Creating Parsing Tables: Methods & Techniques

by Alex Braham 46 views

Hey guys! Ever wondered how compilers understand the code we write? One of the key components is the parsing table. This table acts like a roadmap, guiding the compiler through the syntax of the code. So, let's dive into the methods for creating parsing tables!

What is a Parsing Table?

First off, what exactly is a parsing table? Think of it as a cheat sheet for a parser. A parser is a component of a compiler that takes the source code and checks if it follows the grammar rules of the programming language. The parsing table provides the parser with the necessary information to make these checks efficiently. It essentially tells the parser what action to take based on the current input symbol and the current state of the parser. Without a well-constructed parsing table, the parser would be lost and unable to determine if the code is syntactically correct.

Parsing tables come in different flavors, primarily for top-down and bottom-up parsing techniques. Top-down parsing starts from the start symbol of the grammar and tries to derive the input string. Bottom-up parsing, on the other hand, starts from the input string and tries to reduce it to the start symbol. Each of these approaches requires a specific type of parsing table tailored to its needs. The choice of parsing technique and, consequently, the type of parsing table depends on the characteristics of the grammar and the desired performance of the compiler. Understanding the different types of parsing tables is crucial for anyone looking to delve deeper into compiler design and language processing.

Different types of parsing tables exist, such as LL(k) for top-down parsing and LR(k), SLR(k), and LALR(k) for bottom-up parsing. The 'k' in these notations refers to the number of lookahead symbols used by the parser to make decisions. Lookahead symbols are the next 'k' symbols in the input stream that the parser considers before taking an action. A larger 'k' value allows the parser to handle more complex grammars but also increases the size and complexity of the parsing table. The selection of an appropriate parsing table generation method involves balancing the expressive power needed to handle the grammar with the computational resources available.

The complexity of creating parsing tables arises from the need to handle various grammar constructs, such as ambiguity, left recursion, and common prefixes. Ambiguity occurs when a grammar allows multiple parse trees for the same input string, making it difficult for the parser to choose the correct one. Left recursion occurs when a non-terminal can derive a string that starts with itself, leading to infinite loops in top-down parsing. Common prefixes occur when multiple production rules start with the same symbols, causing the parser to be unsure of which rule to apply. Each of these issues requires specific techniques to resolve when constructing the parsing table.

Top-Down Parsing Table Creation

Top-down parsing table creation involves constructing tables that guide the parser in predicting which production rule to apply based on the current input symbol. The most common type of top-down parsing is LL parsing, where the first 'L' stands for scanning the input from left to right, and the second 'L' stands for producing a leftmost derivation. These tables are often used in LL(k) parsers, where 'k' represents the number of lookahead tokens used to make parsing decisions. The key is to figure out, based on the next input token, which production rule should be used to expand a non-terminal symbol.

To create an LL(1) parsing table (the simplest form), we need to compute two sets for each production rule: FIRST and FOLLOW. The FIRST set of a production rule contains the set of terminal symbols that can start a string derived from the right-hand side of the production. The FOLLOW set of a non-terminal contains the set of terminal symbols that can immediately follow that non-terminal in some sentential form. These sets are crucial for determining which entry in the parsing table should correspond to a given production rule.

The algorithm for constructing an LL(1) parsing table generally follows these steps. First, compute the FIRST and FOLLOW sets for all non-terminals in the grammar. Then, for each production rule A -> α, where A is a non-terminal and α is a string of terminals and non-terminals, do the following: for each terminal 'a' in FIRST(α), add the production A -> α to the parsing table entry M[A, a]. If ε (epsilon, representing an empty string) is in FIRST(α), then for each terminal 'b' in FOLLOW(A), add the production A -> α to the parsing table entry M[A, b]. Finally, if ε is in FIRST(α) and '

(end of input marker) is in FOLLOW(A), add the production A -> α to the parsing table entry M[A, $].

However, not all grammars are suitable for LL(1) parsing. Grammars with left recursion or ambiguity can cause conflicts in the parsing table, making it impossible to construct a valid LL(1) parser. Left recursion, where a non-terminal can derive a string starting with itself, leads to infinite loops in the parser. Ambiguity, where the same input string can have multiple parse trees, results in multiple entries in the same parsing table cell. To handle such grammars, transformations like left recursion elimination and left factoring are often necessary before constructing the parsing table.

Bottom-Up Parsing Table Creation

Bottom-up parsing, in contrast, builds the parse tree from the leaves (input tokens) towards the root (start symbol). This approach uses parsing tables to determine when to shift input tokens onto the stack and when to reduce a sequence of symbols on the stack according to a production rule. LR parsing is a common type of bottom-up parsing, with variations like SLR, LALR, and CLR, each offering different trade-offs between parsing power and table size. These parsing methods are more powerful than LL parsing and can handle a broader range of grammars.

The creation of bottom-up parsing tables revolves around the concept of LR items. An LR item is a production rule with a dot at some position on the right-hand side, indicating how much of the rule has been seen so far. For example, the production A -> XYZ might have LR items like A -> .XYZ, A -> X.YZ, A -> XY.Z, and A -> XYZ.. These items are grouped into sets called LR states, which represent the different possible states the parser can be in while processing the input.

The algorithm for constructing an SLR(1) parsing table involves the following steps. First, augment the grammar by adding a new start symbol S' and a production S' -> S, where S is the original start symbol. Then, construct the set of LR(0) items for the augmented grammar. Next, define the closure operation, which adds all items that can be derived from the current items. The goto operation then determines the transitions between LR states based on the input symbols. Using these operations, build the collection of sets of LR(0) items, which represents the states of the parser.

Once the LR(0) states are constructed, the SLR(1) parsing table is built as follows. For each state 'i' and terminal 'a', if there is a transition from state 'i' to state 'j' on input 'a', then set the action table entry ACTION[i, a] to