 The next session that is non-deterministic finite automata with null transition myself Rashmi Dixit. Learning outcome at the end of this session students will be able to define non-deterministic finite automata with null transition and they are able to find null closure set of particular NFA with null transition. Let us begin the session. Finite automaton may be DFA or may be NFA are characterized because of transition on a particular symbol. In deterministic finite automaton there is a exact one transition for each and every input symbol. Vice versa in non-deterministic finite automaton there may be zero or more than one transition for input symbol. DFA or NFA they both accept the same type of regular language. Any NFA can be converted to DFA using subset construction method. Now the third machine is non-deterministic finite automata with null transition. This particular symbol is used to show null transition or empty string. Now what is this machine? It is extension to NFA a feature called null transition denoted by this particular symbol the empty string. The null transition let us spontaneously take a transition without receiving an input symbol. Mechanism that allows our NFA to be in multiple states at once. Whenever we take a null edge we move to new state. It is sometime convenient but remember one thing it has no more power than normal NFA. Just like an NFA has no more power than a DFA. Now look at this diagram Q is the initial state F is the final or accepting state. Now from Q by processing 0 machine enters to the next state that is R. Now from R without processing anything or by processing a null symbol or empty string machine enters into state S. From S without processing anything machine enters to the state T. From T by processing 1 machine enter into state U and again from U without processing anything machine enters into final or accepting state. So from Q to F 0 null null 1 null that is 0 1 string is accepted by given machine. Yes it is the same machine accept a string 0 1 but in between there is a movement from one state to another without processing anything or by processing an empty string that is NFA with a null transition. This is an definition. NFA with a null transition is the five tuple machine M where Q indicate a finite set of states alphabet a finite set of input symbol Q 0 the initial or a starting state which belongs to Q F a set of final or accepting states which is the subset of Q transition function. Now we will see three differences or differences between the transition function of DFA NFA and NFA with a null. In DFA machine in any state by processing a single symbol at a time machine move to or remains in the same state that is Q. In NFA machine in any state by processing a single symbol machine moves to 2 raise to Q 2 raise to Q is the power set of Q. Now in NFA with null transition machine in any state by processing a symbol or null machine may move to 2 raise to Q 2 raise to Q is the power set of Q that is a subset of sorry the set of all subsets of Q. Here null symbol is included into alphabet it is not an alphabet but by processing null machine move to a new state or remains in the same state. Now what is this null closure of a set of states? Now extended transition function gives us where the machine ends after processing a complete string but in NFA with a null transition machine moves without processing anything. So before studying what is extended transition function or how we are going to find it it is compulsory to see what is null closure of a set of states. Now definition is here suppose m with a phi tuple is an NFA with a null transition and S be any subset of Q. The null closure of S is indicated by this open close bracket with S that can be defined recursively as follow every element of S is an element of a null closure set of that particular state. As we know that machine remains in the same state without processing anything. Now for every Q which belongs to the null closure set of state S there should be every element of that particular transition from Q without processing anything. Now there is a particular algorithm to calculate null closure set of that particular state initialize T null closure set to be S means from which we are starting this is not the starting state do not confuse between this. Make a sequence of passes in each pass considering every Q which belongs to T and adding to T every state in transition Q common null that is not already an element. Stop after the first pass in which T is not changed the final value of T is the null closure set of state. We will see one example so that you will get a perfect particular idea how to find the null closure set of state. Now just look at this diagram Q0 is initial state W is accepting state. Now I want to find the null closure of state S S is here. Now by definition S going to that answer state. Now from S machine move to W without processing anything so W is our next state from W machine move to Q0 without processing anything or by processing null. So S W Q0 so from Q0 machine move to P and T by processing null so P and T both will be included into our set. Now from P there is a no null transition from T also there is a no null transition so our set is complete so null closure of S is S W Q0 P T. Now all students please pause the video a transition table is given for an particular NFA with a null transition this NFA with a null transition having seven state. Now student please try to find answer for null closure of 2 3 and null closure of 1. Take a paper and solve it hope you got the answer. So null closure of 2 3 from 2 sorry 2 3 both will be into set from 2 without processing anything machine move to 5 so 5 from 5 there is a no transition for null so the answer is 2 3 5. Null closure of 1 by definition will be 1 will be into your set from 1 by processing null machine move to 2 from 2 by processing null machine move to 5 so from 5 there is a no transition for null so the answer is 1 2 5. So null closure of a particular state is where the machine move without processing anything from that state and after that we will see how the extended transition function works in NFA with a null transition. Now remember one thing NFA with a null or NFA or DFA all accept the same regular language NFA and NFA with a null transition gives us somewhat relaxation but as NFA with null transition gives us relaxation it also increases number of states as without processing anything we can enter into the new state. So this is the reference.