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Prove that if A is a regular language, a family of branching programs(B1,B2) exists wherein eachBn accepts exactly the strings in A of length n and is bounded in size by a constant times n.

Short Answer

Expert verified

The definition of regular language needs to be understood to solve the above problem i.e., “a formal language that is expressed using a regular expression, is called as a regular language”.

Step by step solution

01

Define Regular Language

Now consider a regular language,A then a family of branching program (B1,B2)in which a string, of length nin language,A is accepted by each Bnand is bounded by a fixed timen in size. It is possible to do so in the manner described below.

02

Understand the Branching Program

Now consider a branching program. A branching program is known as “a directed acyclic graph where labels of all the nodes are maintained by the variables, except for two output nodes which are labelled as 1 or 0 .


The term "query node" refers to all nodes whose labels are kept up to date by the variables. Every query node consists of two outgoing edges: one is labelled 1 and another one is labelled .0 Both output nodes don’t consist outgoing edges.

03

Prove that 1(n-1)*2=O(n) 

The n-input function or a finite regular language can therefore be computed by a branching programme that consists of a constantO(n) size, according to the definitions of branching programme and regular language A as given above.
A circuit n is implemented as a Bubble-sort. Each branching programme in a group is chosen so that they all accept exactly the strings in A of length n.

After carefully comparing, it can be used to compare two bits. Let the inputs bex1,x2and the outputs bey1,y2.


A sub-circuit can be written which accomplishes this asy1=OR(x1,x2) &y2=AND(x1,x2). This circuit contains a size of two.


Now, an array can be mimicked using the action of the bubble-sort algorithm. It can be implemented one step at position to be the n input, n-output subcircuit that passes through all the inputs taken as<k and k+1are unchanged.

Now, the above described compare-swap sub-circuit, on <kandk+1st input is used for the generation of kthand output.k+1st This still has size two. Now, a pass can be implemented as the serial concatenation of steps for each of , k=1,2,,n-1which has a size.(n-1)*2


A bubble-sort can be Proceed to implement as the serial concatenation of one pass. Therefore, this gives a size

1(n-1)*2=O(n)

This means that "a family of branching programmes (B1,B2)in which a string, of length n in language A, is accepted by each member Bn" may be inferred from the reasoning given above.

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