Design a machine that accepts the language of all strings over {1,0,x} of the form uv, where v=1x0 and u consists of any number of copies a block of x's with a 1 on each side. Strings like 1xx11x11x0 or 1xxxx11x0 or 1xxx1111x0 or 1x11x0 or 1x0.

One way to do this is to design a machine that acccepts 1x0,and a machine that accepts zero or more copies of 1 followed by zero or more x's followed by 1, and then tie them together.

This is, of course, too easy! So let's try to make a machine that accepts strings that look like some number of strings accepted by the above machine, all tacked end to end. In other words, I should accept a string if it can be broken up into pieces, each of which starts and ends with the same letter and has length at least 2

Hopefully you're starting to see that these λ-transitions make creating new machines from old ones quite easy. In fact we could have done both of our big examples just using regular old non-dterminism --- i.e. two or more transitions for the same state/character combination. However, the λ-transisions let us do things without having to pick apart the original machines and see how they work. That'll help us later when we construct algorithms on nondeterministic machines.

1. in the nondeterministic case we have "missing" transitions, so our δ wouldn't be defined for all elements of its domain
2. in the nondeterministic case we have more than one possible state for some combinations of state/character, in other words our δ wouldn't be single-valued as a function must be
3. in the nondeterministic case we have λ-transitions which are defined for λ, not for an element of the alphabet - i.e. our δ needs to function outside of its domain

Definition: a nondeterministic finite automaton is an ordered 5-tuple (Q,Σ,Δ,s,W) where

• Q is the set of states (must be finite)
• Σ is the alphabet
• Δ ⊆ Q x (Σ ∪ {λ}) x Q is the transition relation
• s ∈ Q is the start state
• W ⊆ Q is the set of accepting states

This allows us to look at all the same kinds of examples we had for deterministic machines, going back and forth from diagram representation to 5-tuple representation.

5-tuple definition	({q0,q1,q2}, {a,b,c}, {(q0,b,q1),(q1,a,q1),(q1,b,q1), (q1,c,q1),(q1,c,q2)}, q0, {q0,q2})
diagram definition

What about the rules governing how an NDFA computes? Well, we can borrow the idea of configuration, "yields in one step" and "yields in many steps" from DFAs, as long as we add the possibility of λ-transitions to "yields". However, "yields" kind of mean different things for DFAs versus NDFAs. Instead of (p,u) ⇒ (q,v) meaning "if you're in configuration (p,u) then after one step you will be in configuration (q,v)", for NDFAs we may only say that (q,v) is one possibility ... there may well be others.

Definition: For NDFA M = (Q,Σ,Δ,s,W), we say that configuration (p,v) yields in one step configuration (q,w) if there is an x ∈ Σ ∪ {λ} such that v = xw and (p,x,q) ∈ Δ. This is sometimes denoted (p,v) ⇒ (q,w), or (p,v) ⇒_M (q,w) when you need to clearly indicate which machine you're referring to.

Next we will define the "yields" notion ("yields eventually" or "yields in zero of more steps") just as we did for DFAs.

Now the big question is how do we define "accept" and "reject" for NDFAs? Well, we define it the same way as for DFAs, it just means something a little different.

Definition: NDFA M = (Q,Σ,Δ,s,W) accepts string w if (s,w) ⇒* (q,λ) for some q ∈ W.

We can't talk about "the state we're in after all the input has been read", because there may be many possible states to be in after all the input is read (i.e. we have to say a rather than the), or there may be no possible states to be in with all the input read. This uncertainty is captured by the new "yields" definition, which says that (s,w) ⇒* (q,λ) if there is some sequence of configurations .... Note that now $w$ is "rejected" by $M$ when there is no state $q \in W$ for which (s,w) ⇒* (q,λ).