1 / 5

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 =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] 

zeno
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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]  [~p,~q] 4.See next slide

  2. 4. poccurs at every step (strictly) between qandnextr: • Version A - The occurrences of [p,q] are mutually exclusive: • ([~q]*[q][p,~r]*[r]) -- infinite occurrences •  ([~q]*[q][p,~r]*[r])*([~q]*[q])1[~r] •  ([~q]*[q][p,~r]*[r])*[~q] -- finite occurrences • Version B - successive occurrences of[p,r] possibly share end points: • (i) [~q]*[q][p,~r]*(([r,~q][~q]*[q][p,~r]*)*+([r,q][p,~r]*)*)) • (ii)  [~q]*[q][p,~r]*(([r,~q][~q]*[q][p,~r]*)*+([r,q][p,~r]*)*))*([~q]*[q])1[~r] • (iii)  [~q]*[q][p,~r]*(([r,~q][~q]*[q][p,~r]*)*+([r,q][p,~r]*)*))*[~q] • (iv)  [~r]  [~q] • i.infinite many occurrences • finite occurrences: at least one q but after 0 or more occurrences and possibly q, • r does not occur any more • iii. finite occurrences: at least one q but after 0 or more occurrences q does not occur • any more • iv. No r or not q at all possibly r’s here

  3. 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

  4. 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.

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

More Related