CSCI 3136 Principles of Programming Languages Syntactic Analysis - - PowerPoint PPT Presentation

CSCI 3136 Principles of Programming Languages

Syntactic Analysis and Context-Free Grammars - 2

Summer 2013 Faculty of Computer Science Dalhousie University

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Language defined by a CFG

L(G) = {w|S ⇒∗ w} If G is a grammar, the language of the grammar, denoted as L(G), is the set of terminal strings that have derivations from the start symbol.

• Languages defined by CFGs are context-free languages
• Example: Our ‘Jim-ate-cheese’ grammar generates the

following language: L(G) = {“Jim ate cheese”, “Jim ate Jim”, “cheese ate cheese”, “cheese ate Jim”, “big Jim ate cheese”, “big Jim ate Jim”, “big cheese ate cheese”, · · · }

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CF Languages: Example (1)

G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)

• Is ǫ in L(G)?
• Is 01 in L(G)?
• Is 0011 in L(G)?
• Is 0n1n in L(G)?

What language is defined by the following CFG? S → ǫ S → 0S1

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CF Languages: Example (2)

What language is defined by the following CFG? S → ǫ S → 0S0 S → 1S1 Neither of these two languages is regular. There are more context-free languages than regular ones.

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Classes of Chomsky Hierarchy

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Regular grammar (1)

A right regular grammar (or right linear grammar) is a formal grammar (N, Σ, P, S) such that all the production rules in P are of

• ne of the following forms:
• 1. B → a, where B ∈ N and a ∈ Σ
• 2. B → aC, where B, C ∈ N and a ∈ Σ
• 3. B → ǫ, where B ∈ N and ǫ denotes the empty string

In a left regular grammar (or left linear grammar), all rules obey the forms:

• 1. B → a, where B ∈ N and a ∈ Σ
• 2. B → Ca, where B, C ∈ N and a ∈ Σ
• 3. B → ǫ, where B ∈ N and ǫ denotes the empty string

A (context-free) grammar is regular if it is a left or right regular grammar. Regular grammars are too weak to express programming languages.

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Regular grammar (2)

Is the following grammar (with start non-terminal S) regular or context-free? S → 0S S → 1A A → ǫ A → 2A If it is a regular grammar then what is the equivalent regular expression? Otherwise, write down the context-free language defined by the above grammar.

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Regular grammar (3)

What language is defined by the following grammar (with start non-terminal S)? S → 0A A → S1 S → ǫ Is the language defined by the above grammar regular or context-free? Prove your answer.

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Parse Trees

Every derivation can be represented by a parse tree: ———————– S → P V P P → N P → A P A → big|green N → cheese|Jim V → ate ————————

• S ⇒ P V P ⇒ N V P ⇒ N V N ⇒ Jim V N

⇒ Jim ate N ⇒ Jim ate cheese S P N cheese V ate P N Jim

• The root is S
• The children of each node are the symbols (terminals and

non-terminals) it is replaced with.

• The internal nodes are non-terminals. The leaves–called the

yield of the parse-tree–are terminals.

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Parse Tree (another example)

———————– S → P V P P → N P → A P A → big|green N → cheese|Jim V → ate ————————

• S ⇒ P V P ⇒ A P V P ⇒ big P V P ⇒

big N V P ⇒ big Jim V P ⇒ big Jim ate P ⇒ big Jim ate A P ⇒ big Jim ate green P ⇒ big Jim ate green N ⇒ big Jim ate green cheese S P P N cheese A green V ate P P N Jim A big

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Ambiguity: Example

A CFG for arithmetic expressions ———————– E → n E → i E → E + E E → E − E E → E ∗ E E → E/E E → (E) ———————— 2 + 3 ∗ 4 E E E 4 * E 3 + E 2 E E 4 * E E 3 + E 2 Violates precedence rules.

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Ambiguity

• There may be more than one parse tree for the same sentence

generated by a grammar G. If this is the case, we call G ambiguous.

• Problems of ambiguous grammar: one sentence has different

interpretations.

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Some facts about multiple representations

• There are infinitely many context-free grammars generating a

given context-free language. (How?)

• There may be more than one parse tree for the same sentence

generated by a grammar.

• For programming languages, we require that CFG is

unambiguous.

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