L-graphs and what they represent in TOC

In Theory of Computation (TOC), Finite Automata (DFA/NFA) are mathematical models used to recognize Regular Languages. However, not all programming languages can be represented using Finite Automata because they require more computational power than what FA can provide. Languages like C, Pascal, Haskell, and C++ follow specific structures and grammars that can be depicted using simple graphs. Most of these graphs are either Non-deterministic Finite Automata (NFA) or Deterministic Finite Automata (DFA). However, NFA and DFA determine only the simplest group of languages—regular languages—according to Chomsky hierarchy.

This raises the question: what about other types of languages? One of the answers is the Turing Machine, but Turing machines are difficult to visualize. This is where L-graphs come into play.

L-graphs extend finite automata concepts to represent context-sensitive languages. To understand L-graphs better, let’s examine an example of a language that cannot be represented by DFAs, NFAs, or PDAs.

Example Language

The language: L = \{ a^n b^n c^n \mid n \geq 1 \} can be represented using an L-graph but not by simpler finite automata. Corresponding L-graph looks like this:

Structure of L-Graphs

An L-graph possesses two key features:

1. Bracket Groups: L-graphs have up to two independent bracket groups that do not depend on input symbols.
2. Dyck Language Compliance: Both bracket groups must form a valid string from a Dyck language for the input string to be accepted.

Essentially, an L-graph is a modified finite automaton with added bracket groups.

How L-Graphs Accept Strings

Consider the following transitions:

• abc: \{\varepsilon, \varepsilon, \varepsilon\} \rightarrow \{a, (, \varepsilon\} \rightarrow \{ab, (), <\} \rightarrow \{abc, (), <>\}

• a²b²c²: \{\varepsilon, \varepsilon, \varepsilon\} \rightarrow \{a, (, \varepsilon\} \rightarrow \{aa, ((, \varepsilon\} \rightarrow \{aab, ((), <\} \rightarrow \{aabb, (()), <<\} \rightarrow \{aabbc, (()), <<>\} \rightarrow \{aabbcc, (()), <<>>\}

• a⁵b⁵c⁵: \{\varepsilon, \varepsilon, \varepsilon\} \rightarrow \{a, (, \varepsilon\} \rightarrow \dots \rightarrow \{aaaaa, (((((, \varepsilon\} \rightarrow \{aaaaab, (((((), <\} \rightarrow \dots \rightarrow \{aaaaabbbbb, ((((())))), <<<<<\} \rightarrow \{aaaaabbbbbc, ((((())))), <<<<<>\} \rightarrow \dots \rightarrow \{aaaaabbbbbccccc, ((((())))), <<<<<>>>>>\}

Definitions Related to L-Graphs

1. Neutral Path

A path in the L-graph is called neutral if both bracket strings are right-balanced.

2. Nest

If a neutral path T can be represented as: T = T₁T₂T₃ where T₁ and T₃ are cycles and T₂ is a neutral path, then T is called a nest.

3. (ω, d)-Core

A (ω, d)-core in an L-graph G, defined as Core(G, ω, D), is a set of (ω, d)-canons.

A (ω, d)-canon is a path that contains at most:

• m <= ω neutral cycles
• k <= d nests

where each path segment follows: T_{1_1} \dots T_{1_k} T_{2_1} T_{3_1} \dots T_{3_k}

4. Context-Free L-Graph

An L-graph is context-free if it has only one bracket group. All rules in the L-graph follow one of the two structures:

• ['symbol' \mid 'bracket', ?]
• ['symbol' \mid ?, 'bracket']

Dyck Language Definition

A Dyck language consists of two disjoint alphabets, \Sigma_{(} \text{ and } \Sigma_{)}. There exists a bijection: \phi: \Sigma_{(} \rightarrow \Sigma_{)} where the language is defined by the grammar: S \rightarrow \varepsilon \mid aSbS, a \in \Sigma_{(}, b = \phi(a)

Real-Life Example: Organizational Structure

A real-life application of L-graphs can be seen in hierarchical structures such as XML parsing and dependency graphs.

Scenario:

• The main graph represents the company’s departments (Sales, Marketing, Finance, HR, etc.).
• The L-graph represents managerial relationships, where each vertex is a manager and edges show supervisory relationships.

This representation helps in understanding:

• Chain of command
• Decision-making processes