Creating parsing tables is a fundamental aspect of compiler design and language processing. These tables guide the parser in understanding the structure of a program or input text, enabling it to verify syntax and build an internal representation for further processing. In this article, we will explore the various methods and techniques involved in constructing parsing tables, shedding light on their significance and practical applications.

Understanding Parsing Tables

Before we dive into the creation methods, let's first understand what parsing tables are and why they are so important. Parsing tables are essentially lookup tables that a parser uses to determine what action to take based on the current input symbol and the current state of the parser. These tables are typically used in conjunction with a parsing algorithm, such as LL (Left-to-right, Leftmost derivation) or LR (Left-to-right, Rightmost derivation) parsing.

The parser reads the input from left to right, one symbol at a time. The parsing table provides instructions on whether to shift the input symbol onto the stack, reduce a sequence of symbols on the stack according to a grammar rule, accept the input, or report an error. The structure of the parsing table depends on the type of grammar and the parsing algorithm being used. LL parsing tables are used for top-down parsing, while LR parsing tables are used for bottom-up parsing. The choice of parsing method and table structure depends on the characteristics of the language being parsed and the desired performance of the parser.

Parsing tables play a crucial role in ensuring the correctness and efficiency of the parsing process. They provide a systematic way to handle the different possibilities that can arise during parsing, allowing the parser to make informed decisions about how to proceed. Without parsing tables, the parsing process would be ad-hoc and error-prone, making it difficult to reliably process complex languages.

Top-Down Parsing and LL(k) Grammars

Top-down parsing starts from the root of the parse tree and tries to derive the input string by expanding non-terminals according to the grammar rules. LL(k) grammars are a class of context-free grammars that can be parsed using a top-down parser with k symbols of lookahead. The "LL" stands for "Left-to-right, Leftmost derivation," and the "k" indicates the number of lookahead symbols used to make parsing decisions.

Constructing LL(1) Parsing Tables

LL(1) parsing is a special case of LL(k) parsing where only one lookahead symbol is used. Constructing an LL(1) parsing table involves computing two sets for each non-terminal in the grammar: FIRST and FOLLOW. The FIRST set of a non-terminal A is the set of terminals that can start a string derived from A. The FOLLOW set of a non-terminal A is the set of terminals that can immediately follow A in some sentential form.

To construct the LL(1) parsing table, we iterate through each production rule in the grammar. For each rule A → α, where A is a non-terminal and α is a string of terminals and non-terminals, we add entries to the parsing table as follows:

1. For each terminal a in FIRST(α), add the production rule A → α to the table entry T[A, a].
2. If ε (epsilon) is in FIRST(α), then for each terminal b in FOLLOW(A), add the production rule A → α to the table entry T[A, b].
3. If ε is in FIRST(α) and $ (end-of-input marker) is in FOLLOW(A), then add the production rule A → α to the table entry T[A, $].

If any table entry contains more than one production rule, then the grammar is not LL(1), and a different parsing method may be required.

Example of LL(1) Parsing Table Construction

Consider the following grammar for simple arithmetic expressions:

E -> T E'
E' -> + T E' | ε
T -> F T'
T' -> * F T' | ε
F -> ( E ) | id

First, we compute the FIRST and FOLLOW sets for each non-terminal:

• FIRST(E) = {(, id}
• FIRST(E') = {+, ε}
• FIRST(T) = {(, id}
• FIRST(T') = {*, ε}
• FIRST(F) = {(, id}
• FOLLOW(E) = {), $}
• FOLLOW(E') = {), $}
• FOLLOW(T) = {+, ), $}
• FOLLOW(T') = {+, ), $}
• FOLLOW(F) = {*, +, ), $}

Using these sets, we can construct the LL(1) parsing table:

Bottom-Up Parsing and LR Grammars

Bottom-up parsing, also known as shift-reduce parsing, starts from the input string and tries to reduce it to the start symbol of the grammar. LR grammars are a class of context-free grammars that can be parsed using a bottom-up parser. The "LR" stands for "Left-to-right, Rightmost derivation in reverse."

Constructing SLR(1) Parsing Tables

SLR(1) parsing is a simple type of LR parsing that uses a single lookahead symbol. Constructing an SLR(1) parsing table involves the following steps:

1. Augment the Grammar: Add a new start symbol S' and a production rule S' → S, where S is the original start symbol.

2. Compute the Closure of Item Sets: An item is a production rule with a dot (.) at some position in the right-hand side. The closure of an item set I is the set of all items that can be derived from the items in I by repeatedly applying the following rules:

   • If A → α. B β is in I and B → γ is a production rule, then add B → .γ to I.

3. Compute the GOTO Function: The GOTO function maps an item set and a grammar symbol to a new item set. GOTO(I, X) is the closure of the set of all items [A → α X. β] such that [A → α. X β] is in I.

4. Construct the Canonical Collection of Item Sets: Start with the initial item set I₀ = closure({S' → .S}) and repeatedly apply the GOTO function to generate new item sets until no more new item sets can be generated. The resulting collection of item sets is called the canonical collection.

5. Construct the Parsing Table: The parsing table consists of two parts: the ACTION table and the GOTO table. The ACTION table specifies what action the parser should take based on the current state and the input symbol. The GOTO table specifies the next state to go to after a shift or reduce action.

   • For each item set Iᵢ in the canonical collection:
     • If [A → α. a β] is in Iᵢ and GOTO(Iᵢ, a) = Iⱼ, then set ACTION[i, a] to "shift j".
     • If [A → α.] is in Iᵢ, then for each terminal a in FOLLOW(A), set ACTION[i, a] to "reduce A → α".
     • If [S' → S.] is in Iᵢ, then set ACTION[i, $] to "accept".
   • For each item set Iᵢ and non-terminal A, if GOTO(Iᵢ, A) = Iⱼ, then set GOTO[i, A] to j.

If any table entry contains more than one action, then the grammar is not SLR(1), and a more powerful parsing method may be required.

Example of SLR(1) Parsing Table Construction

Consider the following grammar for simple arithmetic expressions:

E -> E + T | T
T -> T * F | F
F -> ( E ) | id

After augmenting the grammar, we have:

E' -> E
E -> E + T | T
T -> T * F | F
F -> ( E ) | id

We can then compute the canonical collection of item sets and construct the SLR(1) parsing table. The resulting table would guide the parser in performing shift and reduce actions to parse input strings according to the grammar.

Other Parsing Techniques

While LL(1) and SLR(1) parsing are widely used, there are other parsing techniques that can handle more complex grammars. These include:

• LALR(1) Parsing: LALR(1) (Look-Ahead LR) parsing is a more powerful variant of LR parsing that merges item sets with the same core but different lookahead sets. This reduces the size of the parsing table compared to canonical LR parsing while still being able to handle a large class of grammars.
• Canonical LR(1) Parsing: Canonical LR(1) parsing is the most general form of LR parsing. It uses the full lookahead information to make parsing decisions, allowing it to handle a wider range of grammars than SLR(1) or LALR(1) parsing. However, the parsing tables for canonical LR(1) parsing can be very large, making it less practical for some applications.
• Recursive Descent Parsing: Recursive descent parsing is a top-down parsing technique that uses a set of recursive procedures to implement the grammar rules. Each non-terminal in the grammar is associated with a procedure that attempts to parse a string derived from that non-terminal. Recursive descent parsing is easy to implement but may not be suitable for all grammars, especially those with left recursion.

Conclusion

Creating parsing tables is a crucial step in building a compiler or interpreter for a programming language. The choice of parsing method and table construction technique depends on the characteristics of the language and the desired performance of the parser. LL(1) and SLR(1) parsing are simple and efficient methods for parsing a large class of grammars, while LALR(1) and canonical LR(1) parsing can handle more complex grammars at the cost of increased table size and complexity. Understanding the different methods for creating parsing tables is essential for any computer scientist or software engineer working with language processing.

By understanding these methods, you can effectively develop parsers that are both accurate and efficient, ensuring that your language processing tools are up to the task of handling complex and varied input.