Temporal logic CTL : syntax. Communication and Concurrency Lecture 6. Φ ::= tt ff Φ 1 Φ 2 Φ 1 Φ 2 [K]Φ K Φ AG Φ EF Φ AF Φ EG Φ A formula can be

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1 Temporl logic CTL : syntx Communiction nd Concurrency Lecture 6 Colin Stirling (cps) Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be School of Informtics 7th October 013 Temporl logic CTL : syntx Temporl logic CTL : syntx Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be formul of Hennessy-Milner logic, Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be formul of Hennessy-Milner logic, formul AG Φ, red s lwys Φ or globlly Φ,

2 Temporl logic CTL : syntx Temporl logic CTL : syntx Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be formul of Hennessy-Milner logic, formul AG Φ, red s lwys Φ or globlly Φ, formul EF Φ, red s possibly Φ, Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be formul of Hennessy-Milner logic, formul AG Φ, red s lwys Φ or globlly Φ, formul EF Φ, red s possibly Φ, formul AF Φ, red s eventully Φ, Temporl logic CTL : syntx Temporl logic CTL : semntics A run (of process E 0 ) is sequence of trnsitions of the form Φ ::= tt ff Φ 1 Φ Φ 1 Φ [K]Φ K Φ A formul cn be formul of Hennessy-Milner logic, formul AG Φ, red s lwys Φ or globlly Φ, formul EF Φ, red s possibly Φ, formul AF Φ, red s eventully Φ, formul EG Φ, red s EG Φ. E E1 3 E which is mximl in the sense tht if it is finite then the finl process is unble to do ny ction.

3 Temporl logic CTL : semntics Intuitive mening A run (of process E 0 ) is sequence of trnsitions of the form E E1 3 E which is mximl in the sense tht if it is finite then the finl process is unble to do ny ction. E 0 = AG Φ mens ll processes rechble from E 0 stisfy Φ. E 0 = AG Φ iff for ll runs E for ll i 0, E i = Φ E 0 = EF Φ iff for some run E for some i 0, E i = Φ E 0 = AF Φ iff for ll runs E for some i 0, E i = Φ E 0 = EG Φ iff for some run E for ll i 0, E i = Φ Intuitive mening Intuitive mening E 0 = AG Φ mens ll processes rechble from E 0 stisfy Φ. E 0 = EF Φ mens some process rechble from E 0 stisfies Φ. E 0 = AG Φ mens ll processes rechble from E 0 stisfy Φ. E 0 = EF Φ mens some process rechble from E 0 stisfies Φ. E 0 = AF Φ mens eventully process will be reched which stisfies Φ.

4 Intuitive mening Exmples E 0 = AG Φ mens ll processes rechble from E 0 stisfy Φ. E 0 = EF Φ mens some process rechble from E 0 stisfies Φ. E 0 = AF Φ mens eventully process will be reched which stisfies Φ. E 0 = EG Φ mens some run lwys stisfies Φ. Exmples Exmples All processes rechble from E 0 cn do some ction. E 0 is dedlock-free. All processes rechble from E 0 cn do some ction. E 0 is dedlock-free. E 0 = AF [ ]ff

5 Exmples Exmples All processes rechble from E 0 cn do some ction. E 0 is dedlock-free. E 0 = AF [ ]ff Eventully process is reched which cnnot execute ny ction. E is gurnteed to terminte. All processes rechble from E 0 cn do some ction. E 0 is dedlock-free. E 0 = AF [ ]ff Eventully process is reched which cnnot execute ny ction. E is gurnteed to terminte. AG [request]af ( grnted tt [ grnted]ff) Exmples Exercise P def =.P + b.q Q def = c.q All processes rechble from E 0 cn do some ction. E 0 is dedlock-free. E 0 = AF [ ]ff Eventully process is reched which cnnot execute ny ction. E is gurnteed to terminte. AG [request]af ( grnted tt [ grnted]ff) All requests will eventully be grnted Does P = Φ hold when Φ is EF c tt AG c tt AF c tt EG c tt AG EF c tt AF EG c tt EF AG c tt EG AF c tt /

6 Exercise Exmple: Peterson s solution to mutul exclusion P def =.P + b.q Q def = c.q Does P = Φ hold when Φ is EF c tt AG c tt AF c tt EG c tt AG EF c tt AF EG c tt EF AG c tt EG AF c tt / B1f = b1rf.b1f + b1wf.b1f + b1wt.b1t B1t = b1rt.b1t + b1wt.b1t + b1wf.b1f Bf = brf.bf + bwf.bf + bwt.bt Bt = brt.bt + bwt.bt + bwf.bf K1 = kr1.k1 + kw1.k1 + kw.k K = kr.k + kw.k + kw1.k1 P1 = b1wt.req1.kw.p11 P11 = brt.p11 + brf.p1 + kr.p11 + kr1.p1 P1 = enter1.exit1.b1wf.p1 P = bwt.req.kw1.p1 P1 = b1rf.p + b1rt.p1 + kr1.p1 + kr.p P = enter.exit.bwf.p Peterson = (P1 P K1 B1f Bf) \L Specifiction: temporl properties Specifiction: temporl properties Mutul exclusion Mutul exclusion Absence of dedlock

7 Specifiction: temporl properties Specifiction: temporl properties Mutul exclusion Absence of dedlock Absence of strvtion Mutul exclusion AG ([exit1]ff [exit]ff) Absence of dedlock Absence of strvtion Specifiction: temporl properties Specifiction: temporl properties Mutul exclusion AG ([exit1]ff [exit]ff) Absence of dedlock AG tt Absence of strvtion Mutul exclusion AG ([exit1]ff [exit]ff) Absence of dedlock AG tt Absence of strvtion (for P1) AG ([req1]af exit1 tt)

8 egtion egtion egtion is lso redundnt in CTL : For every formul Φ of CTL there is formul Φ c such tht for every process E egtion is lso redundnt in CTL : For every formul Φ of CTL there is formul Φ c such tht for every process E E = Φ c iff E = Φ E = Φ c iff E = Φ Φ c is inductively defined s for HML, plus: (AG Φ) c = EF Φ c (EF Φ) c = AG Φ c (AF Φ) c = EG Φ c (EG Φ) c = AF Φ c Proposition For every E 0 nd for every Φ of CTL : E 0 = Φ c iff E 0 = Φ. Proposition For every E 0 nd for every Φ of CTL : E 0 = Φ c iff E 0 = Φ. Proof: By induction on the structure of Φ.

9 Proposition For every E 0 nd for every Φ of CTL : Stisfibility, vlidity, equivlence E 0 = Φ c iff E 0 = Φ. Proof: By induction on the structure of Φ. Cse Φ = AG Φ 1. A formul is stisfible (relisble) if some process stisfies it. E 0 = (AG Φ 1 ) c iff E 0 = EF Φ c 1 iff for some run E for some i 0 s.t. E i = Φ c 1 iff for some run E for some i 0 s.t. E i = Φ 1 iff not for ll run E for ll i 0 s.t. E i = Φ 1 iff E 0 = AG Φ 1 Stisfibility, vlidity, equivlence Stisfibility, vlidity, equivlence A formul is stisfible (relisble) if some process stisfies it. A formul is unstisfible if no process stisfies it. A formul is stisfible (relisble) if some process stisfies it. A formul is unstisfible if no process stisfies it. A formul is vlid ll processes stisfy it.

10 Stisfibility, vlidity, equivlence Which of the following re vlid? / AG Φ AF Φ A formul is stisfible (relisble) if some process stisfies it. A formul is unstisfible if no process stisfies it. A formul is vlid ll processes stisfy it. Two formuls re equivlent if they re stisfied by exctly the sme processes. AF Φ AG Φ AG Φ EG Φ EG Φ AG Φ AF Φ EF Φ EF Φ AF Φ EG Φ EF Φ EF Φ EG Φ AF Φ EG Φ EG Φ AF Φ Which of the following re vlid? AG Φ AF Φ AF Φ AG Φ AG Φ EG Φ EG Φ AG Φ AF Φ EF Φ EF Φ AF Φ EG Φ EF Φ EF Φ EG Φ AF Φ EG Φ EG Φ AF Φ / Exercise Which of the following re equivlent when Φ, Φ nd Φ re rbitrry formuls of CTL? AG (Φ 1 Φ ) AG Φ 1 AG Φ EF (Φ 1 Φ ) EF Φ 1 EF Φ AF (Φ 1 Φ ) AF Φ 1 AF Φ AG AG Φ AF AF Φ EF EF Φ AG EF AG Φ AG EF AG EF Φ AG Φ AF Φ EF Φ AG EF Φ AG EF Φ /

11 Exercise Which of the following re equivlent when Φ, Φ nd Φ re rbitrry formuls of CTL? AG (Φ 1 Φ ) AG Φ 1 AG Φ EF (Φ 1 Φ ) EF Φ 1 EF Φ AF (Φ 1 Φ ) AF Φ 1 AF Φ AG AG Φ AG Φ AF AF Φ AF Φ EF EF Φ EF Φ AG EF AG Φ AG EF Φ AG EF AG EF Φ AG EF Φ /