1. Conformance Simulation Relation ( )Let and be two automata over the same alphabet simulates ( ) if there exists a simulation relation such that implies that Note that simulates but these are not equivalent notions. If may be easier to find a simulation relation than to prove language containment.

2. Language Containment ( ) To show that ) we typically show that (i.e. This requires complementing which may be hard if is non-deterministic (subset construction). So simulation may be easier to check. Equivalence ( ):

3. O1 F I1 most general U2 U1 O2 X I2 Topologies

4. F F I1 I1 one-way cascade two-way cascade U2 U1 U1 O2 O2 X X We can also switch X and F

5. O1 F F I1 Engineering Change U2 U2 U1 U1 O2 rectification X X I2

6. Controller F U2 U1=O1 X I2

7. Solving a language equation Solve where and In particular, find the largest solution X (most general solution). Theorem A: Let A and C be languages over alphabets and respectively. For the equation, the most general solution is Theorem B: Let A and C be languages over alphabets and respectively. For the equation, the most general solution is .

8. Proof: We prove Theorem A. Let . Then means that Thus is the largest solution of The proof of Theorem B is similar.

9. i/u s s’ s’’ i u s s’ Mapping Parallel into Synchronous Suppose F is a FSM on inputs i,v and outputs u,o and S is an FSM on inputs i and outputs o. Transitions are one of the forms (s i/u s’) (s i/o s’) (s v/u s’) or (s v/o s’). For S, its transitions are of the type (q i/o q’). For each, we convert into automata by creating new intermediate states between inputs and outputs. Thus a transition (s i/u s’) becomes two transitions (s i s’’) (s’’ u s’). Similarly for the others. For S a transition of the type (q i/o q’) becomes (q i q’’) (q’’ o q’).

10. Now for each s’’ we add a self loop labeled by s’’ -u i o -u s s’ To expand , for each original state, we add a self loop labeled with –u. denoting any symbol in the alphabet u i/o s s’ . With these conversions, we can do synchronous composition and get the equivalent expanded result of parallel composition Thus we need to implement only one type of compositional method – synchronous, and simply have mapping of each machine into its extended machine to compose in parallel. Finally, we can take the solution and map it back into an FSM.