Ambiguity, closure properties, Overview of ambiguity, and Chomsky s standard form

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1 Ambiguity, closure properties, and Chomsky s normal form Overview of ambiguity, and Chomsky s standard form

2 The notion of ambiguity Convention: we assume that in every substitution, the leftmost remaining variable is the one that is replaced. When this convention is applied to all substitutions in a derivation, we talk about leftmost derivations Definition: A CGF is said to be ambiguous if a string in its language has two different leftmost derivations. In such a case, the string has necessarily two different parsing trees

When this convention is applied to all substitutions in a derivation, we talk about leftmost derivations

3 Equivalence and ambiguity Definition: Two context-free grammars are declared to be equivalent if and only if they produce the same context-free language. Remark: An ambiguous context-free grammar may have an equivalent unambiguous context-free grammar. Definition: Given an ambiguous grammar, the process of identifying an equivalent grammar that is not ambiguous is called disambiguation. If not such grammar exists, the language is called inherently ambiguous 3

Remark: An ambiguous context-free grammar may have an equivalent unambiguous context-free grammar.

4 Example of an ambiguous contextfree grammar Let G = ({, A}, {,}, R, } R = { A λ, A A A } w = L( G) Then, the string has the following two leftmost derivations:.. A A A A A A 4

5 The grammar is not inherently ambiguous The ambiguity stems from the fact that the rule A A can be use in the first or in the second substitution indistinctively. In either case the same string is derived. w = L( G) L(G) However, is not inherently ambiguous. Following is a context free grammar that eliminates the above ambiguity. 5

In either case the same string is derived.

6 Disambiguation Let G R = ({, A, B},{,}, R, ) = { A λ, A A B, B B } Claim: L ( G ) = L ( G ) and under G each string is derived by a unique sequence of leftmost substitutions. This is, the new grammar derives the language unambiguously Proof: exercise! 6

7 Closure properties Theorem 9.: Context-free languages are closed under regular operations L L cketch of the proof: Let and be context-free languages. Let G = ( V, A, R, and G = ( V, A, R, ) be context-free grammars such that L = L( G ) and L ( ) = L G respectively. We may assume without loss of generality that V I = φ. V ) 7

proof: Let and be context-free languages.

8 Proof (cont.) Define the context-free grammars:. U = ({ } UV UV, A, R U R U{ },. C = ({ } UV UV, A, R U R U{ }, 3. T = ({ } U V, A, R U { λ }, ) Here we assume V UV Claim: L ( U ) = L U L L( C) = L L L( T) = L The rest of the proof (i.e. verification of the claim) is left as an exercise. ) ) * 8

9 Applications n. The language { n n n M = : n, n } is context-free, since it is the concatenation n n of L = { : n } with itself. The language = i n n nk nk { : k N ( i =,..., k) n is also context-free, since it is the starclosure of L } 9

10 Chomsky normal form Theorem 9.: (Chomsky normal form) Any context-free grammar is equivalent to a grammar G = ( V, A, R, ) whose productions are either of the form: Z UT Z a λ where Z, U, T V U T ; or where a A ; or

11 Method 9. Transforming a context-free grammar into a grammar in Chomsky normal form Given a context-free grammar G=(V,A,R,). Add a new start variable and add the rule. Remove all rules of the form u λ, u V adding the rules that are necessary to preserve the grammar s productions 3. Eliminate unit rules. These are rules of the form of u v, u, v V and their elimination requires adding the rules u w wherever v w appears, unless was already eliminated 4. Convert rules of the form k to a set of rules of the form u a u, u au,..., uk ak ak where the u i ' s are new variables. 5. Convert rules of the form k to rules of the form of u w u a... a a3 a, k 3 u a... a, k ui ai, wherever ai is a terminal.here, ui is a new variable.

These are rules of the form of u v, u, v V and their elimination requires adding the rules u w wherever v w appears, unless was already eliminated 4.

12 Example The context free grammar: G R = ({ },{,}, R, ) = { λ} is not in Chomsky normal form. We will find an equivalent context-free grammar N, which is in Chomsky normal form, using the previous procedure (Method 9.)

13 Example (cont.). Define a new start variable Old productions λ

14 Example (cont.). Eliminate unitary productions. In our case we have: λ Recall that the removal is performed by eliminating them and adding all rules necessary to preserve the original grammar s derivations

performed by eliminating them and adding all rules

15 Example (cont.) The derivations that involve these unitary productions (and have to be preserved) are: Thus, the new set of rules is: λ

16 Example (cont.) The latter set of rules define an equivalent grammar which is not yet in Chomsky normal form. Indeed no rule, except the one that associate the starting variable with the null string, is in Chomsky normal form. We ll fix this in the next step: 3. Decomposing all remaining productions that are not in Chomsky normal form into sets of productions with two variables or a terminal on the right hand side.

Indeed no rule, except the one that associate the starting variable with the null string, is in Chomsky

17 Example (cont.) This applies to the current productions:,, We transform each of them into a set of productions in Chomsky normal form as follows

18 Example (cont.) Consider first: This production is clearly equivalent to the combined action of the following productions:

19 Example (cont.) As for we have: ) ( 3 3 ince there are here productions that were already defined, we replace the variables: ) ( ) ( : : = =

20 Example (cont.) After replacing we get: 3 3

21 Example (cont.) And similarly, using the already defined productions, we transform and 3 3

22 Example (cont.) In summary, we get:, { ), },{,},,,,, ({ 3 3 R R N = = λ },,,, { R = λ

23 Does it work? Let s derive a few strings: λ 3 3

24 What s so important about Chomsky s normal form? Theorem 9.3: A context-free grammar in Chomsky normal form derives a string of length n in exactly n- substitutions Proof: Let G w = w of rules of of be a context - free grammar in Chomsky normal form and let w such rules. L ( G ).ince each w n i produced by substituting a the form of v But, the formation of the form u v n. Thus, rule of zv, u, z, v such rules involved in the generation of is a terminal, it could only be the form v the derivation of v v n w involves n applications of can only be done by applying variables.there are exactly n applications v i v w n. i in a string of variables

25 A note on context-sensitive grammars Definition: A context-sensitive grammar is a quadruple G = ( V, A, R, ) wherev, A and are just as in a context - free grammar, but the set of rules R includes rules of the form aub cvd, whereu andv are variables, and a, b, c, and d are strings of variables and literals. 5

26 Example Let G=({,R,T,U,V,W},{a, b, c}, R, ) R= { arc, R art b, btc bbcc, btt bbut, UT UU, UUc VUc Vcc, UV VV, bvc bbcc, bvv bbwv, WV WW, WWc TWc Tcc, WT TT } This grammar generates the canonical non-context-free n n n language: {a b c : n } 6

27 A derivation The derivation of the string is: aaabbbccc arc aartc aaarttc aaabttc aaabbutc aaabbuuc aaabbvuc aaabbvcc aaabbbccc 7

28 Note on normal forms Principle: Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent grammar in Kuroda normal form Remark: The normal form will not in general be a context-sensitive grammar, but will be a non-contracting grammar 8

The Halting Problem is Undecidable

The Halting Problem is Undecidable 185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine