RTS Development by the formal approach
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Ex. 1 : Given E={ p , q , r }, let =2 E . Express the behaviors over  that PowerPoint PPT Presentation


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RTS Development by the formal approach Assignment #1. Ex. 1 : Given E={ p , q , r }, let =2 E . 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}] 

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

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Ex 1 given e p q r let 2 e express the behaviors over that

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 1 given e p q r let 2 e express the behaviors over that

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 | p0} = [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


Ex 1 given e p q r let 2 e express the behaviors over that

  • 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 |∃k0 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.


Ex 1 given e p q r let 2 e express the behaviors over that

Ex. 3

Prove that qis semantically equivalent to (q) (namely:  qiff  (q) ).

 q iff iiq -- (semantics of q)

iff iiq -- (semantics of q)

iff i(iq) -- (semantics of )

iff i(iq) -- (semantics of , )

iff i(iq) -- (semantics of )

iff (q) -- (semantics of )

iff  (q) -- (semantics of (q))


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