Prerequisite – Finite automata, L-graphs and what they represent

L-graphs can generate context sensitive languages, but it’s much harder to program a context sensitive language over programming a regular one. This is why I’ve came up with a hypothesis about what kind of L-graphs can generate a regular language. But first, I need to introduce you to what I call an iterating nest.

As you can remember a nest is a neutral path , where and are cycles and path is neutral. We will call an iterating nest, if , and paths print the same string of symbols several times, to be more exact prints , prints , prints , and is a string of input symbols (better, if at least one of ).

From this definition comes out the next hypothesis.

**Hypothesis –** If in a context free L-graph G all nests are iterating, then the language defined by this L-graph G, L(G), is regular.

If this hypothesis will be proven in the near future, it can change a lot in programming that will make creating new easy programming languages much easier than it already is. The hypothesis above leads to the next algorithm of converting context free L-graphs with iterating nests to an NFA.

**Algorithm –** Converting a context free L-graph with iterating complements to a corresponding NFA

**Input –** Context free L-graph with iterating complements

**Output –**

**Step-1:**Languages of the L-graph and NFA must be the same, thusly, we won’t need a new alphabet . (Comment: we build context free L-graph G’’, which is equal to the start graph G’, with no conflicting nests)**Step-2:**Build Core(1, 1) for the graph G.

V’’ := {(v, ) | v V of canon k Core(1, 1), v k}

:= { arcs | start and final states V’’}

For all k Core(1, 1):

Step 1’. v := 1st state of canon k. .

V’’

Step 2’. arc from state followed this arc into new state defined with following rules:

, if the input bracket on this arc ; , if the input bracket is an opening one; , if the input bracket is a closing bracket

v := 2nd state of canon k

V’’

Step 3’. Repeat Step 2’, while there are still arcs in the canon.**Step-3:**Build Core(1, 2).

If the canon has 2 equal arcs in a row: the start state and the final state match; we add the arc from given state into itself, using this arc, to .

Add the remaining in arcs v – u to in the form of**Step-4:**

(Comment: following is an algorithm of converting context free L-graph G’’ into NFA G’)**Step-5:**Do the following to every iterating complement in G’’:

Add a new state v. Create a path that starts in state , equal to . From v into create the path, equal to . Delete cycles and .**Step-6:**G’ = G’’, where arcs are not loaded with brackets.

So that every step above is clear I’m showing you the next example.

Context free L-graph with iterating complements

,

which determines the

Start graph G

Core(1, 1) = { 1 – a – 2 ; 1 – a, (1 – 1 – a – 2 – a, )1 – 2 ; 1 – b, (2 – 2 – c, )2 – 3 }

Core(1, 2) = Core(1, 1) { 1 – a, (1 – 1 – a, (1 – 1 – a – 2 – a, )1 – 2 – a, )1 – 2 }

Step 2: Step 1’ – Step 3’

Intermediate graph G’’

NFA G’

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