Constructing the FIRST Set

Oct 13, 2025 by miodeya.com 93 views

Introduction

In the field of computer science, particularly in the area of compiler design, parsing is a crucial step in the process of analyzing the syntax of a programming language. Predictive parsing is a type of parsing that uses a parsing table to predict the next token in the input stream. In this article, we will discuss how to construct the FIRST and FOLLOW sets for a given grammar and then use these sets to construct the predictive parsing table.

Understanding the Grammar

The given grammar is:

A → DB B → CB | ε C → +A | DB | * D → (A) | a

This grammar defines four non-terminal symbols: A, B, C, and D, and two terminal symbols: +, *, and a.

Constructing the FIRST Set

The FIRST set of a non-terminal symbol is the set of terminal symbols that can appear as the first symbol in a sentential form derived from that non-terminal symbol.

FIRST(A)

To construct the FIRST set of A, we need to consider the production rules that start with A.

A → DB

The first symbol in the sentential form DB is D. Therefore, the FIRST set of A is:

FIRST(A) = {D}

FIRST(B)

To construct the FIRST set of B, we need to consider the production rules that start with B.

B → CB | ε

The first symbol in the sentential form CB is C. Therefore, the FIRST set of B is:

FIRST(B) = {C}

The ε production rule also contributes to the FIRST set of B.

FIRST(B) = {C, ε}

FIRST(C)

To construct the FIRST set of C, we need to consider the production rules that start with C.

C → +A | DB | *

The first symbol in the sentential form +A is +. Therefore, the FIRST set of C is:

FIRST(C) = {+}

The first symbol in the sentential form DB is D. Therefore, the FIRST set of C is:

FIRST(C) = {+, D}

The ε production rule also contributes to the FIRST set of C.

FIRST(C) = {+, D, *}

FIRST(D)

To construct the FIRST set of D, we need to consider the production rules that start with D.

D → (A) | a

The first symbol in the sentential form (A) is (.

Therefore, the FIRST set of D is:

FIRST(D) = {(}

The first symbol in the sentential form a is a.

Therefore, the FIRST set of D is:

FIRST(D) = {(, a}

Constructing the FOLLOW Set

The FOLLOW set of a non-terminal symbol is the set of terminal symbols that can appear after that non-terminal symbol in a sentential form.

FOLLOW(A)

To construct the FOLLOW set of A, we need to consider the production rules that end with A.

A → DB

The production rule ends with A, so the FOLLOW set of A is:

FOLLOW(A) = {)}

FOLLOW(B)

To construct the FOLLOW set of B, we need to consider the production rules that end with B.

B → CB | ε

The production rule ends with B, so the FOLLOW set of B is:

FOLLOW(B) = {)}

FOLLOW(C)

To construct the FOLLOW set of C, we need to consider the production rules that end with C.

C → +A | DB | *

The production rule ends with C, so the FOLLOW set of C is:

FOLLOW(C) = {D, *, a}

FOLLOW(D)

To construct the FOLLOW set of D, we need to consider the production rules that end with D.

D → (A) | a

The production rule ends with D, so the FOLLOW set of D is:

FOLLOW(D) = {)}

Constructing the Predictive Parsing Table

The predictive parsing table is a table that contains the production rules for each non-terminal symbol and the terminal symbols that can appear after that non-terminal symbol.

	+	*	a	)
A	DB
B	CB		ε
C	+A	DB	*
D	(A)		a	)

Conclusion

In this article, we discussed how to construct the FIRST and FOLLOW sets for a given grammar and then use these sets to construct the predictive parsing table. The predictive parsing table is a crucial component of a compiler, as it allows the compiler to predict the next token in the input stream and make decisions about how to parse the input.

References

Aho, A. V., Sethi, R., & Ullman, J. D. (1986). Compilers: Principles, techniques, and tools. Addison-Wesley.
Hopcroft, J. E., Motwani, R., & Ullman, J. D. (2006). Introduction to automata theory, languages, and computation. Addison-Wesley.
Q&A: Predictive Parsing Table and FIRST/FOLLOW Sets =====================================================

Q: What is predictive parsing?

A: Predictive parsing is a type of parsing that uses a parsing table to predict the next token in the input stream. It is a top-down parsing approach that uses the FIRST and FOLLOW sets to determine the next token.

Q: What are FIRST and FOLLOW sets?

A: FIRST and FOLLOW sets are sets of terminal symbols that can appear as the first symbol in a sentential form derived from a non-terminal symbol. The FIRST set contains the terminal symbols that can appear as the first symbol, while the FOLLOW set contains the terminal symbols that can appear after the non-terminal symbol.

Q: How are FIRST and FOLLOW sets constructed?

A: FIRST and FOLLOW sets are constructed by analyzing the production rules of the grammar. For each non-terminal symbol, we consider the production rules that start with that symbol and determine the terminal symbols that can appear as the first symbol or after the symbol.

Q: What is the predictive parsing table?

A: The predictive parsing table is a table that contains the production rules for each non-terminal symbol and the terminal symbols that can appear after that non-terminal symbol. It is used to predict the next token in the input stream.

Q: How is the predictive parsing table constructed?

A: The predictive parsing table is constructed by using the FIRST and FOLLOW sets. For each non-terminal symbol, we consider the production rules that start with that symbol and determine the terminal symbols that can appear as the first symbol or after the symbol. We then use this information to construct the predictive parsing table.

Q: What are the benefits of predictive parsing?

A: Predictive parsing has several benefits, including:

Efficient parsing: Predictive parsing is a top-down approach that uses the FIRST and FOLLOW sets to determine the next token. This makes it more efficient than bottom-up parsing approaches.
Easy to implement: Predictive parsing is relatively easy to implement, especially for simple grammars.
Good for syntax-directed editing: Predictive parsing is well-suited for syntax-directed editing, where the parser needs to predict the next token in the input stream.

Q: What are the limitations of predictive parsing?

A: Predictive parsing has several limitations, including:

Limited applicability: Predictive parsing is not suitable for all types of grammars, especially those with complex or ambiguous productions.
Difficulty with ambiguous grammars: Predictive parsing can struggle with ambiguous grammars, where there are multiple possible parses for a given input.
Difficulty with left-recursive grammars: Predictive parsing can struggle with left-recursive grammars, where a non-terminal symbol appears on the left-hand side of a production rule.

Q: How is predictive parsing used in practice?

A: Predictive parsing is used in a variety of applications, including:

Compilers: Predictive parsing is used in compilers to parse the source code and generate machine code.
Syntax-directed editors: Predictive parsing is used in syntax-directed editors to predict the next token in the input stream and provide syntax-directed editing capabilities.
Language processors: Predictive parsing is used in language processors to parse the input and generate output.

Q: What are some common mistakes to avoid when using predictive parsing?

A: Some common mistakes to avoid when using predictive parsing include:

Not considering the FIRST and FOLLOW sets: Failing to consider the FIRST and FOLLOW sets can lead to incorrect predictions and parsing errors.
Not handling ambiguous grammars correctly: Failing to handle ambiguous grammars correctly can lead to incorrect predictions and parsing errors.
Not handling left-recursive grammars correctly: Failing to handle left-recursive grammars correctly can lead to incorrect predictions and parsing errors.