Example 2.4 I The following CFG handles mathematical expressions G - - PowerPoint PPT Presentation

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Oct 23, 2022 444 likes •583 views

Example 2.4 I The following CFG handles mathematical expressions G 4 = ( V , , R , expr ) V = { expr , term , factor } = { a , + , , ( , ) } R : expr expr + term | term

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Example 2.4 I

The following CFG handles mathematical expressions G4 = (V , Σ, R, expr) V = {expr, term, factor} Σ = {a, +, ×, (, )} R: expr → expr + term | term term → term × factor | factor factor → (expr) | a Fig 2.5: check a + a × a

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Example 2.4 II

E E T F a + T T F a × F a

check (a + a) × a

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Example 2.4 III

E T T F F E E T T F F a ( + a ) × a

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Design Grammars I

{0n1n | n ≥ 0} ∪ {1n0n | n ≥ 0} S1 → 0S11 | ǫ S2 → 1S20 | ǫ S → S1 | S2 CFG versus DFA For a CFG that is regular, it can be recognized by DFA

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Design Grammars II

Rules of CFG can be Ri → aRj if δ(qi, a) = qj Ri → ǫ if qi ∈ F We see the main difference between CFG and DFA CFG: a rule can be like Ri → aRjb

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Design Grammars III

DFA: a rule can only be Ri → aRj, where we treat each Ri as a state and let δ(Ri, a) = Rj

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Ambiguity I

The same string but obtained in different ways For the example of mathematical expressions discussed earlier, what if we consider the following rules? expr → expr + expr | expr × expr | (expr)|a We see the following ways to parse a + a × a

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Ambiguity II

Fig 2.6

E E E a + E a × E a E E a + E E a × E a

This CFG does not give the precedence relation We want that a × a is done first

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Ambiguity III

By the more complicated CFG earlier, the parsing is unambiguous Question: how to formally define the ambiguity? We need to define “leftmost derivation” first

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Leftmost derivation I

Even for an unambiguous CFG we may have the same parse tree, but different derivations For the CFG discussed earlier we can do expr ⇒ expr + term ⇒ expr + term × factor where the second part is expanded. On the other hand, we can handle the first one first. expr ⇒ expr + term ⇒ term + term

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Leftmost derivation II

This is not considered ambiguous Definition of leftmost derivation: at every step the leftmost remaining variable is the one replaced.

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Formal definition of ambiguity I

w ambiguous if there exist two leftmost derivations Some context-free languages can be generated by ambiguous & unambiguous grammars We say a CFG is inherently ambiguous if it only has ambiguous grammars See prob 2.29 in the textbook. Details not given here.

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