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Ex. 1 : Given E={ p , q , r }, let =2 E . Express the behaviors over thatPowerPoint Presentation

Ex. 1 : Given E={ p , q , r }, let =2 E . Express the behaviors over that

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RTS Development by the formal approach

Assignment #1

Ex. 1:

Given E={p,q,r}, let =2E. Express the behaviors over that

satisfy the following properties by proper -regular expressions.

1. initial p is followed by q at the next step: [p][q] [/{p}]

2. p and q never occursimultaneously: [/{p,q}]

3. p cannot occur before q : [/{p,q}]*[q] [/{q}]

4. poccurs at every step (strictly) between qandnextr:

([/{q}]*[q][p/{r}]*[r]) -- infinite occurrences

([/{q}]*[q][p/{r}]*[r])*([/{q}]*[q])1[/{r}]

([/{q}]*[q][p/{r}]*[r])*[/[q]] -- finite occurrences

possibly r here

Ex. 2: Prove that L(), the set of all models of an LTL formula , is an -regular language.

- By induction on the structure of
- tt: Lω(tt) = Σω
- p: Lω(p) = {Σω | p0} = [p]Σω
- , where Lω() is ω-regular:
- Lω() = {Σω | |=}
- = Lω()c -- closure under complementation
- , where Lω(), Lω() are ω-regular:
- Lω(v ) = {Σω | |= or |=}
- = {Σω | |=} U {Σω | |=} = Lω() U Lω(C)
- -- closure under union

- O, where Lω() is ω-regular:
- Lω(O) = {Σω | |=O}
- = {Σω | 1|=} = ΣLω() - by construction/definition
- U, where Lω(), Lω() are ω-regular:
- Lω(U) = {Σω | |= U}
- = {Σω |∃k0 s.t. ∀0≤j<k j|= and k|=}
- = Lω() U (Lω() ∩ (Σ1Lω())
- U (Lω() ∩ Σ1Lω() ∩ Σ2Lω()) U …
- = Uk≥0 ((Σ0Lω() ∩ Σ1Lω() ∩… ∩ Σk-1Lω() ∩ ΣkLω())
- = Uk≥0 (∩0≤j≤k-1ΣjLω()∩ΣkLω())
- -- closure under union and intersection.

Prove that qis semantically equivalent to (q) (namely: qiff (q) ).

q iff iiq -- (semantics of q)

iff iiq -- (semantics of q)

iff i(iq) -- (semantics of )

iff i(iq) -- (semantics of , )

iff i(iq) -- (semantics of )

iff (q) -- (semantics of )

iff (q) -- (semantics of (q))

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