to Theory of Computing
CSCI 2670, Fall 2012 Introduction to Theory of Computing Department of Computer Science University of Georgia Athens, GA 30602 Instructor: Liming Cai www.cs.uga.edu/ cai 0 Lecture Note 3 Context-Free Languages
More informationLecture 12 Simplification of Context-Free Grammars and Normal Forms
Lecture 12 Simplification of Context-Free Grammars and Normal Forms COT 4420 Theory of Computation Chapter 6 Normal Forms for CFGs 1. Chomsky Normal Form CNF Productions of form A BC A, B, C V A a a T
More informationSection 1 (closed-book) Total points 30
CS 454 Theory of Computation Fall 2011 Section 1 (closed-book) Total points 30 1. Which of the following are true? (a) a PDA can always be converted to an equivalent PDA that at each step pops or pushes
More informationProperties of context-free Languages
Properties of context-free Languages We simplify CFL s. Greibach Normal Form Chomsky Normal Form We prove pumping lemma for CFL s. We study closure properties and decision properties. Some of them remain,
More information6.1 The Pumping Lemma for CFLs 6.2 Intersections and Complements of CFLs
CSC4510/6510 AUTOMATA 6.1 The Pumping Lemma for CFLs 6.2 Intersections and Complements of CFLs The Pumping Lemma for Context Free Languages One way to prove AnBn is not regular is to use the pumping lemma
More informationAC68 FINITE AUTOMATA & FORMULA LANGUAGES JUNE 2014
Q.2 a. Show by using Mathematical Induction that n i= 1 i 2 n = ( n + 1) ( 2 n + 1) 6 b. Define language. Let = {0; 1} denote an alphabet. Enumerate five elements of the following languages: (i) Even binary
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 14 Ana Bove May 14th 2018 Recap: Context-free Grammars Simplification of grammars: Elimination of ǫ-productions; Elimination of
More informationProblem Session 5 (CFGs) Talk about the building blocks of CFGs: S 0S 1S ε - everything. S 0S0 1S1 A - waw R. S 0S0 0S1 1S0 1S1 A - xay, where x = y.
CSE2001, Fall 2006 1 Problem Session 5 (CFGs) Talk about the building blocks of CFGs: S 0S 1S ε - everything. S 0S0 1S1 A - waw R. S 0S0 0S1 1S0 1S1 A - xay, where x = y. S 00S1 A - xay, where x = 2 y.
More informationProperties of Context-Free Languages
Properties of Context-Free Languages Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
More informationCFG Simplification. (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions
CFG Simplification (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions 1 Eliminating useless symbols 1. A symbol X is generating if there exists: X * w, for
More informationTheory of Computation - Module 3
Theory of Computation - Module 3 Syllabus Context Free Grammar Simplification of CFG- Normal forms-chomsky Normal form and Greibach Normal formpumping lemma for Context free languages- Applications of
More informationSYLLABUS. Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 3 : REGULAR EXPRESSIONS AND LANGUAGES
Contents i SYLLABUS UNIT - I CHAPTER - 1 : AUT UTOMA OMATA Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 2 : FINITE AUT UTOMA OMATA An Informal Picture of Finite Automata,
More informationCPSC 421: Tutorial #1
CPSC 421: Tutorial #1 October 14, 2016 Set Theory. 1. Let A be an arbitrary set, and let B = {x A : x / x}. That is, B contains all sets in A that do not contain themselves: For all y, ( ) y B if and only
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 7 Last time: Proving a language is not regular Pushdown automata (PDAs) Today: Context-free grammars (CFG) Equivalence of CFGs and PDAs Sofya Raskhodnikova 1/31/2016
More informationFoundations of Informatics: a Bridging Course
Foundations of Informatics: a Bridging Course Week 3: Formal Languages and Semantics Thomas Noll Lehrstuhl für Informatik 2 RWTH Aachen University noll@cs.rwth-aachen.de http://www.b-it-center.de/wob/en/view/class211_id948.html
More informationTheory of Computer Science
Theory of Computer Science C1. Formal Languages and Grammars Malte Helmert University of Basel March 14, 2016 Introduction Example: Propositional Formulas from the logic part: Definition (Syntax of Propositional
More informationComputational Models - Lecture 3
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Computational Models - Lecture 3 Equivalence of regular expressions and regular languages (lukewarm leftover
More informationHomework 4 Solutions. 2. Find context-free grammars for the language L = {a n b m c k : k n + m}. (with n 0,
Introduction to Formal Language, Fall 2016 Due: 21-Apr-2016 (Thursday) Instructor: Prof. Wen-Guey Tzeng Homework 4 Solutions Scribe: Yi-Ruei Chen 1. Find context-free grammars for the language L = {a n
More informationContext Free Languages and Grammars
Algorithms & Models of Computation CS/ECE 374, Fall 2017 Context Free Languages and Grammars Lecture 7 Tuesday, September 19, 2017 Sariel Har-Peled (UIUC) CS374 1 Fall 2017 1 / 36 What stack got to do
More informationContext Free Languages. Automata Theory and Formal Grammars: Lecture 6. Languages That Are Not Regular. Non-Regular Languages
Context Free Languages Automata Theory and Formal Grammars: Lecture 6 Context Free Languages Last Time Decision procedures for FAs Minimum-state DFAs Today The Myhill-Nerode Theorem The Pumping Lemma Context-free
More informationPart 4 out of 5 DFA NFA REX. Automata & languages. A primer on the Theory of Computation. Last week, we showed the equivalence of DFA, NFA and REX
Automata & languages A primer on the Theory of Computation Laurent Vanbever www.vanbever.eu Part 4 out of 5 ETH Zürich (D-ITET) October, 12 2017 Last week, we showed the equivalence of DFA, NFA and REX
More informationVTU QUESTION BANK. Unit 1. Introduction to Finite Automata. 1. Obtain DFAs to accept strings of a s and b s having exactly one a.
VTU QUESTION BANK Unit 1 Introduction to Finite Automata 1. Obtain DFAs to accept strings of a s and b s having exactly one a.(5m )( Dec-2014) 2. Obtain a DFA to accept strings of a s and b s having even
More informationTHEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET
THEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET Regular Languages and FA A language is a set of strings over a finite alphabet Σ. All languages are finite or countably infinite. The set of all languages
More informationCISC4090: Theory of Computation
CISC4090: Theory of Computation Chapter 2 Context-Free Languages Courtesy of Prof. Arthur G. Werschulz Fordham University Department of Computer and Information Sciences Spring, 2014 Overview In Chapter
More informationChapter 16: Non-Context-Free Languages
Chapter 16: Non-Context-Free Languages Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu Please read the corresponding chapter
More informationHarvard CS 121 and CSCI E-207 Lecture 10: CFLs: PDAs, Closure Properties, and Non-CFLs
Harvard CS 121 and CSCI E-207 Lecture 10: CFLs: PDAs, Closure Properties, and Non-CFLs Harry Lewis October 8, 2013 Reading: Sipser, pp. 119-128. Pushdown Automata (review) Pushdown Automata = Finite automaton
More informationContext-Free Grammars: Normal Forms
Context-Free Grammars: Normal Forms Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
More informationCOMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lec. 10 : Context-Free Grammars
COMP-330 Theory of Computation Fall 2017 -- Prof. Claude Crépeau Lec. 10 : Context-Free Grammars COMP 330 Fall 2017: Lectures Schedule 1-2. Introduction 1.5. Some basic mathematics 2-3. Deterministic finite
More informationNotes for Comp 497 (Comp 454) Week 10 4/5/05
Notes for Comp 497 (Comp 454) Week 10 4/5/05 Today look at the last two chapters in Part II. Cohen presents some results concerning context-free languages (CFL) and regular languages (RL) also some decidability
More informationCS375: Logic and Theory of Computing
CS375: Logic and Theory of Computing Fuhua (Frank) Cheng Department of Computer Science University of Kentucky 1 Table of Contents: Week 1: Preliminaries (set algebra, relations, functions) (read Chapters
More informationComputational Models - Lecture 5 1
Computational Models - Lecture 5 1 Handout Mode Iftach Haitner and Yishay Mansour. Tel Aviv University. April 10/22, 2013 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationMTH401A Theory of Computation. Lecture 17
MTH401A Theory of Computation Lecture 17 Chomsky Normal Form for CFG s Chomsky Normal Form for CFG s For every context free language, L, the language L {ε} has a grammar in which every production looks
More informationCS20a: summary (Oct 24, 2002)
CS20a: summary (Oct 24, 2002) Context-free languages Grammars G = (V, T, P, S) Pushdown automata N-PDA = CFG D-PDA < CFG Today What languages are context-free? Pumping lemma (similar to pumping lemma for
More informationNotes for Comp 497 (454) Week 10
Notes for Comp 497 (454) Week 10 Today we look at the last two chapters in Part II. Cohen presents some results concerning the two categories of language we have seen so far: Regular languages (RL). Context-free
More informationINSTITUTE OF AERONAUTICAL ENGINEERING
INSTITUTE OF AERONAUTICAL ENGINEERING DUNDIGAL 500 043, HYDERABAD COMPUTER SCIENCE AND ENGINEERING TUTORIAL QUESTION BANK Course Name : FORMAL LANGUAGES AND AUTOMATA THEORY Course Code : A40509 Class :
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY REVIEW for MIDTERM 1 THURSDAY Feb 6 Midterm 1 will cover everything we have seen so far The PROBLEMS will be from Sipser, Chapters 1, 2, 3 It will be
More informationSheet 1-8 Dr. Mostafa Aref Format By : Mostafa Sayed
Sheet -8 Dr. Mostafa Aref Format By : Mostafa Sayed 09 Introduction Assignment. For = {a, } a) Write 0 strings of the following languages i) All strings with no more than one a,,, a, a, a, a, a, a, a ii)
More informationClosure Properties of Context-Free Languages. Foundations of Computer Science Theory
Closure Properties of Context-Free Languages Foundations of Computer Science Theory Closure Properties of CFLs CFLs are closed under: Union Concatenation Kleene closure Reversal CFLs are not closed under
More informationFinite Automata Theory and Formal Languages TMV026/TMV027/DIT321 Responsible: Ana Bove
Finite Automata Theory and Formal Languages TMV026/TMV027/DIT321 Responsible: Ana Bove Tuesday 28 of May 2013 Total: 60 points TMV027/DIT321 registration VT13 TMV026/DIT321 registration before VT13 Exam
More informationTheory of Computation (Classroom Practice Booklet Solutions)
Theory of Computation (Classroom Practice Booklet Solutions) 1. Finite Automata & Regular Sets 01. Ans: (a) & (c) Sol: (a) The reversal of a regular set is regular as the reversal of a regular expression
More informationCA320 - Computability & Complexity
CA320 - Computability & Complexity Dav Sinclair Context-Free Grammars A context-free grammar (CFG) is a 4-tuple G = (N,Σ,S,P) where N is a set of nonterminal symbols, or variables, Σ is a set of terminal
More informationChapter 4: Context-Free Grammars
Chapter 4: Context-Free Grammars 4.1 Basics of Context-Free Grammars Definition A context-free grammars, or CFG, G is specified by a quadruple (N, Σ, P, S), where N is the nonterminal or variable alphabet;
More informationTheory Of Computation UNIT-II
Regular Expressions and Context Free Grammars: Regular expression formalism- equivalence with finite automata-regular sets and closure properties- pumping lemma for regular languages- decision algorithms
More informationCSC 4181Compiler Construction. Context-Free Grammars Using grammars in parsers. Parsing Process. Context-Free Grammar
CSC 4181Compiler Construction Context-ree Grammars Using grammars in parsers CG 1 Parsing Process Call the scanner to get tokens Build a parse tree from the stream of tokens A parse tree shows the syntactic
More information