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Concurrent Systems/notes/11 - LTSs and Bisimulation.md

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+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