13 Context-Free Languages
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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? i.e., something that will accept (or reject) strings that belong (or do not belong) to the language? 2 Context-Free Languages A language class larger than the class of regular languages Supports natural, recursive notation called “context- free grammar” Applications: Context- Parse trees, compilers Regular free XML (FA/RE) (PDA/CFG) 3 An Example A palindrome is a word that reads identical from both ends E.g., madam, redivider, malayalam, 010010010 Let L = { w | w is a binary palindrome} Is L regular? No. Proof: N N (assuming N to be the p/l constant) Let w=0 10 k By Pumping lemma, w can be rewritten as xyz, such that xy z is also L (for any k≥0) But |xy|≤N and y≠ + ==> y=0 k ==> xy z will NOT be in L for k=0 ==> Contradiction 4 But the language of palindromes… is a CFL, because it supports recursive substitution (in the form of a CFG) This is because we can construct a “grammar” like this: Same as: 1. A ==> A => 0A0 | 1A1 | 0 | 1 | Terminal 2. A ==> 0 3. A ==> 1 Variable or non-terminal Productions 4. A ==> 0A0 5. A ==> 1A1 How does this grammar work? 5 How does the CFG for palindromes work? An input string belongs to the language (i.e., accepted) iff it can be generated by the CFG G: Example: w=01110 A => 0A0 | 1A1 | 0 | 1 | G can generate w as follows: Generating a string from a grammar: 1. -
Ambiguity Detection Methods for Context-Free Grammars
Ambiguity Detection Methods for Context-Free Grammars H.J.S. Basten Master’s thesis August 17, 2007 One Year Master Software Engineering Universiteit van Amsterdam Thesis and Internship Supervisor: Prof. Dr. P. Klint Institute: Centrum voor Wiskunde en Informatica Availability: public domain Contents Abstract 5 Preface 7 1 Introduction 9 1.1 Motivation . 9 1.2 Scope . 10 1.3 Criteria for practical usability . 11 1.4 Existing ambiguity detection methods . 12 1.5 Research questions . 13 1.6 Thesis overview . 14 2 Background and context 15 2.1 Grammars and languages . 15 2.2 Ambiguity in context-free grammars . 16 2.3 Ambiguity detection for context-free grammars . 18 3 Current ambiguity detection methods 19 3.1 Gorn . 19 3.2 Cheung and Uzgalis . 20 3.3 AMBER . 20 3.4 Jampana . 21 3.5 LR(k)test...................................... 22 3.6 Brabrand, Giegerich and Møller . 23 3.7 Schmitz . 24 3.8 Summary . 26 4 Research method 29 4.1 Measurements . 29 4.1.1 Accuracy . 29 4.1.2 Performance . 31 4.1.3 Termination . 31 4.1.4 Usefulness of the output . 31 4.2 Analysis . 33 4.2.1 Scalability . 33 4.2.2 Practical usability . 33 4.3 Investigated ADM implementations . 34 5 Analysis of AMBER 35 5.1 Accuracy . 35 5.2 Performance . 36 3 5.3 Termination . 37 5.4 Usefulness of the output . 38 5.5 Analysis . 38 6 Analysis of MSTA 41 6.1 Accuracy . 41 6.2 Performance . 41 6.3 Termination . 42 6.4 Usefulness of the output . 42 6.5 Analysis . -
Formal Language and Automata Theory (CS21004)
Context Free Grammar Normal Forms Derivations and Ambiguities Pumping lemma for CFLs PDA Parsing CFL Properties Formal Language and Automata Theory (CS21004) Soumyajit Dey CSE, IIT Kharagpur Context Free Formal Language and Automata Theory Grammar (CS21004) Normal Forms Derivations and Ambiguities Pumping lemma Soumyajit Dey for CFLs CSE, IIT Kharagpur PDA Parsing CFL Properties DPDA, DCFL Membership CSL Soumyajit Dey CSE, IIT Kharagpur Formal Language and Automata Theory (CS21004) Context Free Grammar Normal Forms Derivations and Ambiguities Pumping lemma for CFLs PDA Parsing CFL Properties Formal Language and Automata Announcements Theory (CS21004) Soumyajit Dey CSE, IIT Kharagpur Context Free Grammar Normal Forms Derivations and The slide is just a short summary Ambiguities Follow the discussion and the boardwork Pumping lemma for CFLs Solve problems (apart from those we dish out in class) PDA Parsing CFL Properties DPDA, DCFL Membership CSL Soumyajit Dey CSE, IIT Kharagpur Formal Language and Automata Theory (CS21004) Context Free Grammar Normal Forms Derivations and Ambiguities Pumping lemma for CFLs PDA Parsing CFL Properties Formal Language and Automata Table of Contents Theory (CS21004) Soumyajit Dey 1 Context Free Grammar CSE, IIT Kharagpur 2 Normal Forms Context Free Grammar 3 Derivations and Ambiguities Normal Forms 4 Derivations and Pumping lemma for CFLs Ambiguities 5 Pumping lemma PDA for CFLs 6 Parsing PDA Parsing 7 CFL Properties CFL Properties DPDA, DCFL 8 DPDA, DCFL Membership 9 Membership CSL 10 CSL Soumyajit Dey CSE, -
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 Automata 1 Parsing The process of parsing a program is partly about confirming that a given program is well-formed – syntax. But it is also about representing the structure of the program so that it can be executed – semantics. For this purpose, the trail of sentential forms created en route to generating a given sentence is just as important as the question of whether the sentence can be generated or not. COMP 2600 — Context Free Languages and Pushdown Automata 2 The Semantics of Parses Take the code if e1 then if e2 then s1 else s2 where e1, e2 are boolean expressions and s1, s2 are subprograms. Does this mean if e1 then( if e2 then s1 else s2) or if e1 then( if e2 else s1) else s2 We’d better have an unambiguous way to tell which is right, or we cannot know what the program will do at runtime! COMP 2600 — Context Free Languages and Pushdown Automata 3 Ambiguity Recall that we can present CFG derivations as parse trees. Until now this was mere pretty presentation; now it will become important. A context-free grammar G is unambiguous iff every string can be derived by at most one parse tree. G is ambiguous iff there exists any word w 2 L(G) derivable by more than one parse tree. COMP 2600 — Context Free Languages and Pushdown Automata 4 Example: If-Then and If-Then-Else Consider the CFG S ! if bexp then S j if bexp then S else S j prog where bexp and prog stand for boolean expressions and (if-statement free) programs respectively, defined elsewhere. -
Topics in Context-Free Grammar CFG's
Topics in Context-Free Grammar CFG’s HUSSEIN S. AL-SHEAKH 1 Outline Context-Free Grammar Ambiguous Grammars LL(1) Grammars Eliminating Useless Variables Removing Epsilon Nullable Symbols 2 Context-Free Grammar (CFG) Context-free grammars are powerful enough to describe the syntax of most programming languages; in fact, the syntax of most programming languages is specified using context-free grammars. In linguistics and computer science, a context-free grammar (CFG) is a formal grammar in which every production rule is of the form V → w Where V is a “non-terminal symbol” and w is a “string” consisting of terminals and/or non-terminals. The term "context-free" expresses the fact that the non-terminal V can always be replaced by w, regardless of the context in which it occurs. 3 Definition: Context-Free Grammars Definition 3.1.1 (A. Sudkamp book – Language and Machine 2ed Ed.) A context-free grammar is a quadruple (V, Z, P, S) where: V is a finite set of variables. E (the alphabet) is a finite set of terminal symbols. P is a finite set of rules (Ax). Where x is string of variables and terminals S is a distinguished element of V called the start symbol. The sets V and E are assumed to be disjoint. 4 Definition: Context-Free Languages A language L is context-free IF AND ONLY IF there is a grammar G with L=L(G) . 5 Example A context-free grammar G : S aSb S A derivation: S aSb aaSbb aabb L(G) {anbn : n 0} (((( )))) 6 Derivation Order 1. -
Context Free Languages
Context Free Languages COMP2600 — Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2016 Slides by Katya Lebedeva and Ranald Clouston. COMP 2600 — Context Free Languages 1 Ambiguity The definition of CF grammars allow for the possibility of having more than one structure for a given sentence. This ambiguity may make the meaning of a sentence unclear. A context-free grammar G is unambiguous iff every string can be derived by at most one parse tree. G is ambiguous iff there exists any word w 2 L(G) derivable by more than one parse trees. COMP 2600 — Context Free Languages 2 Dangling else Take the code if e1 then if e2 then s1 else s2 where e1, e2 are boolean expressions and s1, s2 are subprograms. Does this mean if e1 then( if e2 then s1 else s2) or if e1 then( if e2 then s1) else s2 The dangling else is a problem in computer programming in which an optional else clause in an “ifthen(else)” statement results in nested conditionals being ambiguous. COMP 2600 — Context Free Languages 3 This is a problem that often comes up in compiler construction, especially parsing. COMP 2600 — Context Free Languages 4 Inherently Ambiguous Languages Not all context-free languages can be given unambiguous grammars – some are inherently ambiguous. Consider the language L = faib jck j i = j or j = kg How do we know that this is context-free? First, notice that L = faibickg [ faib jc jg We then combine CFGs for each side of this union (a standard trick): S ! T j W T ! UV W ! XY U ! aUb j e X ! aX j e V ! cV j e Y ! bYc j e COMP 2600 — Context Free Languages 5 The problem with L is that its sub-languages faibickg and faib jc jg have a non-empty intersection. -
Ambiguity Detection for Context-Free Grammars in Eli
Fakultät für Elektrotechnik, Informatik und Mathematik Bachelor’s Thesis Ambiguity Detection for Context-Free Grammars in Eli Michael Kruse Student Id: 6284674 E-Mail: [email protected] Paderborn, 7th May, 2008 presented to Dr. Peter Pfahler Dr. Matthias Fischer Statement on Plagiarism and Academic Integrity I declare that all material in this is my own work except where there is clear acknowledgement or reference to the work of others. All material taken from other sources have been declared as such. This thesis as it is or in similar form has not been presented to any examination authority yet. Paderborn, 7th May, 2008 Michael Kruse iii iv Acknowledgement I especially whish to thank the following people for their support during the writing of this thesis: • Peter Pfahler for his work as advisor • Sylvain Schmitz for finding some major errors • Anders Møller for some interesting conversations • Ulf Schwekendiek for his help to understand Eli • Peter Kling and Tobias Müller for proofreading v vi Contents 1. Introduction1 2. Parsing Techniques5 2.1. Context-Free Grammars.........................5 2.2. LR(k)-Parsing...............................7 2.3. Generalised LR-Parsing......................... 11 3. Ambiguity Detection 15 3.1. Ambiguous Context-Free Grammars.................. 15 3.1.1. Ambiguity Approximation.................... 16 3.2. Ambiguity Checking with Language Approximations......... 16 3.2.1. Horizontal and Vertical Ambiguity............... 17 3.2.2. Approximation of Horizontal and Vertical Ambiguity..... 22 3.2.3. Approximation Using Regular Grammars........... 24 3.2.4. Regular Supersets........................ 26 3.3. Detection Schemes Based on Position Automata........... 26 3.3.1. Regular Unambiguity...................... 31 3.3.2. -
Lecture 7-8 — Context-Free Grammars and Bottom-Up Parsing
Lecture 7-8 | Context-free grammars and bottom-up parsing • Grammars and languages • Ambiguity and expression grammars • Parser generators • Bottom-up (shift/reduce) parsing • ocamlyacc example • Handling ambiguity in ocamlyacc CS 421 | Class 7{8, 2/7/12-2/9/12 | 1 Grammar for (almost) MiniJava Program -> ClassDeclList ClassDecl -> class id { VarDeclList MethodDeclList } VarDecl -> Type id ; MethodDecl -> Type id ( FormalList ) { VarDeclList StmtList return Exp ; } Formal -> Type id Type -> int [ ] | boolean | int | id Stmt -> { StmtList } | if ( Exp ) Stmt else Stmt | while ( Exp ) Stmt | System.out.println ( Exp ) ; | id = Exp ; | id [ Exp ] = Exp ; Exp -> Exp Op Exp | Exp [ Exp ] | Exp . length | Exp . id ( ExpList ) | integer | true | false | id | this | new int [ Exp ] | new id ( ) | ! Exp | ( Exp ) Op -> && | < | + | - | * ExpList -> Exp ExpRest | ExpRest -> , Exp ExpRest | FormalList -> Type id FormalRest | FormalRest -> , Type id FormalRest | ClassDeclList = ClassDeclList VarDecl | MethodDeclList = MethodDeclList MethodDecl | VarDeclList = VarDeclList VarDecl | StmtList = StmtList Stmt | CS 421 | Class 7{8, 2/7/12-2/9/12 | 2 Grammars and languages • E.g. program class C {int f () { return 0; }} has this parse tree: • Parser converts source file to parse tree. AST is easily calcu- lated from parse tree, or constructed while parsing (without explicitly constructing parse tree). CS 421 | Class 7{8, 2/7/12-2/9/12 | 3 Context-free grammar (cfg) notation • CFG is a set of productions A ! X1X2 :::Xn (n ≥ 0). If n = 0, we may write either -
(Or Simply a Grammar) Is (V, T, P, S), Where (I) V Is a Finite Nonempty Set Whose Elements Are Called Variables
UNIT - III GRAMMARS FORMALISM Definition of Grammar: A phrase-Structure grammar (or Simply a grammar) is (V, T, P, S), Where (i) V is a finite nonempty set whose elements are called variables. (ii) T is finite nonempty set, whose elements are called terminals. (iii) S is a special variable (i.e an element of V ) called the start symbol, and (iv) P is a finite set whose elements are ,where and are strings on V T, has at least one symbol from V. Elements of P are called Productions or production rules or rewriting rules. 1. Regular Grammar: A grammar G=(V,T,P,S) is said to be Regular grammar if the productions are in either right-linear grammar or left-linear grammar. ABx|xB|x 1.1 Right –Linear Grammar: A grammar G=(V,T,P,S) is said to be right-linear if all productions are of the form AxB or Ax. Where A,B V and x T*. 1.2 Left –Linear Grammar: A grammar G=(V,T,P,S) is said to be Left-linear grammar if all productions are of the form ABx or Ax Example: 1. Construct the regular grammar for regular expression r=0(10)*. Sol: Right-Linear Grammar: S0A A10A| Computer Science & Engineering Formal Languages and Automata Theory Left-Linear Grammar: SS10|0 2. Equivalence of NFAs and Regular Grammar 2.1 Construction of -NFA from a right –linear grammar: Let G=(V,T,P,S) be a right-linear grammar .we construct an NFA with -moves, M=(Q,T, ,[S],[ ]) that simulates deviation in ‘G’ Q consists of the symbols [ ] such that is S or a(not necessarily proper)suffix of some Right-hand side of a production in P. -
Principles of Programming Languages Lexics Vs. Syntax Vs. Semantics
Software II: Principles of Programming Languages Lecture 3 – Formal Descriptions of a Programming Language Lexics vs. Syntax Vs. Semantics • Lexics refers to issues regarding the assembly of words that comprise a statement. • Syntax refers to issues regarding the grammar of a statement. • Semantics refers to issues regarding the meaning of a statement. Lexical Structure of Programming Languages • It was believed in the early days of programming language development that it was sufficient to be able specify the syntax of a programming language. We now know that this is not enough. • This led to the development of context-free grammars and Backus-Naur Form. Programming Language Syntax • Syntax is defined as “the arrangement of words as elements in a sentence to show their relationship.” • Syntax provides a great deal of information that we need to understand a program and to guide its translation. • Example 2 + 3 ´ 4 = 14 (not 20 – multiplication takes precedence) Programming Language Semantics • Semantics is defined as “the meaning of a symbol or set of symbols.” • This is equally important in translating a programming correctly and may be more difficult to express unambiguously. Tokens • The lexical structure of program consists of sequence of characters that are assembled into character strings called lexemes which have directly related to tokens, the element of a languages grammar to which they correspond. • Tokens fall into several distinct categories: – reserved words – literals or constants – special symbols such as < = + – identifiers, such as x24, average, balance Reserved Words and Standard Identifiers • Reserved words serve a special purpose within the syntax of a language; for this reason, they are generally not allowed to be used as user-defined identifiers. -
Arxiv:1911.05672V2 [Cs.FL] 4 Feb 2020 Tax of a Language Must Be Defined [2,10,15,29,32]
Resolvable Ambiguity? Viktor Palmkvist1, Elias Castegren1, Philipp Haller1, and David Broman1 KTH Royal Institute of Technology, 100 44 Stockholm, Sweden fvipa,eliasca,dbro,[email protected] Abstract. A common standpoint when designing the syntax of pro- gramming languages is that the grammar definition has to be unambigu- ous. However, requiring up front unambiguous grammars can force lan- guage designers to make more or less arbitrary choices to disambiguate the language. In this paper, we depart from the traditional view of un- ambiguous grammar design, and enable the detection of ambiguities to be delayed until parse time, allowing the user of the language to perform the disambiguation. A natural decision problem follows: given a language definition, can a user always disambiguate an ambiguous program? We introduce and formalize this fundamental problem|called the resolvable ambiguity problem|and divide it into separate static and dynamic re- solvability problems. We provide solutions to the static problem for a restricted language class and sketch proofs of soundness and complete- ness. We also provide a sound and complete solution to the dynamic problem for a much less restricted class of languages. The approach is evaluated through two separate case studies, covering both a large ex- isting programming language, and the composability of domain-specific languages. Keywords: Syntax · Ambiguity · Grammars · Parsing · Domain-specific languages 1 Introduction Ever since the early 60s, it has been known that determining whether a context- free grammar is ambiguous is undecidable [7]. As a consequence, a large number of restricted grammars have been developed to guarantee that language defi- nitions are unambiguous. -
A Complex Measure for Linear Grammars‡
1 Demonstratio Mathematica, Vol. 38, No. 3, 2005, pp 761-775 A Complex Measure for Linear Grammars‡ Ishanu Chattopadhyay Asok Ray [email protected] [email protected] The Pennsylvania State University University Park, PA 16802, USA Keywords: Language Measure; Discrete Event Supervisory Control; Linear Grammars Abstract— The signed real measure of regular languages, introduced supervisors that satisfy their own respective specifications. Such and validated in recent literature, has been the driving force for a partially ordered set of sublanguages requires a quantitative quantitative analysis and synthesis of discrete-event supervisory (DES) measure for total ordering of their respective performance. To control systems dealing with finite state automata (equivalently, regular languages). However, this approach relies on memoryless address this issue, a signed real measures of regular languages state-based tools for supervisory control synthesis and may become has been reported in literature [7] to provide a mathematical inadequate if the transitions in the plant dynamics cannot be captured framework for quantitative comparison of controlled sublanguages by finitely many states. From this perspective, the measure of regular of the unsupervised plant language. This measure provides a languages needs to be extended to that of non-regular languages, total ordering of the sublanguages of the unsupervised plant such as Petri nets or other higher level languages in the Chomsky hierarchy. Measures for non-regular languages has not apparently and formalizes a procedure for synthesis of DES controllers for been reported in open literature and is an open area of research. finite state automaton plants, as an alternative to the procedure This paper introduces a complex measure of linear context free of Ramadge and Wonham [5].