The grammar S → SS | (S)| ϵ is not suitable for predictive parsing because the grammar is:
Ambiguous
Understanding Predictive Parsing and Grammar Suitability
Predictive parsing is a type of top-down parsing that does not require backtracking. It's a deterministic parsing method often implemented using a parsing table. For a grammar to be suitable for standard predictive parsing (specifically, LL(1) parsing), it must satisfy certain properties:
- It must be unambiguous.
- It must not contain left recursion.
- It must satisfy the LL(1) condition (First/Follow sets condition) to enable deterministic selection of productions with a single lookahead token.
Analyzing the Given Grammar: S → SS | (S)| ϵ
The grammar provided is: \( S \to SS \mid (S) \mid \varepsilon \)
Let's examine this grammar against the requirements for predictive parsing.
Issue 1: Left Recursion
The production rule \( S \to SS \) exhibits left recursion because the leftmost symbol on the right-hand side (\(S\)) is the same as the non-terminal on the left-hand side. Standard predictive parsers cannot handle left recursion directly as it leads to infinite loops during parsing. Left recursion must be eliminated from a grammar for it to be suitable for predictive parsing.
Issue 2: Ambiguity
A grammar is considered ambiguous if there exists at least one string in the language generated by the grammar that has more than one distinct leftmost derivation (or more than one distinct parse tree). Let's consider the string \(()()\) which is generated by this grammar.
We can show two distinct leftmost derivations for the string \(()()\):
| Derivation 1 | Explanation |
|---|---|
| \( S \Rightarrow SS \) | Apply \( S \to SS \) |
| \( \Rightarrow (S)S \) | Apply \( S \to (S) \) to the first \(S\) (leftmost) |
| \( \Rightarrow (\varepsilon)S \) | Apply \( S \to \varepsilon \) to the first \(S\) (leftmost) |
| \( \Rightarrow (\varepsilon)(S) \) | Apply \( S \to (S) \) to the second \(S\) (leftmost) |
| \( \Rightarrow (\varepsilon)(\varepsilon) \) | Apply \( S \to \varepsilon \) to the second \(S\) (leftmost) |
| Result: \( ()() \) |
| Derivation 2 | Explanation |
|---|---|
| \( S \Rightarrow SS \) | Apply \( S \to SS \) |
| \( \Rightarrow S(S) \) | Apply \( S \to (S) \) to the second \(S\) (leftmost remaining non-terminal) |
| \( \Rightarrow S(\varepsilon) \) | Apply \( S \to \varepsilon \) to the second \(S\) (leftmost remaining non-terminal) |
| \( \Rightarrow (S)(\varepsilon) \) | Apply \( S \to (S) \) to the first \(S\) (leftmost) |
| \( \Rightarrow (\varepsilon)(\varepsilon) \) | Apply \( S \to \varepsilon \) to the first \(S\) (leftmost) |
| Result: \( ()() \) |
Since the string \(()()\) has two distinct leftmost derivations, the grammar \( S \to SS \mid (S) \mid \varepsilon \) is ambiguous.
Why Ambiguity Affects Predictive Parsing
Predictive parsers are deterministic. At each step, based on the current non-terminal and the lookahead token, the parser must uniquely determine which production rule to apply. If a grammar is ambiguous, a single string has multiple valid parse trees or derivations. A deterministic parser cannot choose between these multiple possibilities, making the grammar unsuitable for predictive parsing.
Evaluating the Options
- An Operator Grammar: This grammar generates sequences of balanced parentheses, not typical arithmetic expressions requiring operator precedence parsing. It's not primarily known as an operator grammar, and being one doesn't inherently prevent predictive parsing (though operator precedence parsing is a different method).
- Right Recursive: The grammar is not primarily right recursive in a way that prevents predictive parsing (like \(A \to xA\)). It has left recursion (\(S \to SS\)) and a recursive rule involving parentheses (\(S \to (S)\)).
- Left Recursive: The grammar is left recursive due to \(S \to SS\). This is indeed a reason it's unsuitable for predictive parsing.
- Ambiguous: As demonstrated above, the grammar is ambiguous. This is also a reason it's unsuitable for predictive parsing.
Both left recursion and ambiguity make a grammar unsuitable for predictive parsing. However, ambiguity is a more fundamental issue that cannot always be resolved by simple mechanical transformations like left recursion removal while preserving the language structure in a way suitable for LL(1). Given the options, ambiguity is a definitive property that prevents predictive parsing.
Conclusion on Grammar Suitability
The grammar \( S \to SS \mid (S) \mid \varepsilon \) is not suitable for predictive parsing primarily because it is ambiguous. While it is also left-recursive, ambiguity presents a fundamental challenge to deterministic parsing requiring multiple interpretations for a single input string.
Revision Table: Key Grammar Concepts
| Concept | Description | Issue for Predictive Parsing |
|---|---|---|
| Left Recursion | A rule like \( A \to A\alpha \). Leftmost symbol on RHS is the same as LHS. | Leads to infinite loops in top-down parsing. Must be eliminated. |
| Right Recursion | A rule like \( A \to \alpha A \). Rightmost symbol on RHS is the same as LHS. | Generally acceptable for top-down parsing. |
| Ambiguity | A grammar generating a string with multiple distinct parse trees or leftmost derivations. | Parser cannot make a unique deterministic choice at parsing steps. Unsuitable for deterministic parsers like predictive parsers. |
| LL(1) Condition | Based on FIRST and FOLLOW sets, ensuring a unique production choice for each non-terminal and lookahead token. | Failure to satisfy this condition means the parser cannot predict the correct production. |
Additional Information: Resolving Grammar Issues for Parsing
Compilers use various techniques to handle grammars that are not directly suitable for a chosen parsing method:
- Left Recursion Elimination: Mechanical transformation techniques can remove immediate and indirect left recursion, converting rules like \( A \to A\alpha \mid \beta \) into non-left-recursive forms like \( A \to \beta A' \) and \( A' \to \alpha A' \mid \varepsilon \).
- Factoring: Left factoring \( A \to \alpha\beta_1 \mid \alpha\beta_2 \) into \( A \to \alpha A' \) and \( A' \to \beta_1 \mid \beta_2 \) helps satisfy the LL(1) condition when multiple productions for a non-terminal start with the same symbols.
- Ambiguity Resolution: This is often the most difficult. Sometimes, rewriting the grammar can remove ambiguity while preserving the language. Other times, parser generators might use precedence and associativity rules (especially for arithmetic expressions) to resolve shift-reduce conflicts that arise from ambiguity in bottom-up parsers, but this doesn't make the underlying grammar unambiguous. For top-down (predictive) parsing, unambiguous grammar is generally required. If a grammar is inherently ambiguous (no equivalent unambiguous grammar exists), then a deterministic parser like a standard predictive parser cannot be used.
The grammar \( S \to SS \mid (S) \mid \varepsilon \), while generating balanced parentheses sequences, is a classic example of an ambiguous grammar, making it unsuitable for predictive parsing.
Important Questions from Parser
Consider the following statements:
Statement I: LALR parser is more powerful than canonical LR Parser.
Statement II: SLR parser is more powerful than LALR
Which of the following is correct?
Given below are two statements
Statement I : LL(1) and LR are examples of Bottom‐up parsers.
Statement II : Recursive descent parser and SLR are examples of Top‐down parsers
In light of the above statements, choose the correct answer from the options given below
Consider the following Grammar G:
S ➝ A | B
A➝ a | c
B➝ b | c
Where {S, A, B} is the set of non-terminals, {a, b, c} is the set of terminals.
Which of the following statement(s) is/are correct?
S 1 : LR(1) can parse all strings that are generated using grammar G.
S 2 : LL(1) can parse all strings that are generated using grammar G.- Arrange the following parsers in increasing order of their power of handling grammars i.e. from the least powerful parser to the most powerful parser.
A. LR(0)
B. LR(1)
C. LALR(1)
D. LL(0)
E. SLR
Choose the correct answer from the options given below: - Which of the following statements is/are true?
