Parse Tree in Automata Theory

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Parsing is an important part in context of automata and compiler design perspective. Parsing refers to the process of analyzing a string of symbols, either in natural language or programming language, according to the rules of a formal grammar. This is done by the concept called parser, which determines if a given string belongs to a language defined by a specific grammar.

The parser often generates a graphical representation of the derivation process. In this chapter, we will explain the concept of parsing and parse trees through examples.

Basics of Context-Free Grammar (CFG)

Before getting the idea of parse trees, it is essential to understand Context-Free Grammar (CFG). A CFG consists of a set of production rules that describe all possible strings in a given formal language. These production rules define how the symbols in the language can be combined and transformed to generate valid strings.

A Context-Free Grammar $\mathrm{G}$ is defined by four components −

• $\mathrm{V}$: A finite set of variables (non-terminal symbols).
• $\mathrm{T}$: A finite set of terminal symbols.
• $\mathrm{S}$: A start symbol, which is a special non-terminal symbol from $\mathrm{V}$.
• $\mathrm{P}$: A finite set of production rules, each having a form $\mathrm{A \rightarrow \alpha A}$, where $\mathrm{A}$ is a non-terminal and $\mathrm{\alpha}$ is a string consisting of terminals and/or non-terminals.

Parse Trees and Their Role in Parsing

A parse tree, also known as derivation tree is a graphical representation of the derivation of a string according to the production rules of a CFG. It shows how a start symbol of a CFG can be transformed into a terminal string, using applying production rules in a sequence.

Each node in a parse tree represents a symbol of the grammar, while the edges represent the application of production rules.

• Tree Nodes − The nodes in a parse tree represent either terminal or non-terminal symbols of the grammar.
• Tree Edges − The edges represent the application of production rules leading from one node to another.

Types of Parsers

Parsers can be broadly categorized into two types based on how they construct the parse tree −

• Top-down Parsers − These parsers build the parse tree starting from the root and proceed towards the leaves. They typically perform a leftmost derivation. They expand the leftmost non-terminal at each step.
• Bottom-up Parsers − These parsers start from the leaves and move towards the root, performing a rightmost derivation in reverse order.

Example of Parse Tree Construction

Let us consider an example of a parse tree construction using a simple grammar −

Given grammar $\mathrm{G}$ −

$$\mathrm{E \:\rightarrow \: E \:+\:E \:| \:E\: \times \:E\:|(E)|\:|\:-\:E|\:id}$$

Deriving the String "-(id + id)"

To determine whether the string "-(id + id)" is a valid sentence in this grammar, we can construct a parse tree as follows −

• Start with $\mathrm{E}$.
• Apply the production $\mathrm{E \:\rightarrow \:-\: EE}$ to get the intermediate string "\mathrm{- E}".
• Apply $\mathrm{E \:\rightarrow\: (E)}$ to derive "$\mathrm{-(E)}$".
• Apply $\mathrm{E \:\rightarrow\: E \:+\: E}$ to derive "$\mathrm{-(E \:+\: E)}$".
• Finally, apply $\mathrm{E \:\rightarrow\: id}$ to get the final string "$\mathrm{-(id \:+\: id)}$".

Here is the corresponding parse tree −

This representation shows the hierarchical structure of the derivation, with the root representing the start symbol and the leaves representing the terminal symbols of the string.

Another Example of Parse Tree

Consider a different grammar for a second example −

Given grammar $\mathrm{G}$

$$\mathrm{S \:\rightarrow\: SS\: |\: aSb\: |\: \varepsilon}$$

To derive the string "aabb", we proceed as follows −

• Start with $\mathrm{S}$.
• Apply $\mathrm{S \:\rightarrow\: aSb}$ to get "$\mathrm{aSb}$".
• Apply $\mathrm{S \:\rightarrow\: asb}$ again within the non-terminal $\mathrm{S}$ to derive "$\mathrm{aaSbb}$".
• Finally, apply $\mathrm{S \:\rightarrow\: \varepsilon}$ to replace the last $\mathrm{S}$, resulting in "$\mathrm{aabb}$".

The corresponding parse tree can be represented like below −

Ambiguity in Parse Trees

A grammar is said to be ambiguous if there exists at least one string that can be derived in more than one way. So there are multiple distinct parse trees. Ambiguity poses challenges when parsing as it can lead to multiple interpretations of the same string.

For instance, consider the following grammar −

$$\mathrm{S \:\rightarrow\:S \:+\: S\:|\:S\:\times\:S\:|\:id}$$

The string "id + id × id" can have two different parse trees depending on the order of operations −

First Parse Tree

Second Parse Tree

Conclusion

Parse trees are a basic concept in Theory of Computation and Compiler Design. Parse trees are used to parse context-free languages. They provide a visual representation of how a string is derived from the grammars start symbol. It offers an insight into the structure and hierarchy of the language.