# Ambiguous Grammars and Compactification

Size: px
Start display at page:

Transcription

1 Ambiguous Grammars and Compactification Mridul Aanjaneya Stanford University July 17, 2012 Mridul Aanjaneya Automata Theory 1/ 44

2 Midterm Review Mathematical Induction and Pigeonhole Principle Finite Automata (DFA s, NFA s, ε-nfa s) Equivalence of DFA s, NFA s and ε-nfa s Regular Languages and Closure Properties Union, Concatenation, Kleene Closure, Intersection, Difference, Complement, Reversal, (Inverse) Homomorphisms Regular Expressions and Equivalence with Automata Pumping Lemma for DFA s Efficient State Minimization for DFA s Context-Free Languages and Parse Trees Mridul Aanjaneya Automata Theory 2/ 44

3 Homework #1 Problem 1 Given 8 distinct natural numbers, none greater than 15, show that at least three pairs of them have the same positive difference. The pairs need not be disjoint as sets. 8 distinct natural numbers 28 pairs. 14 possible differences. The difference 14 can only be achieved by 1 pair PHP! Common Mistakes: 1 Not mentioning PHP. Mridul Aanjaneya Automata Theory 3/ 44

4 Homework #1 Problem 2 Prove that the number (243 ones) is divisible by 243. Let α n denote the number (3 n ones). Proof by induction. α n+1 = α n n + α n 10 3n + α n n and 10 3n leave remainder 1 modulo 3. α n+1 3 α n (mod 3) 0(mod 3 n+1 ) Common Mistakes: 1 Using divisibility of sum of digits modulo 3 without proof. 2 Not stating general claim implies n = 5. Mridul Aanjaneya Automata Theory 4/ 44

5 Homework #1 Problem 3 Show that the language L consisting of runs of even numbers of 0 s and runs of odd numbers of 1 s is regular. 1 0,1 0 A B 1 C 0 A 1 0 0,1 1 B C 0 D 0 1 Common Mistakes: 1 Not having a reject state that loops on all inputs. 2 Inconsistent use of empty string. Mridul Aanjaneya Automata Theory 5/ 44

6 Homework #1 Problem 4 Construct an NFA for the language L over {0,1} such that each string has two 0 s separated by a number of positions that is a nonzero multiple of 5. 0,1 A 0 0,1 0,1 B C D 0,1 1 0,1 H 0 G 0,1 F 0,1 E Common Mistakes: 1 Looping back incorrectly. 2 Not enforcing that the first and last transitions are zeroes. Mridul Aanjaneya Automata Theory 6/ 44

7 Homework #1 Problem 5 Define an NFA N 2 from a given NFA N 1 by reversing the final/nonfinal states. Is L(N 2 ) the complement of L(N 1 )? A Common Mistakes: 1 Overcomplicated thoughts. 2 Proving its true! Mridul Aanjaneya Automata Theory 7/ 44

8 Recap: Context-Free Grammars Terminals: symbols of the alphabet of the language being defined. Variables (nonterminals): a finite set of other symbols, each of which represents a language. Start symbol: the variable whose language is the one being defined. Mridul Aanjaneya Automata Theory 8/ 44

9 Example: Formal CFG Here is a formal CFG for {0 n 1 n n 1}. Terminals = {0,1}. Variables = {S}. Start symbol = S. Productions = S 01 S 0S1 Mridul Aanjaneya Automata Theory 9/ 44

10 Recap: Leftmost Derivations Say waα lm wβα if w is a string of terminals only and A β is a production. Also, α lm β if α becomes β by a sequence of zero or more lm steps. Balanced parantheses grammar: S SS (S) () S lm SS lm (S)S lm (())S lm (())() Thus, S lm (())() Mridul Aanjaneya Automata Theory 10/ 44

11 Recap: Rightmost Derivations Say αaw rm αβw if w is a string of terminals only and A β is a production. Also, α rm β if α becomes β by a sequence of zero or more rm steps. Balanced parantheses grammar: S SS (S) () S rm SS rm S() rm (S)() rm (())() Thus, S rm (())() Mridul Aanjaneya Automata Theory 11/ 44

12 Recap: Parse Trees Parse trees are trees labeled by symbols of a particular CFG. Leaves: labeled by a terminal or ε. Interior nodes: labeled by a variable. Children are labeled by the right side of a production for the parent. Root: must be labeled by the start symbol. Mridul Aanjaneya Automata Theory 12/ 44

13 Example: Parse Tree S SS (S) () S S S ( S ) ( ) ( ) Mridul Aanjaneya Automata Theory 13/ 44

14 Recap: Ambiguous Grammars A CFG is ambiguous is there is a string in the language that is the yield of two or more parse trees. Example: S SS (S) () Two parse trees for ()()()! Mridul Aanjaneya Automata Theory 14/ 44

15 Ambiguous Grammars Ambiguity is a property of grammars not languages. For the balanced parentheses language, here is another CFG which is unambiguous: B (RB ε R ) (RR Note: R generates strings that have one more right parentheses than left. Mridul Aanjaneya Automata Theory 15/ 44

16 Unambiguous Grammars B (RB ε R ) (RR Construct a unique leftmost derivation for a given balanced string of parentheses by scanning the string from left to right. If we need to expand B, then use B (RB if the next symbol is ( and ε if at the end. If we need to expand R, use R ) if the next symbol is ) and (RR if it is (. Mridul Aanjaneya Automata Theory 16/ 44

17 The Parsing Process (())() B (RB ((RRB (()RB (())B (())(RB (())()B (())() Mridul Aanjaneya Automata Theory 17/ 44

18 LL(1) Grammars As an aside, a grammar such as B (RB ε R ) (RR where you can always figure out the production to use in a leftmost derivation by scanning the given string left-to-right and looking only at the next one symbol is called LL(1). Leftmost derivation, left-to-right scan, one symbol of lookahead. Most programming languages have LL(1) grammars. LL(1) grammars are never ambiguous. Mridul Aanjaneya Automata Theory 18/ 44

19 Inherent Ambiguity It would be nice if for every ambiguous grammar, there was some way to fix the ambiguity, as we did for the balanced parentheses grammar. Unfortunately, certain CFL s are inherently ambiguous, meaning that every grammar for the language is ambiguous. Mridul Aanjaneya Automata Theory 19/ 44

20 Example: Inherent Ambiguity The language L = {0 i 1 j 2 k i = j or j = k} Intuitively, at least some of the strings of the form 0 n 1 n 2 n must be generated by two different parse trees, one based on checking the 0 s and 1 s, the other based on checking the 1 s and 2 s. Mridul Aanjaneya Automata Theory 20/ 44

21 One Possible Ambiguous Grammar S AB CD A 0A1 01 B 2B 2 C 0C 0 D 1D2 12 A generates equal numbers 0 s and 1 s. B generates any number of 2 s. C generates any number of 0 s. D generates equal numbers 1 s and 2 s. Mridul Aanjaneya Automata Theory 21/ 44

22 One Possible Ambiguous Grammar S AB CD A 0A1 01 B 2B 2 C 0C 0 D 1D2 12 And there are two derivations of every string with equal numbers of 0 s, 1 s and 2 s. e.g.: S AB 01B 012 S CD 0D 012 Mridul Aanjaneya Automata Theory 22/ 44

23 Example #1 Problem Show that the language of all palindromes over {0,1} is context-free. Mridul Aanjaneya Automata Theory 23/ 44

24 Example #1 Problem Show that the language of all palindromes over {0,1} is context-free. P ε 0 1 0P0 1P1 Mridul Aanjaneya Automata Theory 24/ 44

25 Example #2 Problem Give a context-free language for the regular expression 0 1(0+1). Mridul Aanjaneya Automata Theory 25/ 44

26 Example #2 Problem Give a context-free language for the regular expression 0 1(0+1). S A1B A 0A ε B 0B 1B ε Mridul Aanjaneya Automata Theory 26/ 44

27 Variables that derive nothing Consider: S AB, A aa, B AB Although A derives all strings of a s, B derives no terminal strings. Thus, S derives nothing, and the language is empty! Mridul Aanjaneya Automata Theory 27/ 44

28 Testing if a variable derives a terminal string Basis: If there is a production A w, where w has no variables, then A derives a terminal string. Induction: If there is a production A α, where α consists only of terminals and variables known to derive a terminal string, then A derives a terminal string. Mridul Aanjaneya Automata Theory 28/ 44

29 Testing Eventually, we can find no more variables. An easy induction on the order in which variables are discovered shows that each one truly derives a terminal string. Conversely, any variable that derives a terminal string will be discovered by this algorithm. Mridul Aanjaneya Automata Theory 29/ 44

30 Proof of Converse The proof is an induction on the height of the least-height parse tree by which a variable A derives a terminal string. Basis: Height = 1. Tree looks like: a A... a 1 n Then the basis of the algorithm tells us that A will be discovered. Mridul Aanjaneya Automata Theory 30/ 44

31 Induction for Converse Assume IH for parse trees of height < h, and suppose A derives a terminal string via a parse tree of height h: X A... X 1 n w 1 w n By IH, those X i s that are variables are discovered. Thus, A will also be discovered, because it has a right side of terminals and/or discovered variables. Mridul Aanjaneya Automata Theory 31/ 44

32 Algorithm to Eliminate Variables that Derive Nothing 1 Discover all variables that derive terminal strings. 2 For all other variables, remove all productions in which they appear, either on the left or on the right. Mridul Aanjaneya Automata Theory 32/ 44

33 Example: Eliminate Variables S AB C, A aa a, B bb, C c Basis: A and c are identified because of A a and C c. Induction: S is identified because of S C. Nothing else can be identified. Result: S C, A aa a, C c Mridul Aanjaneya Automata Theory 33/ 44

34 Unreachable Symbols Another way a terminal or a variable deserves to be eliminated is if it cannot appear in any derivation from the start symbol. Basis: We can reach S (the start symbol). Induction: If we can reach A, and there is a production A α, then we can reach all symbols of α. Mridul Aanjaneya Automata Theory 34/ 44

35 Unreachable Symbols Easy inductions in both directions show that when we can discover no more symbols, then we have all and only the symbols that appear in derivations from S. Left as exercises. Algorithm: Remove from the grammar all symbols not discovered reachable from S and all productions that involve these symbols. Mridul Aanjaneya Automata Theory 35/ 44

36 Eliminating Useless Symbols A symbol is useful if it appears in some derivation of some terminal string from the start symbol. Otherwise it is useless. Eliminate all useless symbols by: 1 Eliminating symbols that derive no terminal string. 2 Eliminating unreachable symbols. Mridul Aanjaneya Automata Theory 36/ 44

37 Useless Symbols S AB, A C, C c, B bb If we eliminated unreachable symbols first, we would find everything is reachable. A, C and c would never get eliminated. Mridul Aanjaneya Automata Theory 37/ 44

38 Why It Works After step (1), every symbol remaining derives some terminal string. After step (2), the only symbols remaining are all derivable from S. In addition, they still derive a terminal string, because such a derivation can only involve symbols reachable from S. Mridul Aanjaneya Automata Theory 38/ 44

39 Epsilon Productions We can almost avoid using productions of the form A ε (called ε-productions). The problem is that ε cannot be in the language of any grammar that has no ε-productions. Theorem If L is a CFL, then L {ε} has a CFG with no ε-productions. Mridul Aanjaneya Automata Theory 39/ 44

40 Nullable Symbols To eliminate ε-productions, we first need to discover the nullable symbols = variables A such that A ε. Basis: If there is a production A ε, then A is nullable. Induction: If there is a production A α, and all symbols of α are nullable, then A is nullable. Mridul Aanjaneya Automata Theory 40/ 44

41 Example: Nullable Symbols S AB, A aa ε, B bb A Basis: A is nullable because of A ε. Induction: B is nullable because of B A. Then, S is nullable because of S AB. Mridul Aanjaneya Automata Theory 41/ 44

42 Proof of Algorithm: Nullable Symbols Proof is very much like that for the algorithm for testing variables that derive terminal strings. Left to the imagination! Mridul Aanjaneya Automata Theory 42/ 44

43 Eliminating ε-productions Key idea: turn each production A X 1...X n into a family of productions. For each subset of nullable X s, there is one production with those eliminated from the right side in advance. Except, if all X s are nullable, do not make a production with ε as the right hand side. Mridul Aanjaneya Automata Theory 43/ 44

44 Example: Eliminating ε-productions S ABC, A aa ε, B bb ε, C ε A, B, C and S are all nullable. New grammar: S ABC AB AC BC A B C A aa a B bb b Note: C is now useless, eliminate its productions. Mridul Aanjaneya Automata Theory 44/ 44

### Context-Free Languages and Parse Trees

Context-Free Languages and Parse Trees Mridul Aanjaneya Stanford University July 12, 2012 Mridul Aanjaneya Automata Theory 1/ 41 Context-Free Grammars A context-free grammar is a notation for describing

### Normal Forms for CFG s. Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form

Normal Forms for CFG s Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form 1 Variables That Derive Nothing Consider: S -> AB, A -> aa a, B -> AB Although A derives

### Context-Free Grammars

Context-Free Grammars 1 Informal Comments A context-free grammar is a notation for describing languages. It is more powerful than finite automata or RE s, but still cannot define all possible languages.

### Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 11 Ana Bove April 26th 2018 Recap: Regular Languages Decision properties of RL: Is it empty? Does it contain this word? Contains

### JNTUWORLD. Code No: R

Code No: R09220504 R09 SET-1 B.Tech II Year - II Semester Examinations, April-May, 2012 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science and Engineering) Time: 3 hours Max. Marks: 75 Answer any five

### MA513: Formal Languages and Automata Theory Topic: Context-free Grammars (CFG) Lecture Number 18 Date: September 12, 2011

MA53: Formal Languages and Automata Theory Topic: Context-free Grammars (CFG) Lecture Number 8 Date: September 2, 20 xercise: Define a context-free grammar that represents (a simplification of) expressions

### Final Course Review. Reading: Chapters 1-9

Final Course Review Reading: Chapters 1-9 1 Objectives Introduce concepts in automata theory and theory of computation Identify different formal language classes and their relationships Design grammars

### QUESTION BANK. Formal Languages and Automata Theory(10CS56)

QUESTION BANK Formal Languages and Automata Theory(10CS56) Chapter 1 1. Define the following terms & explain with examples. i) Grammar ii) Language 2. Mention the difference between DFA, NFA and εnfa.

### R10 SET a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA?

R1 SET - 1 1. a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA? 2. a) Design a DFA that accepts the language over = {, 1} of all strings that

### Context-Free Languages & Grammars (CFLs & CFGs) Reading: Chapter 5

Context-Free Languages & Grammars (CFLs & CFGs) Reading: Chapter 5 1 Not all languages are regular So what happens to the languages which are not regular? Can we still come up with a language recognizer?

### CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Sipser Ch 1.2, 1.3 Design NFA recognizing a given language Convert an NFA (with or without

### Homework. Context Free Languages. Before We Start. Announcements. Plan for today. Languages. Any questions? Recall. 1st half. 2nd half.

Homework Context Free Languages Homework #2 returned Homework #3 due today Homework #4 Pg 133 -- Exercise 1 (use structural induction) Pg 133 -- Exercise 3 Pg 134 -- Exercise 8b,c,d Pg 135 -- Exercise

### Context Free Grammars. CS154 Chris Pollett Mar 1, 2006.

Context Free Grammars CS154 Chris Pollett Mar 1, 2006. Outline Formal Definition Ambiguity Chomsky Normal Form Formal Definitions A context free grammar is a 4-tuple (V, Σ, R, S) where 1. V is a finite

### Complexity Theory. Compiled By : Hari Prasad Pokhrel Page 1 of 20. ioenotes.edu.np

Chapter 1: Introduction Introduction Purpose of the Theory of Computation: Develop formal mathematical models of computation that reflect real-world computers. Nowadays, the Theory of Computation can be

### HKN CS 374 Midterm 1 Review. Tim Klem Noah Mathes Mahir Morshed

HKN CS 374 Midterm 1 Review Tim Klem Noah Mathes Mahir Morshed Midterm topics It s all about recognizing sets of strings! 1. String Induction 2. Regular languages a. DFA b. NFA c. Regular expressions 3.

VALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year & Semester : III Year, V Semester Section : CSE - 1 & 2 Subject Code : CS6503 Subject

### Chapter Seven: Regular Expressions

Chapter Seven: Regular Expressions Regular Expressions We have seen that DFAs and NFAs have equal definitional power. It turns out that regular expressions also have exactly that same definitional power:

### UNIT I PART A PART B

OXFORD ENGINEERING COLLEGE (NAAC ACCREDITED WITH B GRADE) DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING LIST OF QUESTIONS YEAR/SEM: III/V STAFF NAME: Dr. Sangeetha Senthilkumar SUB.CODE: CS6503 SUB.NAME:

### CS5371 Theory of Computation. Lecture 8: Automata Theory VI (PDA, PDA = CFG)

CS5371 Theory of Computation Lecture 8: Automata Theory VI (PDA, PDA = CFG) Objectives Introduce Pushdown Automaton (PDA) Show that PDA = CFG In terms of descriptive power Pushdown Automaton (PDA) Roughly

### Optimizing Finite Automata

Optimizing Finite Automata We can improve the DFA created by MakeDeterministic. Sometimes a DFA will have more states than necessary. For every DFA there is a unique smallest equivalent DFA (fewest states

### QUESTION BANK. Unit 1. Introduction to Finite Automata

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 )(Jun-Jul 10) 2. Obtain a DFA to accept strings of a s and b s having even

### Context-Free Grammars and Languages (2015/11)

Chapter 5 Context-Free Grammars and Languages (2015/11) Adriatic Sea shore at Opatija, Croatia Outline 5.0 Introduction 5.1 Context-Free Grammars (CFG s) 5.2 Parse Trees 5.3 Applications of CFG s 5.4 Ambiguity

### Where We Are. CMSC 330: Organization of Programming Languages. This Lecture. Programming Languages. Motivation for Grammars

CMSC 330: Organization of Programming Languages Context Free Grammars Where We Are Programming languages Ruby OCaml Implementing programming languages Scanner Uses regular expressions Finite automata Parser

### (a) R=01[((10)*+111)*+0]*1 (b) ((01+10)*00)*. [8+8] 4. (a) Find the left most and right most derivations for the word abba in the grammar

Code No: R05310501 Set No. 1 III B.Tech I Semester Regular Examinations, November 2008 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science & Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE

### CS525 Winter 2012 \ Class Assignment #2 Preparation

1 CS525 Winter 2012 \ Class Assignment #2 Preparation Ariel Stolerman 2.26) Let be a CFG in Chomsky Normal Form. Following is a proof that for any ( ) of length exactly steps are required for any derivation

### Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016 Lecture 15 Ana Bove May 23rd 2016 More on Turing machines; Summary of the course. Overview of today s lecture: Recap: PDA, TM Push-down

### (Refer Slide Time: 0:19)

Theory of Computation. Professor somenath Biswas. Department of Computer Science & Engineering. Indian Institute of Technology, Kanpur. Lecture-15. Decision Problems for Regular Languages. (Refer Slide

### Multiple Choice Questions

Techno India Batanagar Computer Science and Engineering Model Questions Subject Name: Formal Language and Automata Theory Subject Code: CS 402 Multiple Choice Questions 1. The basic limitation of an FSM

### Derivations of a CFG. MACM 300 Formal Languages and Automata. Context-free Grammars. Derivations and parse trees

Derivations of a CFG MACM 300 Formal Languages and Automata Anoop Sarkar http://www.cs.sfu.ca/~anoop strings grow on trees strings grow on Noun strings grow Object strings Verb Object Noun Verb Object

### Finite Automata. Dr. Nadeem Akhtar. Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur

Finite Automata Dr. Nadeem Akhtar Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur PhD Laboratory IRISA-UBS University of South Brittany European University

### MIT Specifying Languages with Regular Expressions and Context-Free Grammars

MIT 6.035 Specifying Languages with Regular essions and Context-Free Grammars Martin Rinard Laboratory for Computer Science Massachusetts Institute of Technology Language Definition Problem How to precisely

### CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages Context Free Grammars and Parsing 1 Recall: Architecture of Compilers, Interpreters Source Parser Static Analyzer Intermediate Representation Front End Back

### Theory of Computation, Homework 3 Sample Solution

Theory of Computation, Homework 3 Sample Solution 3.8 b.) The following machine M will do: M = "On input string : 1. Scan the tape and mark the first 1 which has not been marked. If no unmarked 1 is found,

### Lexical Analysis. Dragon Book Chapter 3 Formal Languages Regular Expressions Finite Automata Theory Lexical Analysis using Automata

Lexical Analysis Dragon Book Chapter 3 Formal Languages Regular Expressions Finite Automata Theory Lexical Analysis using Automata Phase Ordering of Front-Ends Lexical analysis (lexer) Break input string

### CT32 COMPUTER NETWORKS DEC 2015

Q.2 a. Using the principle of mathematical induction, prove that (10 (2n-1) +1) is divisible by 11 for all n N (8) Let P(n): (10 (2n-1) +1) is divisible by 11 For n = 1, the given expression becomes (10

### Compiler Construction

Compiler Construction Exercises 1 Review of some Topics in Formal Languages 1. (a) Prove that two words x, y commute (i.e., satisfy xy = yx) if and only if there exists a word w such that x = w m, y =

### MIT Specifying Languages with Regular Expressions and Context-Free Grammars. Martin Rinard Massachusetts Institute of Technology

MIT 6.035 Specifying Languages with Regular essions and Context-Free Grammars Martin Rinard Massachusetts Institute of Technology Language Definition Problem How to precisely define language Layered structure

### 9.5 Equivalence Relations

9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same

Automata Theory TEST 1 Answers Max points: 156 Grade basis: 150 Median grade: 81% 1. (2 pts) See text. You can t be sloppy defining terms like this. You must show a bijection between the natural numbers

### CMSC 330: Organization of Programming Languages. Architecture of Compilers, Interpreters

: Organization of Programming Languages Context Free Grammars 1 Architecture of Compilers, Interpreters Source Scanner Parser Static Analyzer Intermediate Representation Front End Back End Compiler / Interpreter

### Languages and Compilers

Principles of Software Engineering and Operational Systems Languages and Compilers SDAGE: Level I 2012-13 3. Formal Languages, Grammars and Automata Dr Valery Adzhiev vadzhiev@bournemouth.ac.uk Office:

### 1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1}

CSE 5 Homework 2 Due: Monday October 6, 27 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in

### Formal Languages and Automata

Mobile Computing and Software Engineering p. 1/3 Formal Languages and Automata Chapter 3 Regular languages and Regular Grammars Chuan-Ming Liu cmliu@csie.ntut.edu.tw Department of Computer Science and

### T.E. (Computer Engineering) (Semester I) Examination, 2013 THEORY OF COMPUTATION (2008 Course)

*4459255* [4459] 255 Seat No. T.E. (Computer Engineering) (Semester I) Examination, 2013 THEY OF COMPUTATION (2008 Course) Time : 3 Hours Max. Marks : 100 Instructions : 1) Answers to the two Sections

### Homework. Announcements. Before We Start. Languages. Plan for today. Chomsky Normal Form. Final Exam Dates have been announced

Homework Homework #3 returned Homework #4 due today Homework #5 Pg 169 -- Exercise 4 Pg 183 -- Exercise 4c,e,i Pg 184 -- Exercise 10 Pg 184 -- Exercise 12 Pg 185 -- Exercise 17 Due 10 / 17 Announcements

### Learn Smart and Grow with world

Learn Smart and Grow with world All Department Smart Study Materials Available Smartkalvi.com TABLE OF CONTENTS S.No DATE TOPIC PAGE NO. UNIT-I FINITE AUTOMATA 1 Introduction 1 2 Basic Mathematical Notation

### CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages Context Free Grammars 1 Architecture of Compilers, Interpreters Source Analyzer Optimizer Code Generator Abstract Syntax Tree Front End Back End Compiler

### Chapter Seven: Regular Expressions. Formal Language, chapter 7, slide 1

Chapter Seven: Regular Expressions Formal Language, chapter 7, slide The first time a young student sees the mathematical constant π, it looks like just one more school artifact: one more arbitrary symbol

### Decision Properties for Context-free Languages

Previously: Decision Properties for Context-free Languages CMPU 240 Language Theory and Computation Fall 2018 Context-free languages Pumping Lemma for CFLs Closure properties for CFLs Today: Assignment

### CIT3130: Theory of Computation. Regular languages

ƒ CIT3130: Theory of Computation Regular languages ( M refers to the first edition of Martin and H to IALC by Hopcroft et al.) Definitions of regular expressions and regular languages: A regular expression

### CS 432 Fall Mike Lam, Professor. Finite Automata Conversions and Lexing

CS 432 Fall 2017 Mike Lam, Professor Finite Automata Conversions and Lexing Finite Automata Key result: all of the following have the same expressive power (i.e., they all describe regular languages):

### Architecture of Compilers, Interpreters. CMSC 330: Organization of Programming Languages. Front End Scanner and Parser. Implementing the Front End

Architecture of Compilers, Interpreters : Organization of Programming Languages ource Analyzer Optimizer Code Generator Context Free Grammars Intermediate Representation Front End Back End Compiler / Interpreter

### University of Nevada, Las Vegas Computer Science 456/656 Fall 2016

University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen

### Regular Languages. MACM 300 Formal Languages and Automata. Formal Languages: Recap. Regular Languages

Regular Languages MACM 3 Formal Languages and Automata Anoop Sarkar http://www.cs.sfu.ca/~anoop The set of regular languages: each element is a regular language Each regular language is an example of a

### Skyup's Media. PART-B 2) Construct a Mealy machine which is equivalent to the Moore machine given in table.

Code No: XXXXX JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY, HYDERABAD B.Tech II Year I Semester Examinations (Common to CSE and IT) Note: This question paper contains two parts A and B. Part A is compulsory

### CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages Context Free Grammars 1 Architecture of Compilers, Interpreters Source Analyzer Optimizer Code Generator Abstract Syntax Tree Front End Back End Compiler

### CS154 Midterm Examination. May 4, 2010, 2:15-3:30PM

CS154 Midterm Examination May 4, 2010, 2:15-3:30PM Directions: Answer all 7 questions on this paper. The exam is open book and open notes. Any materials may be used. Name: I acknowledge and accept the

### Post's Correspondence Problem. An undecidable, but RE, problem that. appears not to have anything to do with. TM's.

Post's Correspondence Problem An undecidable, but RE, problem that appears not to have anything to do with TM's. Given two lists of ëcorresponding" strings èw 1 ;w 2 ;:::;w n è and x 1 ;x 2 ;:::;x n è,

### CMSC 330: Organization of Programming Languages. Context-Free Grammars Ambiguity

CMSC 330: Organization of Programming Languages Context-Free Grammars Ambiguity Review Why should we study CFGs? What are the four parts of a CFG? How do we tell if a string is accepted by a CFG? What

### 1. [5 points each] True or False. If the question is currently open, write O or Open.

University of Nevada, Las Vegas Computer Science 456/656 Spring 2018 Practice for the Final on May 9, 2018 The entire examination is 775 points. The real final will be much shorter. Name: No books, notes,

### CMSC 330: Organization of Programming Languages. Context Free Grammars

CMSC 330: Organization of Programming Languages Context Free Grammars 1 Architecture of Compilers, Interpreters Source Analyzer Optimizer Code Generator Abstract Syntax Tree Front End Back End Compiler

### Closure Properties of CFLs; Introducing TMs. CS154 Chris Pollett Apr 9, 2007.

Closure Properties of CFLs; Introducing TMs CS154 Chris Pollett Apr 9, 2007. Outline Closure Properties of Context Free Languages Algorithms for CFLs Introducing Turing Machines Closure Properties of CFL

### Context-Free Grammars

Context-Free Grammars Describing Languages We've seen two models for the regular languages: Finite automata accept precisely the strings in the language. Regular expressions describe precisely the strings

### Fall Compiler Principles Context-free Grammars Refresher. Roman Manevich Ben-Gurion University of the Negev

Fall 2016-2017 Compiler Principles Context-free Grammars Refresher Roman Manevich Ben-Gurion University of the Negev 1 xample grammar S S ; S S id := S print (L) id num + L L L, shorthand for Statement

Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,

### TOPIC PAGE NO. UNIT-I FINITE AUTOMATA

TABLE OF CONTENTS SNo DATE TOPIC PAGE NO UNIT-I FINITE AUTOMATA 1 Introduction 1 2 Basic Mathematical Notation Techniques 3 3 Finite State systems 4 4 Basic Definitions 6 5 Finite Automaton 7 6 DFA NDFA

### CMSC 330: Organization of Programming Languages. Context Free Grammars

CMSC 330: Organization of Programming Languages Context Free Grammars 1 Architecture of Compilers, Interpreters Source Analyzer Optimizer Code Generator Abstract Syntax Tree Front End Back End Compiler

### Syntax Analysis. Prof. James L. Frankel Harvard University. Version of 6:43 PM 6-Feb-2018 Copyright 2018, 2015 James L. Frankel. All rights reserved.

Syntax Analysis Prof. James L. Frankel Harvard University Version of 6:43 PM 6-Feb-2018 Copyright 2018, 2015 James L. Frankel. All rights reserved. Context-Free Grammar (CFG) terminals non-terminals start

### Regular Languages and Regular Expressions

Regular Languages and Regular Expressions According to our definition, a language is regular if there exists a finite state automaton that accepts it. Therefore every regular language can be described

### CSE 431S Scanning. Washington University Spring 2013

CSE 431S Scanning Washington University Spring 2013 Regular Languages Three ways to describe regular languages FSA Right-linear grammars Regular expressions Regular Expressions A regular expression is

### Programming Languages (CS 550) Lecture 4 Summary Scanner and Parser Generators. Jeremy R. Johnson

Programming Languages (CS 550) Lecture 4 Summary Scanner and Parser Generators Jeremy R. Johnson 1 Theme We have now seen how to describe syntax using regular expressions and grammars and how to create

### 14.1 Encoding for different models of computation

Lecture 14 Decidable languages In the previous lecture we discussed some examples of encoding schemes, through which various objects can be represented by strings over a given alphabet. We will begin this

### 3. Syntax Analysis. Andrea Polini. Formal Languages and Compilers Master in Computer Science University of Camerino

3. Syntax Analysis Andrea Polini Formal Languages and Compilers Master in Computer Science University of Camerino (Formal Languages and Compilers) 3. Syntax Analysis CS@UNICAM 1 / 54 Syntax Analysis: the

### DVA337 HT17 - LECTURE 4. Languages and regular expressions

DVA337 HT17 - LECTURE 4 Languages and regular expressions 1 SO FAR 2 TODAY Formal definition of languages in terms of strings Operations on strings and languages Definition of regular expressions Meaning

### Context Free Languages and Pushdown Automata

Context Free Languages and Pushdown Automata COMP2600 Formal Methods for Software Engineering Ranald Clouston Australian National University Semester 2, 2013 COMP 2600 Context Free Languages and Pushdown

### CS 2210 Sample Midterm. 1. Determine if each of the following claims is true (T) or false (F).

CS 2210 Sample Midterm 1. Determine if each of the following claims is true (T) or false (F). F A language consists of a set of strings, its grammar structure, and a set of operations. (Note: a language

### Last lecture CMSC330. This lecture. Finite Automata: States. Finite Automata. Implementing Regular Expressions. Languages. Regular expressions

Last lecture CMSC330 Finite Automata Languages Sets of strings Operations on languages Regular expressions Constants Operators Precedence 1 2 Finite automata States Transitions Examples Types This lecture

### Foundations of Computer Science Spring Mathematical Preliminaries

Foundations of Computer Science Spring 2017 Equivalence Relation, Recursive Definition, and Mathematical Induction Mathematical Preliminaries Mohammad Ashiqur Rahman Department of Computer Science College

### Midterm Exam. CSCI 3136: Principles of Programming Languages. February 20, Group 2

Banner number: Name: Midterm Exam CSCI 336: Principles of Programming Languages February 2, 23 Group Group 2 Group 3 Question. Question 2. Question 3. Question.2 Question 2.2 Question 3.2 Question.3 Question

### Non-deterministic Finite Automata (NFA)

Non-deterministic Finite Automata (NFA) CAN have transitions on the same input to different states Can include a ε or λ transition (i.e. move to new state without reading input) Often easier to design

### Finite Automata Part Three

Finite Automata Part Three Friday Four Square! Today at 4:15PM, Outside Gates. Announcements Problem Set 4 due right now. Problem Set 5 out, due next Friday, November 2. Play around with finite automata

### SLR parsers. LR(0) items

SLR parsers LR(0) items As we have seen, in order to make shift-reduce parsing practical, we need a reasonable way to identify viable prefixes (and so, possible handles). Up to now, it has not been clear

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### COP4020 Programming Languages. Syntax Prof. Robert van Engelen

COP4020 Programming Languages Syntax Prof. Robert van Engelen Overview n Tokens and regular expressions n Syntax and context-free grammars n Grammar derivations n More about parse trees n Top-down and

### Zhizheng Zhang. Southeast University

Zhizheng Zhang Southeast University 2016/10/5 Lexical Analysis 1 1. The Role of Lexical Analyzer 2016/10/5 Lexical Analysis 2 2016/10/5 Lexical Analysis 3 Example. position = initial + rate * 60 2016/10/5

### Announcements. Written Assignment 1 out, due Friday, July 6th at 5PM.

Syntax Analysis Announcements Written Assignment 1 out, due Friday, July 6th at 5PM. xplore the theoretical aspects of scanning. See the limits of maximal-munch scanning. Class mailing list: There is an

### Syntax Analysis Check syntax and construct abstract syntax tree

Syntax Analysis Check syntax and construct abstract syntax tree if == = ; b 0 a b Error reporting and recovery Model using context free grammars Recognize using Push down automata/table Driven Parsers

### Finite Automata Part Three

Finite Automata Part Three Recap from Last Time A language L is called a regular language if there exists a DFA D such that L( D) = L. NFAs An NFA is a Nondeterministic Finite Automaton Can have missing

### Answer All Questions. All Questions Carry Equal Marks. Time: 20 Min. Marks: 10.

Code No: 134BD Set No. 1 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech. II Year II Sem., I Mid-Term Examinations, February - 2018 FORMAL LANGUAGES AND AUTOMATA THEORY Objective Exam Name:

### CS 315 Programming Languages Syntax. Parser. (Alternatively hand-built) (Alternatively hand-built)

Programming languages must be precise Remember instructions This is unlike natural languages CS 315 Programming Languages Syntax Precision is required for syntax think of this as the format of the language

### Introduction to Syntax Analysis. The Second Phase of Front-End

Compiler Design IIIT Kalyani, WB 1 Introduction to Syntax Analysis The Second Phase of Front-End Compiler Design IIIT Kalyani, WB 2 Syntax Analysis The syntactic or the structural correctness of a program

### Syntax Analysis: Context-free Grammars, Pushdown Automata and Parsing Part - 4. Y.N. Srikant

Syntax Analysis: Context-free Grammars, Pushdown Automata and Part - 4 Department of Computer Science and Automation Indian Institute of Science Bangalore 560 012 NPTEL Course on Principles of Compiler

### 4. Lexical and Syntax Analysis

4. Lexical and Syntax Analysis 4.1 Introduction Language implementation systems must analyze source code, regardless of the specific implementation approach Nearly all syntax analysis is based on a formal

### Section 17. Closed Sets and Limit Points

17. Closed Sets and Limit Points 1 Section 17. Closed Sets and Limit Points Note. In this section, we finally define a closed set. We also introduce several traditional topological concepts, such as limit

### CS143 Handout 20 Summer 2011 July 15 th, 2011 CS143 Practice Midterm and Solution

CS143 Handout 20 Summer 2011 July 15 th, 2011 CS143 Practice Midterm and Solution Exam Facts Format Wednesday, July 20 th from 11:00 a.m. 1:00 p.m. in Gates B01 The exam is designed to take roughly 90

### Properties of Regular Expressions and Finite Automata

Properties of Regular Expressions and Finite Automata Some token patterns can t be defined as regular expressions or finite automata. Consider the set of balanced brackets of the form [[[ ]]]. This set

### Decision Properties of RLs & Automaton Minimization

Decision Properties of RLs & Automaton Minimization Martin Fränzle formatics and Mathematical Modelling The Technical University of Denmark Languages and Parsing MF Fall p./ What you ll learn. Decidable

### Formal Languages and Compilers Lecture IV: Regular Languages and Finite. Finite Automata

Formal Languages and Compilers Lecture IV: Regular Languages and Finite Automata Free University of Bozen-Bolzano Faculty of Computer Science POS Building, Room: 2.03 artale@inf.unibz.it http://www.inf.unibz.it/

### Theory of Computation Dr. Weiss Extra Practice Exam Solutions

Name: of 7 Theory of Computation Dr. Weiss Extra Practice Exam Solutions Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try to be