A (finite non-deterministic) automaton is a quintuple M = (Q,Act,q0,F,T), where:
- Q is the set of states
- Act is the set of actions
- q0 is the starting state
- F is the set of final states
- T is the transition relation (T ⊆ Q × Act × Q)

Automata Behaviour: language equivalence
	(where L(M) is the set of all the sequences of input characters that bring the automaton M from its starting state to a final one)

>[!note] Language equivalence
>M1 and M2 are *language equivalent* if and only if L(M1)=L(M2)

![](../../Pasted%20image%2020250408091924.png)
By considering the starting states as also final, they both generate the same language, i.e.:
$$(20.(tea + 20.coffee))∗ = (20.tea + 20.20.coffee)∗$$


But, do they behave the same from the point of view of an external observer??
![](../../Pasted%20image%2020250408092853.png)
The essence of the difference is WHEN the decision to branch is taken
-  language equivalence gets rid of branching points
	- it is too coarse for our purposes!

### LTSs
In concurrency theory, we don’t use finite automata but Labeled Transition System (LTS). The main differences between the two formalisms are:
- automata usually rely on a finite number of states, whereas states of an LTS can be infinite
- automata fix one starting state, whereas in an LTS every state can be considered as initial (this corresponds to different possible behaviors of the process)
- automata rely on final states for describing the language accepted, whereas in LTS this notion is not very informative

>[!note] LTS formal definition
>Fix a set of action names (or, simply, actions), written N.
>
>A Labeled Transition System (LTS) is a pair (Q, T), where Q is the set of states and T is the transition relation (T ⊆ Q × N × Q).

We shall usually write s –a–> s′ instead of ⟨s,a,s′⟩ ∈ T.

### Bisimulation