L-graphs and what they represent in TOC
Prerequisite – Finite automata introduction
All programming languages can be represented as a finite automata. C, Paskal, Haskell, C++, all of them have a specific structure, grammar, that can be represented by a simple graph. Most of the graphs are NFA’s or DFA’s. But NFA’s and DFA’s determine the simplest possible language group: group of regular languages [Chomsky’s hierarchy]. This leaves us with a question: what about all other types of languages? One of the answers is Turing machine, but a Turing machine is hard to visualize. This is why in this article I will tell you about a type of finite automata called an L-graph.
In order to understand how L-graphs work we need to know what type of languages L-graphs determine. To put it simply, L-graphs represent context-sensitive type of languages [and every other type that the context-sensitive group contains]. If you don’t know what “context-sensitive” means, let me show you an example of a language that can be represented by an L-graph and not by any easier type of finite automata.
This language is . Corresponding L-graph looks like this:
As you can see the brackets after the symbol ‘|’ control the numbers of symbols that come after the symbols ‘a’. This leads us to the two features that all L-graphs possess: all L-graphs have up to two independent from each other and from input symbols bracket groups, both bracket groups have to be right [string from a Dyck language] in order for the string of input symbols to be accepted by the given L-graph.
You can see that an L-graph is just a version of finite automata with an added couple of bracket groups. To help you get an understanding of why the languages determined by L-graphs are context-sensitive, check what strings the L-graph shown above has to accept.
To conclude, I would like to add three other definitions that I’ll be using in the future. These definitions are very important for the hypothesis [and its future proof or disproof]. Refer – Hypothesis (language regularity) and algorithm (L-graph to NFA)
We will call a path in the L-graph neutral, if both bracket strings are right. If a neutral path T can be represented like this, T = , where and are cycles and is a neutral path (, or can be empty), T is called a nest. We can also say that the three (, , ) is a nest or that and form a nest in the path T.
(, d)-core in an L-graph G, defined as Core(G, , d), is a set of (, d)-canons. (, d)-canon, where and d are positive whole numbers, is a path that contains at most m, , neutral cycles and at most k, k d, nests that can be represented this way: is part of the path T, , i = 1 or 3, , are cycles, every path is a nest, where = , .
The last definition is about a context free L-graph. An L-graph G is called context free if G has only one bracket group (all rules in the L-graph have only one look of these two: [‘symbol’ | ‘bracket’, ?] or [‘symbol’ | ?, ‘bracket’]).
[Definition of a Dyck language. and are disjoint alphabets. There exists a bijection (function that for every element from the 1st set matches one and only one element from the 2nd set) . Then the language defined by the grammar , , we will call a Dyck language.]