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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}
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CSCI 3136 Principles of Programming Languages Syntactic Analysis - - PowerPoint PPT Presentation
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 1 / 102 Language defined by a CFG L ( G ) = { w | S w } 2 / 102 Language
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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.
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SLIDE 4
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
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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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SLIDE 6
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
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SLIDE 7
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)?
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SLIDE 8
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
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SLIDE 9
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)?
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SLIDE 10
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
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SLIDE 11
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)?
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SLIDE 12
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)? because S ⇒ 0S1 ⇒ 00S11 ⇒ 0011
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SLIDE 13
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)? because S ⇒ 0S1 ⇒ 00S11 ⇒ 0011
- Is 0n1n in L(G)?
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SLIDE 14
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)? because S ⇒ 0S1 ⇒ 00S11 ⇒ 0011
- Is 0n1n in L(G)? because S ⇒∗ 0n1n
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SLIDE 15
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)? because S ⇒ 0S1 ⇒ 00S11 ⇒ 0011
- Is 0n1n in L(G)? because S ⇒∗ 0n1n
What language is defined by the following CFG? S → ǫ S → 0S1
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SLIDE 16
CF Languages: Example (1)
G = ({S}, {0, 1}, {S → 0S1|ǫ}, S)
- Is ǫ in L(G)? because S ⇒ ǫ
- Is 01 in L(G)? because S ⇒ 0S1 ⇒ 01
- Is 0011 in L(G)? because S ⇒ 0S1 ⇒ 00S11 ⇒ 0011
- Is 0n1n in L(G)? because S ⇒∗ 0n1n
What language is defined by the following CFG? S → ǫ S → 0S1 L(G) = {0n1n|n ≥ 0}
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CF Languages: Example (2)
What language is defined by the following CFG? S → ǫ S → 0S0 S → 1S1
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CF Languages: Example (2)
What language is defined by the following CFG? S → ǫ S → 0S0 S → 1S1 L(G) = {wwR|w ∈ {0, 1}∗} where wR is w written backwards
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SLIDE 19
CF Languages: Example (2)
What language is defined by the following CFG? S → ǫ S → 0S0 S → 1S1 L(G) = {wwR|w ∈ {0, 1}∗} where wR is w written backwards Neither of these two languages is regular.
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SLIDE 20
CF Languages: Example (2)
What language is defined by the following CFG? S → ǫ S → 0S0 S → 1S1 L(G) = {wwR|w ∈ {0, 1}∗} where wR is w written backwards Neither of these two languages is regular. There are more context-free languages than regular ones.
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SLIDE 21
Classes of Chomsky Hierarchy
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SLIDE 22
Regular grammar (1)
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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:
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SLIDE 24
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
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SLIDE 25
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
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SLIDE 26
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
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SLIDE 27
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:
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SLIDE 28
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
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SLIDE 29
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.
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SLIDE 30
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
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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 (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. 0∗12∗
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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
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SLIDE 35
Regular grammar (3)
What language is defined by the following grammar (with start non-terminal S)? S → 0A A → S1 S → ǫ L(G) = {0n1n|n ≥ 0}
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SLIDE 36
Regular grammar (3)
What language is defined by the following grammar (with start non-terminal S)? S → 0A A → S1 S → ǫ L(G) = {0n1n|n ≥ 0} Is the language defined by the above grammar regular or context-free? Prove your answer.
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SLIDE 37
Parse Trees
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SLIDE 38
Parse Trees
Every derivation can be represented by a parse tree:
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SLIDE 39
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 ————————
- 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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SLIDE 40
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
- 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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SLIDE 41
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
S P V P
- 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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SLIDE 42
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
S P V P
- 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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SLIDE 43
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
S P V P N
- 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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SLIDE 44
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
S P V P N
- 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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SLIDE 45
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
S P V P N
- 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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SLIDE 46
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
S P N V P N
- 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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SLIDE 47
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
S P N V P N
- 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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SLIDE 48
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
S P N V P N
- 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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SLIDE 49
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
S P N V 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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SLIDE 50
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
S P N V 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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SLIDE 51
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
S P N V 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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SLIDE 52
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 S P N 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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SLIDE 53
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 S P N 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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SLIDE 54
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 S P N 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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SLIDE 55
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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SLIDE 56
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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SLIDE 57
Parse Tree (another example)
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SLIDE 58
Parse Tree (another example)
———————– S → P V P P → N P → A P A → big|green N → cheese|Jim V → ate ————————
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SLIDE 59
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
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SLIDE 60
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
S P V P
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SLIDE 61
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
S P V P
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SLIDE 62
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
S P V P P A
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SLIDE 63
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
S P V P P A
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SLIDE 64
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
S P V P P A
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SLIDE 65
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
S P V P P A big
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SLIDE 66
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
S P V P P A big
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SLIDE 67
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
S P V P P A big
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SLIDE 68
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 S P V P P N A big
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SLIDE 69
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 S P V P P N A big
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SLIDE 70
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 S P V P P N A big
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SLIDE 71
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 S P V P P N Jim A big
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SLIDE 72
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 S P V P P N Jim A big
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SLIDE 73
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 S P V P P N Jim A big
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SLIDE 74
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 S P V ate P P N Jim A big
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SLIDE 75
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 S P V ate P P N Jim A big
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SLIDE 76
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 S P V ate P P N Jim A big
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SLIDE 77
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 S P P A V ate P P N Jim A big
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SLIDE 78
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 S P P A V ate P P N Jim A big
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SLIDE 79
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 S P P A V ate P P N Jim A big
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SLIDE 80
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 S P P A green V ate P P N Jim A big
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SLIDE 81
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 S P P A green V ate P P N Jim A big
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SLIDE 82
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 S P P A green V ate P P N Jim A big
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SLIDE 83
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 S P P N A green V ate P P N Jim A big
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SLIDE 84
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 S P P N A green V ate P P N Jim A big
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SLIDE 85
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 S P P N A green V ate P P N Jim A big
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SLIDE 86
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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SLIDE 87
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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SLIDE 88
Ambiguity: Example
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SLIDE 89
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) ————————
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SLIDE 90
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
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SLIDE 91
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
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SLIDE 92
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
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SLIDE 93
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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SLIDE 94
Ambiguity
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SLIDE 95
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.
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SLIDE 96
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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SLIDE 97
Some facts about multiple representations
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SLIDE 98
Some facts about multiple representations
- There are infinitely many context-free grammars generating a
given context-free language.
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SLIDE 99
Some facts about multiple representations
- There are infinitely many context-free grammars generating a
given context-free language. (How?)
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SLIDE 100
Some facts about multiple representations
- There are infinitely many context-free grammars generating a
given context-free language. (How?)
- Just add arbitrary non-terminals to the right-hand sides of
productions and then add ǫ-productions for these non-terminals.
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SLIDE 101
Some facts about multiple representations
- There are infinitely many context-free grammars generating a
given context-free language. (How?)
- Just add arbitrary non-terminals to the right-hand sides of
productions and then add ǫ-productions for these non-terminals.
- There may be more than one parse tree for the same sentence
generated by a grammar.
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SLIDE 102
Some facts about multiple representations
- There are infinitely many context-free grammars generating a
given context-free language. (How?)
- Just add arbitrary non-terminals to the right-hand sides of
productions and then add ǫ-productions for these non-terminals.
- 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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