DOCSLIB.ORG

Context-Free Grammars

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Context-Free Grammars Chapter 3 Context-Free Grammars By Dr Zalmiyah Zakaria •Context-Free Grammars and Languages •Regular Grammars Formal Definition of Context-Free Grammars (CFG) A CFG can be formally defined by a quadruple of (V, , P, S) where: – V is a finite set of variables (non-terminal) – (the alphabet) is a finite set of terminal symbols , where V = – P is a finite set of rules (production rules) written as: A for A V, (v )*. – S is the start symbol, S V 2 Formal Definition of CFG • We can give a formal description to a particular CFG by specifying each of its four components, for example, G = ({S, A}, {0, 1}, P, S) where P consists of three rules: S → 0S1 S → A A → Sept2011 Theory of Computer Science 3 Context-Free Grammars • Terminal symbols – elements of the alphabet • Variables or non-terminals – additional symbols used in production rules • Variable S (start symbol) initiates the process of generating acceptable strings. 4 Terminal or Variable ? • S → (S) | S + S | S × S | A • A → 1 | 2 | 3 • The terminal symbols are { (, ), +, ×, 1, 2, 3} • The variable symbols are S and A Sept2011 Theory of Computer Science 5 Context-Free Grammars • A rule is an element of the set V (V )*. • An A rule: [A, w] or A w • A null rule or lambda rule: A 6 Context-Free Grammars • Grammars are used to generate strings of a language. • An A rule can be applied to the variable A whenever and wherever it occurs. • No limitation on applicability of a rule – it is context free 8 Context-Free Grammars • CFG have no restrictions on the right-hand side of production rules. All of the following are valid CFG production rules: S aSbS S c S λ S xyz S ABcDEF S MZK • Notice that there can be a λ, one or more terminals, one or more non-terminals and any combination of terminals and non-terminals on the right-hand side of a production rule. Sept2011 Theory of Computer Science 9 Generating Strings with CFG • Generate a string by applying rules – Start with the initial symbol – Repeat: • Pick any non-terminal in the string • Replace that non-terminal with the right-hand side of some rule that has that non-terminal as a left-hand side • Repeat until all elements in the string are terminals • E.g. : P: S uAv A w We can derived string uwv as: S ⇒ uAv ⇒ uwv 10 Generating Strings with CFG P: S aS S Bb B cB B • Generating a string: S replace S with aS aS replace S with Bb aBb replace B with cB acBb replace B with acb Final String Sept2011 Theory of Computer Science 11 Generating Strings with CFG P: S aS S Bb B cB B • Generating a string: S replace S with aS aS replace S with aS aaS replace S with Bb aaBb replace B with cB aacBb replace B with cB aaccBb replace B with aaccb Final String Sept2011 Theory of Computer Science 12 Generating Strings with CFG P: S aS S Bb B cB B • Regular expression equivalent to this CFG: a∗c∗b • Shortest string: S ⇒ Bb ⇒ b. Sept2011 Theory of Computer Science 13 Derivation • A derivation is a listing of how a string is generated – showing what the string looks like after every replacement. S AB S ⇒ AB A aA | ⇒ aAB B bB | ⇒ aAbB ⇒ abB ⇒ abbB ⇒ abb Sept2011 Theory of Computer Science 14 Derivation • A terminal string is in the language of the grammar if it can be derived from the start symbol S using the rules of the grammar: • Example: S AASB | AAB A a B bbb 15 Derivation Derivation Rule Applied S ⇒ AASB S AASB ⇒ AAAASBB S AASB ⇒ AAAAAASBBB S AASB ⇒ AAAAAAAABBBB S AAB ⇒ aaaaaaaaBBBB A a ⇒ aaaaaaaabbbbbbbbbbbb B bbb 16 Derivation • Let G be the grammar : S aS | aA A bA | • The derivation of aabb is as shown: S aS aaA aabA aabbA aabb aabb 17 Example • Let G be the grammar : S aSb | ab • The derivation of aabb is as shown: S aSb aaSbb aaaSbbb aaaaSbbbb aaaabbbb Set notation equivalent to this CFG is n n L = {a b | n > 0} 18 CFG – Generating strings • E.g. : Formal description of G = ({S}, {a, b}, P, S) P: S aS | bS | • The following strings can be derived: S ⇒ S ⇒ aS ⇒ a ⇒ a S ⇒ bS ⇒ b ⇒ b S ⇒ aS ⇒ aaS ⇒ aa ⇒ aa S ⇒ aS ⇒ abS ⇒ ab ⇒ ab S ⇒ bS ⇒ baS ⇒ ba ⇒ ba S ⇒ aS ⇒ abS ⇒ abbS ⇒ abb ⇒ abb 19 CFG – Generating strings • E.g. : G = ({S}, {a,b}, P, S) P: S aS | bS | • The language above can also be defined using regular expression: L(G) = (a + b)* 20 CFG – Generating strings • E.g. : G = ({S, A}, {a, b}, P, S) P: S AA A AAA | bA | Ab | a • The following strings can be derived: S ⇒ AA S ⇒ aA rule [A a] Shortest S ⇒ aAAA rule [A AAA] string ? S ⇒ abAAA rule [A bA] S ⇒ abaAA rule [A a] S ⇒ abaAbA rule [A Ab] S ⇒ abaabA rule [A a] S ⇒ abaaba rule [A a] 21 CFG – Generating strings G = (V, , P, S), V = {S, A}, = {a, b}, P: S AA A AAA | bA | Ab | a • Four distinct derivations of ababaa in G: (a) S ⇒ AA (b) S ⇒ AA (c) S ⇒ AA (d) S ⇒ AA ⇒ aA ⇒ AAAA ⇒ Aa ⇒ aA ⇒ aAAA ⇒ aAAA ⇒ AAAa ⇒ aAAA ⇒ abAAA ⇒ abAAA ⇒ AAbAa ⇒ aAAa ⇒ abaAA ⇒ abaAA ⇒ AAbaa ⇒ abAAa ⇒ ababAA ⇒ ababAA ⇒ AbAbaa ⇒ abAbAa ⇒ ababaA ⇒ ababaA ⇒ Ababaa ⇒ ababAa ⇒ ababaa ⇒ ababaa ⇒ ababaa ⇒ ababaa (a) & (b) leftmost, (c) rightmost, (d) arbitrary 22 Context-Free Grammars S ⇒ aS ⇒ aSa ⇒ aba • Sentencial forms are the strings derivable from start symbol of the grammar. • Sentences are forms that contain only terminal symbols. • A set of strings over is context-free language if there is a context-free grammar that generates it. 23 Derivation Tree, DT G = ({S, A}, {a, b}, P, S) P: S AA A AAA | bA | Ab | a The derivation tree for S ⇒ AA ⇒ aA ⇒ abA ⇒ abAb ⇒ abab 24 Derivation Tree S ⇒ aS ⇒ aSa ⇒ aba S ⇒ Sa ⇒ aSa ⇒ aba 25 Derivation Tree • A CFG G is ambiguous if there exist more than 1 DT for n, where n L(G). • Example: G = ({S}, {a, b}, P, S) P: S aS | Sa | b The string aba can be derived as: S ⇒ aS ⇒ aSa ⇒ aba or S ⇒ Sa ⇒ aSa ⇒ aba 26 Example • Let G be the grammar given by the production S aSa | aBa B bB | b • Then L(G) = {anbman | n > 0, m > 0} 27 Example • Let L(G) = {anbmcmd2n | n≥0, m>0} • Then the production rules for this grammar is: S aSdd | A A bAc | bc 28 Example • A string w is a palindrome if w = wR • The set of palindrome over {a, b} can be derived using rules: S a | b | S aSa | bSb 29 More Examples • What is the language of this grammar ? • S aA a(a + b)*b A aA | bA | b • S aA aa*bb* A aA | bB B bB | • S aS | bA a*bba* A bB B aB | 30 How to convert RE to CFG • General rules: • a + b S a | b • ab S aA A b • a* S → aS | λ Sept2011 Theory of Computer Science 31 Regular Grammars • Regular grammars are an important subclass of context-free grammars that play an important role in the lexical analysis and parsing of programming languages. • Regular grammars is a subset of CFG. • Regular grammars are obtained by placing restrictions on the form of the right hand side of the rules. 33 Definition • A regular grammar is a CFG in which each rule has one of the following form: 1. A a 2. A aB 3. A λ where A, B V, and a , A, B is a non- terminal and a can be one or more terminals 34 Regular Grammars • There is at most ONE variable in a sentential form – the rightmost symbol in the string. • Each rule application adds ONE terminal to the derived string. Derivation is terminated by rules: A a OR A λ 35 Example • Consider the grammar: G : S abSA | A Aa | • The equivalent regular grammar: Gr : S aB | B bS | bA A aA | 36 Example Syntax of Pascal in Backus-Naur Form <assign> <var> := <exp> <var> A B C <exp> <var> + <exp> <var> - <exp> (<exp>) <var> * <exp> <var> 37 Example Is A := B*(A+C) Syntactically correct? <assign> <var> := <expr> A := <expr> A := <var>*<expr> A := B*<expr> A := B*(<var>+<expr>) A := B*(A+<expr>) A := B*(A+<var>) A := B*(A+C) 38 Example Is A := B*(A+C) Syntactically correct? 39 How to construct RG from a RE • Use a new non-terminal for every new character • Each loop state turns into a recursive definition on a non-terminal • Example: R.E. ab*ab RG S aA A bA A aB B b Sept2011 Theory of Computer Science 40 Reg Expression to Reg Grammar • RE a(a + b)*b RG S → aA A → aA A → bA A → b • RE ab*a RG S → aA A → bA A → a Sept2011 Theory of Computer Science 41 How to construct RG from a RE (cont.) • Eg. R.E. a(a + b)*b Regular Grammar S aA A aA A bA A b Sept2011 Theory of Computer Science 42 RE ⇒ RG • Consider the regular expression: a+b* • The regular grammar is: S aS aR R bR 43 RE ⇒ RG ⇒ RL • The regular language L = a+b* can be defined as: L = ( V, , P, S) where: V = {S, R} = {a, b} P: S aS aR R bR 44 Non-Regular Language • The language L = a+b* can also be defined as: L = ( V, , P, S) where: V = {S, A, B} = {a, b} P: S AB non-regular A aA | a grammar B bB | 45 Non-RG to RG • Since every RG defines a RL and every RL must have a RE, it would be easier to go ahead and figure out what the RE is and convert it to the RG.
Load more

Recommended publications
  • Theory of Computer Science
    Theory of Computer Science MODULE-1 Alphabet A finite set of symbols is denoted by ∑. Language A language is defined as a set of strings of symbols over an alphabet. Language of a Machine Set of all accepted strings of a machine is called language of a machine. Grammar Grammar is the set of rules that generates language. CHOMSKY HIERARCHY OF LANGUAGES Type 0-Language:-Unrestricted Language, Accepter:-Turing Machine, Generetor:-Unrestricted Grammar Type 1- Language:-Context Sensitive Language, Accepter:- Linear Bounded Automata, Generetor:-Context Sensitive Grammar Type 2- Language:-Context Free Language, Accepter:-Push Down Automata, Generetor:- Context Free Grammar Type 3- Language:-Regular Language, Accepter:- Finite Automata, Generetor:-Regular Grammar Finite Automata Finite Automata is generally of two types :(i)Deterministic Finite Automata (DFA)(ii)Non Deterministic Finite Automata(NFA) DFA DFA is represented formally by a 5-tuple (Q,Σ,δ,q0,F), where: Q is a finite set of states. Σ is a finite set of symbols, called the alphabet of the automaton. δ is the transition function, that is, δ: Q × Σ → Q. q0 is the start state, that is, the state of the automaton before any input has been processed, where q0 Q. F is a set of states of Q (i.e. F Q) called accept states or Final States 1. Construct a DFA that accepts set of all strings over ∑={0,1}, ending with 00 ? 1 0 A B S State/Input 0 1 1 A B A 0 B C A 1 * C C A C 0 2. Construct a DFA that accepts set of all strings over ∑={0,1}, not containing 101 as a substring ? 0 1 1 A B State/Input 0 1 S *A A B 0 *B C B 0 0,1 *C A R C R R R R 1 NFA NFA is represented formally by a 5-tuple (Q,Σ,δ,q0,F), where: Q is a finite set of states.
    [Show full text]
  • Cs 61A/Cs 98-52
    CS 61A/CS 98-52 Mehrdad Niknami University of California, Berkeley Mehrdad Niknami (UC Berkeley) CS 61A/CS 98-52 1 / 23 Something like this? (Is this good?) def find(string, pattern): n= len(string) m= len(pattern) for i in range(n-m+ 1): is_match= True for j in range(m): if pattern[j] != string[i+ j] is_match= False break if is_match: return i What if you were looking for a pattern? Like an email address? Motivation How would you find a substring inside a string? Mehrdad Niknami (UC Berkeley) CS 61A/CS 98-52 2 / 23 def find(string, pattern): n= len(string) m= len(pattern) for i in range(n-m+ 1): is_match= True for j in range(m): if pattern[j] != string[i+ j] is_match= False break if is_match: return i What if you were looking for a pattern? Like an email address? Motivation How would you find a substring inside a string? Something like this? (Is this good?) Mehrdad Niknami (UC Berkeley) CS 61A/CS 98-52 2 / 23 What if you were looking for a pattern? Like an email address? Motivation How would you find a substring inside a string? Something like this? (Is this good?) def find(string, pattern): n= len(string) m= len(pattern) for i in range(n-m+ 1): is_match= True for j in range(m): if pattern[j] != string[i+ j] is_match= False break if is_match: return i Mehrdad Niknami (UC Berkeley) CS 61A/CS 98-52 2 / 23 Motivation How would you find a substring inside a string? Something like this? (Is this good?) def find(string, pattern): n= len(string) m= len(pattern) for i in range(n-m+ 1): is_match= True for j in range(m): if pattern[j] != string[i+
    [Show full text]
  • Regular Languages and Finite Automata for Part IA of the Computer Science Tripos
    N Lecture Notes on Regular Languages and Finite Automata for Part IA of the Computer Science Tripos Prof. Andrew M. Pitts Cambridge University Computer Laboratory c 2012 A. M. Pitts Contents Learning Guide ii 1 Regular Expressions 1 1.1 Alphabets,strings,andlanguages . .......... 1 1.2 Patternmatching................................. .... 4 1.3 Somequestionsaboutlanguages . ....... 6 1.4 Exercises....................................... .. 8 2 Finite State Machines 11 2.1 Finiteautomata .................................. ... 11 2.2 Determinism, non-determinism, and ε-transitions. .. .. .. .. .. .. .. .. .. 14 2.3 Asubsetconstruction . .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... 17 2.4 Summary ........................................ 20 2.5 Exercises....................................... .. 20 3 Regular Languages, I 23 3.1 Finiteautomatafromregularexpressions . ............ 23 3.2 Decidabilityofmatching . ...... 28 3.3 Exercises....................................... .. 30 4 Regular Languages, II 31 4.1 Regularexpressionsfromfiniteautomata . ........... 31 4.2 Anexample ....................................... 32 4.3 Complementand intersectionof regularlanguages . .............. 34 4.4 Exercises....................................... .. 36 5 The Pumping Lemma 39 5.1 ProvingthePumpingLemma . .... 40 5.2 UsingthePumpingLemma . ... 41 5.3 Decidabilityoflanguageequivalence . ........... 44 5.4 Exercises....................................... .. 45 6 Grammars 47 6.1 Context-freegrammars . ..... 47 6.2 Backus-NaurForm ................................
    [Show full text]
  • Neural Edit Operations for Biological Sequences
    Neural Edit Operations for Biological Sequences Satoshi Koide Keisuke Kawano Toyota Central R&D Labs. Toyota Central R&D Labs. [email protected] [email protected] Takuro Kutsuna Toyota Central R&D Labs. [email protected] Abstract The evolution of biological sequences, such as proteins or DNAs, is driven by the three basic edit operations: substitution, insertion, and deletion. Motivated by the recent progress of neural network models for biological tasks, we implement two neural network architectures that can treat such edit operations. The first proposal is the edit invariant neural networks, based on differentiable Needleman-Wunsch algorithms. The second is the use of deep CNNs with concatenations. Our analysis shows that CNNs can recognize regular expressions without Kleene star, and that deeper CNNs can recognize more complex regular expressions including the insertion/deletion of characters. The experimental results for the protein secondary structure prediction task suggest the importance of insertion/deletion. The test accuracy on the widely-used CB513 dataset is 71.5%, which is 1.2-points better than the current best result on non-ensemble models. 1 Introduction Neural networks are now used in many applications, not limited to classical fields such as image processing, speech recognition, and natural language processing. Bioinformatics is becoming an important application field of neural networks. These biological applications are often implemented as a supervised learning model that takes a biological string (such as DNA or protein) as an input, and outputs the corresponding label(s), such as a protein secondary structure [13, 14, 15, 18, 19, 23, 24, 26], protein contact maps [4, 8], and genome accessibility [12].
    [Show full text]
  • Theory of Computation
    Theory of Computation Alexandre Duret-Lutz [email protected] September 10, 2010 ADL Theory of Computation 1 / 121 References Introduction to the Theory of Computation (Michael Sipser, 2005). Lecture notes from Pierre Wolper's course at http://www.montefiore.ulg.ac.be/~pw/cours/calc.html (The page is in French, but the lecture notes labelled chapitre 1 to chapitre 8 are in English). Elements of Automata Theory (Jacques Sakarovitch, 2009). Compilers: Principles, Techniques, and Tools (A. Aho, R. Sethi, J. Ullman, 2006). ADL Theory of Computation 2 / 121 Introduction What would be your reaction if someone came at you to explain he has invented a perpetual motion machine (i.e. a device that can sustain continuous motion without losing energy or matter)? You would probably laugh. Without looking at the machine, you know outright that such the device cannot sustain perpetual motion. Indeed the laws of thermodynamics demonstrate that perpetual motion devices cannot be created. We know beforehand, from scientic knowledge, that building such a machine is impossible. The ultimate goal of this course is to develop similar knowledge for computer programs. ADL Theory of Computation 3 / 121 Theory of Computation Theory of computation studies whether and how eciently problems can be solved using a program on a model of computation (abstractions of a computer). Computability theory deals with the whether, i.e., is a problem solvable for a given model. For instance a strong result we will learn is that the halting problem is not solvable by a Turing machine. Complexity theory deals with the how eciently.
    [Show full text]
  • A Representation-Based Approach to Connect Regular Grammar and Deep Learning
    The Pennsylvania State University The Graduate School A REPRESENTATION-BASED APPROACH TO CONNECT REGULAR GRAMMAR AND DEEP LEARNING A Dissertation in Information Sciences and Technology by Kaixuan Zhang ©2021 Kaixuan Zhang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2021 The dissertation of Kaixuan Zhang was reviewed and approved by the following: C. Lee Giles Professor of College of Information Sciences and Technology Dissertation Adviser Chair of Committee Kenneth Huang Assistant Professor of College of Information Sciences and Technology Shomir Wilson Assistant Professor of College of Information Sciences and Technology Daniel Kifer Associate Professor of Department of Computer Science and Engineering Mary Beth Rosson Professor of Information Sciences and Technology Director of Graduate Programs, College of Information Sciences and Technology ii Abstract Formal language theory has brought amazing breakthroughs in many traditional areas, in- cluding control systems, compiler design, and model verification, and continues promoting these research directions. As recent years have witnessed that deep learning research brings the long-buried power of neural networks to the surface and has brought amazing break- throughs, it is crucial to revisit formal language theory from a new perspective. Specifically, investigation of the theoretical foundation, rather than a practical application of the con- necting point obviously warrants attention. On the other hand, as the spread of deep neural networks (DNN) continues to reach multifarious branches of research, it has been found that the mystery of these powerful models is equally impressive as their capability in learning tasks. Recent work has demonstrated the vulnerability of DNN classifiers constructed for many different learning tasks, which opens the discussion of adversarial machine learning and explainable artificial intelligence.
    [Show full text]
  • 6.035 Lecture 2, Specifying Languages with Regular Expressions and Context-Free Grammars
    MIT 6.035 Specifying Languages with Regular Expressions and Context-Free Grammars Martin Rinard Laboratory for Computer Science Massachusetts Institute of Technology Language Definition Problem • How to precisely define language • Layered struc ture of langua ge defifinitiition • Start with a set of letters in language • Lexical structurestructure - identifies “words ” in language (each word is a sequence of letters) • Syntactic sstructuretructure - identifies “sentences ” in language (each sentence is a sequence of words) • Semantics - meaning of program (specifies what result should be for each input) • Today’s topic: lexical and syntactic structures Specifying Formal Languages • Huge Triumph of Computer Science • Beautiful Theoretical Results • Practical Techniques and Applications • Two Dual Notions • Generative approach ((grammar or regular expression) • Recognition approach (automaton) • Lots of theorems about convertingconverting oonene approach automatically to another Specifying Lexical Structure Using Regular ExpressionsExpressions • Have some alphabet ∑ = set of letters • R egular expressions are bu ilt from: • ε -empty string • AAnyn yl lettere tte rf fromr oma alphabetlp ha be t ∑ •r1r2 – regular expression r1 followed by r2 (sequence) •r1| r2 – either regular expression r1 or r2 (choice) • r* - iterated sequence and choice ε | r | rr | … • Parentheses to indicate grouping/precedence Concept of Regular Expression Generating a StringString Rewrite regular expression until have only a sequence of letters (string) left Genera
    [Show full text]
  • Constraints for Membership in Formal Languages Under Systematic Search and Stochastic Local Search
    Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1027 Constraints for Membership in Formal Languages under Systematic Search and Stochastic Local Search JUN HE ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-8617-4 UPPSALA urn:nbn:se:uu:diva-196347 2013 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, Friday, April 26, 2013 at 13:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Abstract He, J. 2013. Constraints for Membership in Formal Languages under Systematic Search and Stochastic Local Search. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1027. 74 pp. Uppsala. ISBN 978-91-554-8617-4. This thesis focuses on constraints for membership in formal languages under both the systematic search and stochastic local search approaches to constraint programming (CP). Such constraints are very useful in CP for the following three reasons: They provide a powerful tool for user- level extensibility of CP languages. They are very useful for modelling complex work shift regulation constraints, which exist in many shift scheduling problems. In the analysis, testing, and verification of string-manipulating programs, string constraints often arise. We show in this thesis that CP solvers with constraints for membership in formal languages are much more suitable than existing solvers used in tools that have to solve string constraints. In the stochastic local search approach to CP, we make the following two contributions: We introduce a stochastic method of maintaining violations for the regular constraint and extend our method to the automaton constraint with counters.
    [Show full text]
  • CIT 425- AUTOMATA THEORY, COMPUTABILITY and FORMAL LANGUAGES LECTURE NOTE by DR. OYELAMI M. O. Introduction • This Course Cons
    CIT 425- AUTOMATA THEORY, COMPUTABILITY AND FORMAL LANGUAGES LECTURE NOTE BY DR. OYELAMI M. O. Status: Core Description: Words and String. Concatenation, word Length; Language Definition. Regular Expression, Regular Language, Recursive Languages; Finite State Automata (FSA), State Diagrams; Pumping Lemma, Grammars, Applications in Computer Science and Engineering, Compiler Specification and Design, Text Editor and Implementation, Very Large Scale Integrated (VLSI) Circuit Specification and Design, Natural Language Processing (NLP) and Embedded Systems. Introduction This course constitutes the theoretical foundation of computer science. Loosely speaking we can think of automata, grammars, and computability as the study of what can be done by computers in principle, while complexity addresses what can be done in practice. This course has applications in the following areas: o Digital design, o Programming languages o Compilers construction Languages Dictionaries define the term informally as a system suitable for the expression of certain ideas, facts, or concepts, including a set of symbols and rules for their manipulation. While this gives us an intuitive idea of what a language is, it is not sufficient as a definition for the study of formal languages. We need a precise definition for the term. A formal language is an abstraction of the general characteristics of programming languages. Languages can be specified in various ways. One way is to list all the words in the language. Another is to give some criteria that a word must satisfy to be in the language. Another important way is to specify a language through the use of some terminologies: 1 Alphabet: A finite, nonempty set Σ of symbols.
    [Show full text]
  • Regular Languages and Finite Automata for Part IA of the Computer Science Tripos
    N Lecture Notes on Regular Languages and Finite Automata for Part IA of the Computer Science Tripos Prof. Andrew M. Pitts Cambridge University Computer Laboratory c 2010 A. M. Pitts First Edition 1998. Revised 1999, 2000, 2001, 2002, 2003, 2005, 2006, 2007, 2008, 2009, 2010. Contents Learning Guide ii 1 Regular Expressions 1 1.1 Alphabets, strings, and languages . ...... 1 1.2 Patternmatching ................................ 4 1.3 Some questions about languages . ... 6 1.4 Exercises .................................... 8 2 Finite State Machines 11 2.1 Finiteautomata ................................. 11 2.2 Determinism, non-determinism, and ε-transitions . 14 2.3 Asubsetconstruction . .... .... .... ..... .... .... ... 17 2.4 Summary .................................... 20 2.5 Exercises .................................... 20 3 Regular Languages, I 23 3.1 Finite automata from regular expressions . ........ 23 3.2 Decidabilityofmatching . 28 3.3 Exercises .................................... 30 4 Regular Languages, II 31 4.1 Regular expressions from finite automata . ..... 31 4.2 Anexample ................................... 32 4.3 Complement and intersection of regular languages . ......... 34 4.4 Exercises .................................... 36 5 The Pumping Lemma 39 5.1 ProvingthePumpingLemma. 40 5.2 UsingthePumpingLemma.. .... .... ..... .... .... .... 41 5.3 Decidability of language equivalence . ....... 44 5.4 Exercises .................................... 45 6 Grammars 47 6.1 Context-freegrammars . 47 6.2 Backus-NaurForm ..............................
    [Show full text]
  • QUESTION BANK SOLUTION Unit 1 Introduction to Finite Automata
    FLAT 10CS56 QUESTION BANK SOLUTION 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 number of a’s and b’s.( 5m )(Jun-Jul 10) L = {Œ,aabb,abab,baba,baab,bbaa,aabbaa,---------} 3. Give Applications of Finite Automata. (5m )(Jun-Jul 10) String Processing Consider finding all occurrences of a short string (pattern string) within a long string (text string). This can be done by processing the text through a DFA: the DFA for all strings that end with the pattern string. Each time the accept state is reached, the current position in the text is output. Finite-State Machines A finite-state machine is an FA together with actions on the arcs. Statecharts Statecharts model tasks as a set of states and actions. They extend FA diagrams. Lexical Analysis Dept of CSE, SJBIT 1 FLAT 10CS56 In compiling a program, the first step is lexical analysis. This isolates keywords, identifiers etc., while eliminating irrelevant symbols. A token is a category, for example “identifier”, “relation operator” or specific keyword. 4. Define DFA, NFA & Language? (5m)( Jun-Jul 10) Deterministic finite automaton (DFA)—also known as deterministic finite state machine—is a finite state machine that accepts/rejects finite strings of symbols and only produces a unique computation (or run) of the automaton for each input string. 'Deterministic' refers to the uniqueness of the computation. Nondeterministic finite automaton (NFA) or nondeterministic finite state machine is a finite state machine where from each state and a given input symbol the automaton may jump into several possible next states.
    [Show full text]
  • Formal Languages
    CSCE 222 Discrete Structures for Computing ! Formal Languages Hyunyoung Lee ! ! ! ! ! ! Based on slides by Andreas Klappenecker !1 Motivation The syntax of programming languages (such as Algol, C, Java) are described by phrase structure grammars. The grammars are also used in the specification of data exchange formats. A grammar specifies a formal language. The advantage is that a fairly small grammar can describe a fairly complex language. Formal languages are also a convenient tool in computational complexity theory, as we will see. !2 Notation Let V be a set. We denote by V* the set of strings over V. ! A language over the alphabet V is a subset of V*. ! The empty string is denoted by λ. !3 Phrase-Structure Grammars A phrase-structure grammar G consists of - a set V called the vocabulary, - a subset T of V consisting of terminal symbols, [N=V\T is called the set of nonterminal symbols] - a distinguished nonterminal element S in N, called start symbol - and a finite set of productions (or rules). We denote this data by G=(V,T,S,P). !4 Productions The productions are term-rewriting rules that specify how a part of string can be modified. A production rule is of the form A -> B where A is string in V* containing at least one nonterminal symbol, and B is a string in V*. The production rule A -> B specifies that A can be replaced by B within a string. !5 Example Grammar Rules (Part of C++ Grammar) selection-statement: A.5 Statements if ( condition ) statement statement: Words such as if ( condition ) statement else statement labeled-statement
    [Show full text]