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Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Reference 3Computer Sciences Department

4 objectives -Describing Languages -A Formal Definition Of Context-free Grammars -The Properties Of Context-free Languages -A PushDown Automata PDA -Equivalence -The Pumping Lemma

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.G.)  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. 5Computer Sciences Department

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. 6Computer Sciences Department

Context free languages  The collection of languages associated with context- free grammars are called the context-free languages. 7Computer Sciences Department

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. 8Computer Sciences Department

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. 9Computer Sciences Department

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.). 10Computer Sciences Department

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 11 start variable terminal variable symbols Computer Sciences Department

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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. 13Computer Sciences Department

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. 14Computer Sciences Department

Derivation  The sequence of substitutions to obtain a string is called a derivation.  A derivation of string 000#111 in grammar G1 is: 15 parse tree Computer Sciences Department

 Grammar G2 has ??  rules?? - variables ?? - terminals??. 16Computer Sciences Department

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. 17Computer Sciences Department

Derivation  Each of these strings has a derivation in grammar G2. The following is a derivation of the first string on this list. 18Computer Sciences Department

FORMAL DEFINITION OF A CONTEXT- FREE GRAMMAR 19Computer Sciences Department

Example Grammar G1 20 In grammar G2 Computer Sciences Department

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. 21Computer Sciences Department

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?????????????????? 22Computer Sciences Department

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) 23 Derivation tree for the sentence abcd -1&2 Derivation tree for the sentence abcd -3 true or false????? c Computer Sciences Department

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. 24Computer Sciences Department

Example For example, consider grammar G5: This grammar generates the string a+axa ambiguously?????????????? Yes (two different parse trees) 25Computer Sciences Department

DEFINITION 2.7  A string ω 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. 26Computer Sciences Department

Chomsky normal form 27Computer Sciences Department

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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. 29Computer Sciences Department

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Context-Free Languages {0 n 1 n I n > 0} is not regular??? 33 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 Computer Sciences Department

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 34 Write a derivation of string (a + b)*c Computer Sciences Department

35 Write a derivation of string (a + b)*c The derivation of string (a + b)*c: Computer Sciences Department

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. 36Computer Sciences Department

37 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 Computer Sciences Department

38 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 Computer Sciences Department

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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. 40Computer Sciences Department

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. 41Computer Sciences Department

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. 42Computer Sciences Department

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. 43Computer Sciences Department

FORMAL DEFINITION OF A PUSHDOWN AUTOMATON 44Computer Sciences Department

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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 Computer Sciences Department

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)) Computer Sciences Department

50 Input string = 0011 Input string = Computer Sciences Department

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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. Computer Sciences Department

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Computer Sciences Department56 EQUIVALENCE WITH CONTEXT-FREE GRAMMARS

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Proof Computer Sciences Department58

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Computer Sciences Department62 THE PUMPING LEMMA FOR CONTEXT-FREE LANGUAGES

Computer Sciences Department63 PROOF

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Computer Sciences Department65 Explanation

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Computer Sciences Department69  The one difference is that the PDA can decide at any point to stop reading ω and begin reading ω R.

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