Chomsky and Greibach Normal Forms. Chomsky and Greibach Normal Forms p.1/24

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1 Chomsky and Greibach Normal Forms Chomsky and Greibach Normal Forms p.1/24

2 Simplifying a CFG It is often convenient to simplify CFG Chomsky and Greibach Normal Forms p.2/24

3 Simplifying a CFG It is often convenient to simplify CFG One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form Chomsky and Greibach Normal Forms p.2/24

4 Simplifying a CFG It is often convenient to simplify CFG One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form Another normal form usually used in algebraic specifications is Greibach normal form Chomsky and Greibach Normal Forms p.2/24

5 Simplifying a CFG It is often convenient to simplify CFG One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form Another normal form usually used in algebraic specifications is Greibach normal form Note the difference between grammar cleaning and simplification Chomsky and Greibach Normal Forms p.2/24

6 Note Normal forms are useful when more advanced topics in computation theory are approached, as we shall see further Chomsky and Greibach Normal Forms p.3/24

7 Definition A context-free grammar every rule is of the form: is in Chomsky normal form if where is a terminal, are nonterminals, and may not be the start variable (the axiom) Chomsky and Greibach Normal Forms p.4/24

8 Note The rule, where is the start variable, is not excluded from a CFG in Chomsky normal form. Chomsky and Greibach Normal Forms p.5/24

9 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Chomsky and Greibach Normal Forms p.6/24

10 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Chomsky and Greibach Normal Forms p.6/24

11 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Show that any CFG Chomsky normal form can be converted into a CFG in Chomsky and Greibach Normal Forms p.6/24

12 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Show that any CFG Chomsky normal form can be converted into a CFG in Conversion procedure has several stages where the rules that violate Chomsky normal form conditions are replaced with equivalent rules that satisfy these conditions Chomsky and Greibach Normal Forms p.6/24

13 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Show that any CFG Chomsky normal form can be converted into a CFG in Conversion procedure has several stages where the rules that violate Chomsky normal form conditions are replaced with equivalent rules that satisfy these conditions Order of transformations: (1) add a new start variable, (2) eliminate all -rules, (3) eliminate unit-rules, (4) convert other rules Chomsky and Greibach Normal Forms p.6/24

14 Theorem 2.9 Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Show that any CFG Chomsky normal form can be converted into a CFG in Conversion procedure has several stages where the rules that violate Chomsky normal form conditions are replaced with equivalent rules that satisfy these conditions Order of transformations: (1) add a new start variable, (2) eliminate all -rules, (3) eliminate unit-rules, (4) convert other rules Check that the obtained CFG defines the same language Chomsky and Greibach Normal Forms p.6/24

15 Proof Let be the original CFG. Chomsky and Greibach Normal Forms p.7/24

16 Proof Let be the original CFG. Step 1: add a new start symbol to, and the rule to Chomsky and Greibach Normal Forms p.7/24

17 Proof Let be the original CFG. Step 1: add a new start symbol to, and the rule to Note: this change guarantees that the start symbol of does not occur on the of any rule Chomsky and Greibach Normal Forms p.7/24

18 Step 2: eliminate -rules Repeat 1. Eliminate the rule from where is not the start symbol 2. For each occurrence of on the with that occurrence of deleted Example: replace by replace by of a rule, add a new rule to ; 3. Replace the rule, (if it is present) by rule has been previously eliminated unless the until all rules are eliminated Chomsky and Greibach Normal Forms p.8/24

19 Step 3: remove unit rules Repeat 1. Remove a unit rule 2. For each rule, add the rule was a unit rule previously removed until all unit rules are eliminated to, unless Note: is a string of variables and terminals Chomsky and Greibach Normal Forms p.9/24

20 Convert all remaining rules Repeat 1. Replace a rule,, where each is a variable or a terminal, by:,,, where,,, are new variables,, 2. If rule replace any terminal with a new variable and add the until no rules of the form with remain Chomsky and Greibach Normal Forms p.10/24

23 Example CFG conversion Consider the grammar whose rules are: Notation: symbols removed are green and those added are red. After first step of transformation we get: Chomsky and Greibach Normal Forms p.11/24

26 Chomsky and Greibach Normal Forms p.12/24 Removing rules : Removing : Removing

29 Removing unit rule Removing : Removing : Chomsky and Greibach Normal Forms p.13/24

32 More unit rules Removing : Removing : Chomsky and Greibach Normal Forms p.14/24

35 Converting remaining rules Chomsky and Greibach Normal Forms p.15/24

38 Note The conversion procedure produces several variables along with several rules. Since all these represent the same rule, we may simplify the result using a single variable and a single rule Chomsky and Greibach Normal Forms p.16/24

41 Greibach Normal Form A context-free grammar Greibach normal form if each rule property:,,. is in has the and Note: Greibach normal form provides a justification of operator prefix-notation usually employed in algebra. Chomsky and Greibach Normal Forms p.17/24

44 Greibach Theorem Every CFL where can be generated by a CFG in Greibach normal form. Proof idea: Let is in Chomsky normal form be a CFG generating. Assume that Let be an ordering of nonterminals. Construct the Greibach normal form from Chomsky normal form Chomsky and Greibach Normal Forms p.18/24

47 Construction 1. Modify the rules in so that if then 2. Starting with and proceeding to this is done as follows: (a) Assume that productions have been modified so that for, only if (b) If is a production with productions substituting for the production, generate a new set of the rhs of each (c) Repeating (b) at most, times we obtain rules of the form (d) Replace rules by removing left-recursive rules Chomsky and Greibach Normal Forms p.19/24

50 Removing left-recursion Left-recursion can be eliminated by the following scheme: If nonterminal, say left recursive rules, and are all remaining -rules then chose a new are all Add the new -rules, Replace the -rules by, This construction preserve the language. Chomsky and Greibach Normal Forms p.20/24

53 21-1

54 More on Greibach NF See Introduction to Automata Theory, Languages, and Computation, J.E, Hopcroft and J.D Ullman, Addison-Wesley 1979, p Chomsky and Greibach Normal Forms p.22/24

55 Chomsky and Greibach Normal Forms p.23/24 Example Convert the CFG where into Greibach normal form.

58 Solution 1. Step 1: ordering the rules: (Only rules violate ordering conditions, hence only rules need to be changed). Following the procedure we replace rules by: 2. Eliminating left-recursion we get:, 3. All rules start with a terminal. We use them to replace. This introduces the rules 4. Use production to make them start with a terminal Chomsky and Greibach Normal Forms p.24/24