A partial DFA is a 5tuple M Q Sigma delta q0 F where the on

A partial DFA is a 5-tuple M = (Q, Sigma, delta, q_0, F), where the only difference from a DFA is that the transition function is some delta: U rightarrow Q for some U SubsetEqual Q times Sigma. That is, a partial DFA is a DFA if and only if U = Q times Sigma. Are partial DFAs and DFAs equivalent? That is, for every partial DFA M, is there a DFA M\' such that L(M) = L(M\')?^1 Why or why not?

Solution

Deterministic Finite Automaton Formal definition :-
A deterministic finite automaton M is a 5-tuple machine consisting of (Q, , , q0, F),

Q:-a finite set of states
:-a finite set of input symbols
:-a transition function
q0:-an starting state (q0 Q)
F:-a set of accept states (F Q)

In other words,the machine starts with the start state q0. The given each character of string w, the machine will make a transition from one state to another state according to the
transition function . The machine accepts w the string if the last input of w causes the machine to halt in one of the accepting states.
Otherwise, it is said that the automaton rejects the string. The set of strings that M accepts is the language recognized by M and this language is denoted by L(M).

A deterministic finite automaton without accept states and without a starting state is known as a partial transition system.

According to the definition,
1.deterministic finite automata are always complete:
2.they define a transition for each state and each input symbol.

This is the most common definition, but some may use the term deterministic finite automaton for a slightly different notion:
1.an automaton that defines at most one transition for each state and each input symbol
2.the transition function is allowed to be partial.
3.When no transition is defined, such an automaton halts.

Hence by these definitions we can say that the DFAs and Partails DFAs are not equivalent.