In the grammars shown below, upper case letters denote non-terminal symbols, lower case letters denote terminal symbols, the start symbol is always S, and & denotes the empty string. For each grammar: (1) Describe the language generated by the grammar concisely in English. For example: "All strings of zeros and ones which contain at least 5 zeros." And (2) say whether or not the grammar is ambiguous. If the grammar is ambiguous, find a string that demonstrates the ambiguity, and draw two distinct parse trees for this string. (a) SaSa | bSb | cSc | a | b | c | & (b) S &ab | ba | aSb | bSa (c) SaSb | CC &c | Cc
In the grammars shown below, upper case letters denote non-terminal symbols, lower case letters denote terminal symbols, the start symbol is always S, and & denotes the empty string. For each grammar: (1) Describe the language generated by the grammar concisely in English. For example: "All strings of zeros and ones which contain at least 5 zeros." And (2) say whether or not the grammar is ambiguous. If the grammar is ambiguous, find a string that demonstrates the ambiguity, and draw two distinct parse trees for this string. (a) SaSa | bSb | cSc | a | b | c | & (b) S &ab | ba | aSb | bSa (c) SaSb | CC &c | Cc
Computer Networking: A Top-Down Approach (7th Edition)
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![### Understanding Ambiguity in Context-Free Grammars
In the grammars shown below, upper case letters denote non-terminal symbols, lower case letters denote terminal symbols, the start symbol is always `S`, and `ε` denotes the empty string.
For each grammar:
1. **Describe the Language Generated by the Grammar**: Provide a concise description of the strings generated by the grammar in English.
2. **Determine if the Grammar is Ambiguous**: Indicate whether or not the grammar is ambiguous. If the grammar is ambiguous, find a string that demonstrates the ambiguity and draw two distinct parse trees for this string.
#### Grammar Descriptions
**(a)**
\[ S \rightarrow aSa \ | \ bSb \ | \ cSc \ | \ a \ | \ b \ | \ c \ | \ ε \]
- **Language Description**: The language consists of all odd-length palindromes (strings that read the same forwards and backwards) formed over the alphabet {a, b, c}, including the empty string.
- **Ambiguity**: This grammar is ambiguous. An example of ambiguity can be demonstrated with the string `ε` (the empty string), which can be derived from multiple different productions:
- Using \( S \rightarrow ε \)
- Using \( S \rightarrow aSa \) followed by \( S \rightarrow bSb \) and then each \( S \) leads to \( ε \).
Here, parsing this string can lead to different parse trees, confirming the ambiguity.
**(b)**
\[ S \rightarrow ε \ | \ ab \ | \ ba \ | \ aSb \ | \ bSa \]
- **Language Description**: The language consists of all strings over the alphabet {a, b} that have even length and are symmetric about their center.
- **Ambiguity**: This grammar is also ambiguous. For example, the string `ab` can be derived in multiple ways:
- Directly from \( S \rightarrow ab \)
- Or using \( S \rightarrow aSb \) where the inner \( S \) leads to \( ε \).
This results in different parse trees indicating ambiguity.
**(c)**
\[ S \rightarrow aSb \ | \ C \]
\[ C \rightarrow ε \ | \ c \ | \ Cc \]
- **Language Description**: This language consists of strings over the alphabet {a,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11ad9986-04b0-4336-8340-a3038312afe8%2F9df101c0-e89f-445e-a7c3-9bb1785b249d%2Firqwxpm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Ambiguity in Context-Free Grammars
In the grammars shown below, upper case letters denote non-terminal symbols, lower case letters denote terminal symbols, the start symbol is always `S`, and `ε` denotes the empty string.
For each grammar:
1. **Describe the Language Generated by the Grammar**: Provide a concise description of the strings generated by the grammar in English.
2. **Determine if the Grammar is Ambiguous**: Indicate whether or not the grammar is ambiguous. If the grammar is ambiguous, find a string that demonstrates the ambiguity and draw two distinct parse trees for this string.
#### Grammar Descriptions
**(a)**
\[ S \rightarrow aSa \ | \ bSb \ | \ cSc \ | \ a \ | \ b \ | \ c \ | \ ε \]
- **Language Description**: The language consists of all odd-length palindromes (strings that read the same forwards and backwards) formed over the alphabet {a, b, c}, including the empty string.
- **Ambiguity**: This grammar is ambiguous. An example of ambiguity can be demonstrated with the string `ε` (the empty string), which can be derived from multiple different productions:
- Using \( S \rightarrow ε \)
- Using \( S \rightarrow aSa \) followed by \( S \rightarrow bSb \) and then each \( S \) leads to \( ε \).
Here, parsing this string can lead to different parse trees, confirming the ambiguity.
**(b)**
\[ S \rightarrow ε \ | \ ab \ | \ ba \ | \ aSb \ | \ bSa \]
- **Language Description**: The language consists of all strings over the alphabet {a, b} that have even length and are symmetric about their center.
- **Ambiguity**: This grammar is also ambiguous. For example, the string `ab` can be derived in multiple ways:
- Directly from \( S \rightarrow ab \)
- Or using \( S \rightarrow aSb \) where the inner \( S \) leads to \( ε \).
This results in different parse trees indicating ambiguity.
**(c)**
\[ S \rightarrow aSb \ | \ C \]
\[ C \rightarrow ε \ | \ c \ | \ Cc \]
- **Language Description**: This language consists of strings over the alphabet {a,
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