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C O N T E X T - F R E E LANGUAGES ( use a grammar to describe a language) 1

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Context-free grammars Ch.1 introduced two different, though equivalent, methods of describing languages: finite automata and regular expressions. Languages can be described in this way but that some simple languages, such as (0 n 1 n | n > 0}, cannot. CONTEXT-FREE GRAMMARS (C.F.L.) powerful method of describing languages. used in the study of human languages. understanding the relationship of terms such as noun, verb, and preposition. An important application of context-free grammars occurs in the specification and compilation of programming languages. 3

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Context-free grammars (cont.) A grammar for a programming language often appears as a reference for people trying to learn the language syntax. Designers of compilers and interpreters for programming languages often start by obtaining a grammar for the language. Most compilers and interpreters contain a component called a parser that extracts the meaning of a program prior to generating the compiled code or performing the interpreted execution. 4

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Context free languages The collection of languages associated with context-free grammars are called the context- free languages. 5

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Objectives To give a formal definition of context-free grammars and study the properties of context-free languages. To introduce pushdown automata, a class of machines recognizing the context-free languages. 6

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CONTEXT-FREE GRAMMARS The following is an example of a context-free grammar, which we call G1: A grammar consists of a collection of substitution rules, also called productions. Each rule appears as a line in the grammar, comprising a symbol and a string separated by an arrow. 7

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A → β The symbol is called a variable. The string consists of variables and other symbols called terminals. The variable symbols often are represented by capital letters. The terminals are (input alphabet) often represented by lowercase letters, numbers, or special symbols. One variable is designated as the start variable (left-hand side of the topmost rule.). 8

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Rule Any production of the form A → β. β can therefore be any string of terminal and non- terminal elements. Example: A → BC A → a 9 start variable terminals variable symbols

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Example Example grammar G1: Grammar G1 contains three rules. G1 's variables are A and B, where A is the start variable. Its terminals are 0, 1, and #. 10

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Describing a language a grammar is used to describe a language by generating each string of that language in the following manner: 1.Write down the start variable. (left-hand side of the top rule, unless specified otherwise). 2.Find a variable that is written down and a rule that starts with that variable. Replace the written down variable with the right-hand side of that rule. 3.Repeat step 2 until no variables remain. 11

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Abbreviation abbreviate several rules with the same left- hand variable, such as A 0A1 and A B, into a single line A 0A1 I B, using the symbol " I " as an "or.“ A 0A1 or B. 12

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Derivation The sequence of substitutions to obtain a string is called a derivation. A derivation of string 000#111 in grammar G1 is: 13 parse tree

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Grammar G2 has ?? – rules?? - variables ?? - terminals??. 14

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Grammar G2 Grammar G2 has: 10 variables (the capitalized grammatical terms written inside brackets); 27 terminals (the standard English alphabet plus a space character); 18 rules. 15

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Derivation Each of these strings has a derivation in grammar G2. The following is a derivation of the first string on this list. 16

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FORMAL DEFINITION OF A CONTEXT- FREE GRAMMAR 17

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Example Grammar G1 18 In grammar G2

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Ambiguity The generation of a sentence by a context-free grammar can be represented by a tree diagram. Not only one way in which a sentence can be derived. 19

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Example Let G be a context-free grammar with the following productions: 1. S → AB 5. Β → Sd 2. S → CD 6. C → aS 3. S → bc 7. D → d 4. A → a The sentence abcd can be derived from this grammar?????????????????? 20

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solution & Derivation tree The sentence abcd can be derived from this grammar as follows: 1. S ⇒ AB ⇒ aB ⇒ aSd ⇒ abcd. 2. S ⇒ AB ⇒ ASd ⇒ Abcd ⇒ abcd, 3. S ⇒ CD ⇒ aSD ⇒ abcD ⇒ abcd. (Alternative) 21 Derivation tree for the sentence abcd -1&2 Derivation tree for the sentence abcd -3 true or false????? c

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Ambiguity (cont.) If a grammar generates the same string in several different ways, we say that the string is derived ambiguously in that grammar. If a grammar generates some string ambiguously we say that the grammar is ambiguous. 22

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Example For example, consider grammar G5: This grammar generates the string a+axa ambiguously?????????????? Yes (two different parse trees) 23

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DEFINITION 2.7 A string w is derived ambiguously in context- free grammar G if it has two or more different leftmost derivations. Grammar G is ambiguous if it generates some string ambiguously. 24

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Chomsky normal form 25

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EXAMPLE 2.10 Let G6 be the following CFG and convert it to Chomsky normal form by using the conversion procedure just given. The series of grammars presented illustrates the steps in the conversion. Rules shown in bold have just been added. Rules shown in gray have just been removed. 27

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Context-Free Languages {0 n 1 n I n > 0} is not regular??? 31 Example: G1 A → 0A1 A → B B → # A ⇒ 0A1 ⇒ 00A11 ⇒ 000A111 ⇒ 000B111 ⇒ 000#111 L(G1) = {0 n #1 n | n ≥ 0 } parse tree Can the following grammar generate the subsequent language? 000#111

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A Grammar for Arithmetic Expressions Let : X = {E, T, F, id, +, -,*,/,(,), a, b, c} T = {a, b, c, +, -,*,/,(,)}. The start symbol S is E and the productions are as follows 32 Write a derivation of string (a + b)*c

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33 Write a derivation of string (a + b)*c The derivation of string (a + b)*c:

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PUSHDOWN AUTOMATA It’s a new type of computational model called pushdown automata. These automata are like nondeterministic finite automata but have an extra component called a stack. The stack provides additional memory beyond the finite amount available in the control. 34

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35 The following figure is a schematic representation of a finite automaton. The control represents the states and transition function, the tape contains the input string, and the arrow represents the input head, pointing at the next input symbol to be read. Schematic of a finite automaton

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36 Schematic of a pushdown automaton With the addition of a stack component we obtain a schematic representation of a pushdown automaton, as shown in the following figure

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PUSHDOWN AUTOMATA A pushdown automaton (PDA) can write symbols on the stack and read them back later. Writing a symbol "pushes down" all the other symbols on the stack. At any time the symbol on the top of the stack can be read and removed. Writing a symbol on the stack is often referred to as pushing the symbol, and removing a symbol is referred to as popping it. 38

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PUSHDOWN AUTOMATA a stack is a "last in, first out" storage device. If certain information is written on the stack and additional information is written afterward, the earlier information becomes inaccessible until the later information is removed. A stack is valuable because it can hold an unlimited amount of information. 39

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PUSHDOWN AUTOMATA Recall that a finite automaton is unable to recognize the language {0 n 1 n I n > 0} because it cannot store very large numbers in its finite memory. A PDA is able to recognize this language because it can use its stack to store the number of Os it has seen. Thus the unlimited nature of a stack allows the PDA to store numbers of unbounded size. 40

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Informal description The following informal description shows how the automaton for this language works: –Read symbols from the input. As each 0 is read, push it onto the stack. –Pop a 0 off the stack for each 1 read. –If reading the input is finished exactly when the stack becomes empty of 0 s, accept the input. –If the stack becomes empty while is remain or if the is are finished while the stack still contains 0 s or if any 0 s appear in the input following is, reject the input. 41

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FORMAL DEFINITION OF A PUSHDOWN AUTOMATON 42

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Example aaabba = reject?? 46

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Example ab#ab = accept?? Note that it merely records a's and b's on the stack until it reaches the marker (#) and then checks them off against the remainder of the input. 47

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48 Let P be defined through X = { a, b, c }, Z = { z A = z 1, z 2, z 3 }, S = { S A, S, }, Z F = { z 3 } and, finally the state transitions

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Example a 3 bc 3 = aaabccc?? 49 ((a, z 1, S A ),(z 1, SS A )) ((a, z 1, S),(z 1, SS)) ((b, z 1, S),(z 2,λ)) ((c, z 2, S),(z 2, λ)) ((c, z 2, S A ),(z 3, λ))

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50 Input string = 0011 Input string = 00111

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52 PDA in Figure - test empty stack by initially placing a special symbol, $, on the stack If ever it sees $ again on the stack, it knows that the stack is effectively empty.

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THEOREM 2.20 A language is context free if and only if some pushdown automaton recognizes it. LEMMA 2.21 If a language is context free, then some pushdown automaton recognizes it. 54

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Pages 115 – 122 read only “EQUIVALENCE WITH CONTEXT-FREE GRAMMARS” Home work pages 123-127 NON-CONTEXT-FREE LANGUAGES 55

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11 CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 7 School of Innovation, Design and Engineering Mälardalen University 2012.

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