Question 2 (10 pt): In this question we look at an extension to DFAs. A composable-reset DFA (CR-DFA) is a five-tuple, (Q,S,d,q0,F) where:

– Q is the set of states,

– S is the alphabet,

– d:Q×(S?{?})?Qisthetransitionfunction, – q0 ? Q is the start state, and

– F ? Q is the set of accept states.

Every CR-DFA must satisfy one additional property:

When running a CR-DFA one can take a ?-transition if and only if the input has already been exhausted, and d cannot have any cycles that have a ?-transition.

A CR-DFA differs from a DFA by the addition of a new symbol denoted ? which can only be used by the transition function. This symbol is not part of the alphabet of the DFA.

The run function for a CR-DFA is defined as follows:

dˆ 0 : Q × S * × S * ? Q dˆ0(q,e,w1) = q

if d(q, ?) is undefined. dˆ0(q, e, w1) = dˆ0(q', w1, w1)

if d(q, ?) = q'

dˆ0(q, aw, w1) = dˆ0(q', w, w1)

if d(q, a) = q' dˆ : Q × S * ? Q

dˆ ( q , w ) = dˆ ( q , w , w ) 0

1

We can see that the run function, dˆ, is defined interms of an auxiliary function called dˆ0. The latter takes three arguments: i. the current state, the input word, and a second input word called w1. The second input word is called an accumulator, and it will be used to remember the original input to the run function, but when defining the auxiliary run function we leave this arbitrary.

The definition of the auxiliary run function follows the definition of the run function for DFAs, but in the case where the input word has been exhausted we check to see if the transition function allows the input to be reset to w1, and if it does, then we call dˆ0 on the next state given by d, and the input word is reset to w1. If when the input is exhausted and the transition function does not allow a ?-transition, then we proceed as usual.

Note that the definition of acceptance for a CR-DFA is the same as for DFAs.

We now define an interesting language. Suppose S = {a, b, c, d, ?, ?} is an alphabet. The symbol ? represents a binary operation, and the symbols a, b, c, d, and ? represent inputs to the binary operation ?. The language L is defined by the following:

i. a,b,c,d,? ? L

ii. Foranyei ?S,thewordw=e1?e2?e3?···?en ?L

iii. For any w ? L, any well-balanced parenthesization of w is a member of L

iv. There are no other words in L.

The following are some example words in L:

a

b

c

d

?

(a?b) (a?(b?c)) (a?(b?(c?d))) a?b?c (a?b)?c

So the words of L are all the possible associations of applications of the binary operation ?. Define a CR-DFA in the diagrammatic from used with DFAs that recognizes the language L as defined above. In addition, describe why CR-DFAs are bad in practice.

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