ThI Theoretische Informtik Complexity in Modl Tem Logic Julin-Steffen Müller Theoretische Informtik 18. Jnur 2012 Theorietg 2012
Theoretische Informtik Inhlt 1 Preliminries 2 Closure properties 3 Model Checking 4 Results 5 Conclusion Preliminries Closure properties Model Checking Results Conclusion Seite 2
Theoretische Informtik Motivtion Modl Dependence Logic MDL descries tomic dependencies etween vriles. Originl introduced y Vänäänen for first-order logic. Studying computle cses. Modl Tem Logic MDL cnnot express tht certin dependence etween vriles does not hold. Tem semtics without dependence tom. Preliminries Closure properties Model Checking Results Conclusion Seite 3
Theoretische Informtik Semntic of MTL Split junction M, T = ϕ ϕ T 1 T 2 = T : M, T 1 = ϕ nd M, T 2 = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Split junction M, T = ϕ ϕ T 1 T 2 = T : M, T 1 = ϕ nd M, T 2 = ϕ Exmples M, {, } = Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl disjunction M, T = ϕ ϕ M, T = ϕ or M, T = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl disjunction M, T = ϕ ϕ M, T = ϕ or M, T = ϕ Exmples M, {, } = Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl disjunction M, T = ϕ ϕ M, T = ϕ or M, T = ϕ Exmples M, {, } = M, {,, } = Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Modl exists M, T = ϕ ˆT T : M, ˆT = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Modl exists M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, {, } = ( ) Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Modl next M, T = ϕ M, R[T ] = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Modl next M, T = ϕ M, R[T ] = ϕ Exmples M, {, } = ( ) Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Modl next M, T = ϕ M, R[T ] = ϕ Exmples M, {, } = ( ) M, {, } = ( ) Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Dependence tom M, T = dep(p 1,..., p n 1 ; p n ) ˆT T : M, ˆT = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Dependence tom M, T = dep(p 1,..., p n 1 ; p n ) ˆT T : M, ˆT = ϕ Exmples M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Dependence tom M, T = dep(p 1,..., p n 1 ; p n ) ˆT T : M, ˆT = ϕ Exmples M, {, } = dep(; ) M, {,, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl negtion M, T = ϕ M, T = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl negtion M, T = ϕ M, T = ϕ Exmples M, { } = Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Semntic of MTL Clssicl negtion M, T = ϕ M, T = ϕ Exmples M, { } = M, {, } = Preliminries Closure properties Model Checking Results Conclusion Seite 4
Theoretische Informtik Dul opertor to opertor Let ϕ e MTL formul. Then ϕ is defined y ϕ. Preliminries Closure properties Model Checking Results Conclusion Seite 5
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Semntic of MTL Modl forll M, T = ϕ ˆT T : M, ˆT = ϕ Exmples M, { } = dep() M, {, } = dep(; ) Preliminries Closure properties Model Checking Results Conclusion Seite 6
Theoretische Informtik Closure properties Downwrds closure M, T = ϕ T T : M, T = ϕ Union closure M, T = ϕ nd M, T = ϕ M, T T = ϕ Fltness M, T = ϕ w T : M, w = ϕ Fltness follows from downwrds nd union closure! Preliminries Closure properties Model Checking Results Conclusion Seite 7
Theoretische Informtik Closure properties Downwrds closure ML MDL MTL M, T = ϕ T T : M, T = ϕ Union closure M, T = ϕ nd M, T = ϕ M, T T = ϕ Fltness M, T = ϕ w T : M, w = ϕ Fltness follows from downwrds nd union closure! Preliminries Closure properties Model Checking Results Conclusion Seite 7
Theoretische Informtik Closure properties Downwrds closure ML MDL MTL M, T = ϕ T T : M, T = ϕ Union closure ML MDL MTL M, T = ϕ nd M, T = ϕ M, T T = ϕ Fltness M, T = ϕ w T : M, w = ϕ Fltness follows from downwrds nd union closure! Preliminries Closure properties Model Checking Results Conclusion Seite 7
Theoretische Informtik Closure properties Downwrds closure ML MDL MTL M, T = ϕ T T : M, T = ϕ Union closure ML MDL MTL M, T = ϕ nd M, T = ϕ M, T T = ϕ Fltness ML MDL MTL M, T = ϕ w T : M, w = ϕ Fltness follows from downwrds nd union closure! Preliminries Closure properties Model Checking Results Conclusion Seite 7
Theoretische Informtik ϕ ϕ Theorem Let ϕ e downwrds closed MTL formul. Then ϕ is equivlent to ϕ. Preliminries Closure properties Model Checking Results Conclusion Seite 8
Theoretische Informtik Model Checking Prolem MTL-MC Instnce: MTL formul ϕ, Kripke Modell M, MTL tem T Question: Is ϕ vlid under M nd T (M, T = ϕ)? MTL-SAT Instnce: MTL formul ϕ Question: Is there Kripke Model M nd tem T which stisfies ϕ? Preliminries Closure properties Model Checking Results Conclusion Seite 9
Theoretische Informtik Known Results for MDL Modl dependence logic (MDL) is defined nlogous to modl tem logic, ut without the clssicl negtion. Complexity of Model Checking It ws shown y Eing nd Lohmnn in 2011 tht the model checking prolem for MDL is NP-complete. Complexity of Stisfyility It ws shown y Lohmnn nd Vollmer in 2010 tht the stisfiility prolem for MDL is NEXPTIME-complete. How expressive is MTL in comprsion to MDL? Preliminries Closure properties Model Checking Results Conclusion Seite 10
Theoretische Informtik Result overview k dep Complexity + + + PSPACE-complete + + + PSPACE-complete + + + k 0 + Σ p k+1 -complete + + + k 0 Σ p k -complete + + + P NP[1] -complete + + P-complete + + + + + NC 1 -complete Telle: Complexity results for model checking in MTL Preliminries Closure properties Model Checking Results Conclusion Seite 11
Theoretische Informtik MTL-MC is in PSPACE Non determinism in MDL Let M e Kripke Model nd T tem over M. 1 M, T = ϕ ϕ T 1, T 2 with T = T 1 T 2 : T 1 = ϕ nd T 2 = ϕ 2 M, T = ϕ ˆT T : M ˆT = ϕ 3 M, T = ϕ M, T = ϕ In worst cse ech su formul uses the non determinism to serch through ll worlds. This is polynomil ounded in spce. Preliminries Closure properties Model Checking Results Conclusion Seite 12
Theoretische Informtik MTL-MC is PSPACE-hrd Theorem Let ϕ = x 1 x 2... x m n i=1 C i. Then ϕ 3QBF (M, T, δ 1 ) MTL-MC. Kripke Model M p 1 1 p 1 1 p 1 1 p 1 n 0... n w n n d 1 0... d n 0... d n n 0... n w n n p 0 1 p 0 1 p 0 1 p 0 n Preliminries Closure properties Model Checking Results Conclusion Seite 13
Theoretische Informtik MTL-MC is PSPACE-hrd Theorem Let ϕ = x 1 x 2... x m n i=1 C i. Then ϕ 3QBF (M, T, δ 1 ) MTL-MC. Kripke Model M, strting tem T p 1 1 p 1 1 p 1 1 p 1 n 0... n w n n d 1... d 0 0 n... d n n 0... n w n n p 0 1 p 0 1 p 0 1 p 0 n Preliminries Closure properties Model Checking Results Conclusion Seite 13
Theoretische Informtik MTL-MC is PSPACE-hrd Existentil prt (i is odd) δ i = ((p 1 i p 0 i ) δ i+1) Universl prt (i is even) δ i = ((p 1 i p 0 i ) δ i+1) Evlution (i = n + 1) δ i = m i=1 ( l i1 l i2 l i3 ) Preliminries Closure properties Model Checking Results Conclusion Seite 14
Theoretische Informtik Restricted clssicl negtion Wht hppens if the nesting of clssicl negtion is restricted? Equivilnt with restricting the lterntion of existentil nd universl quntifictions. 3QBF k. MTL-MC(,,, ) is Σ p k -complete. MTL-MC(,,,, dep) is Σ p k+1 -complete. Preliminries Closure properties Model Checking Results Conclusion Seite 15
Theoretische Informtik MTL-MC (, dep, ) is in P NP[1] Ech MTL formul over {, dep, } is of the form ϕ = ϕ or ϕ = ϕ, where: ϕ = i 1 i 2... i k λ λ {dep} VAR ϕ cn e solved with the NP-complete prolem MDL-MC. Orcle Answer is negted if ϕ = ϕ. Preliminries Closure properties Model Checking Results Conclusion Seite 16
Theoretische Informtik MTL-MC(, dep, ) is P NP[1] hrd A generl P NP[1] mchine cn e reduced to: input x f SAT 1 0 cc rej f SAT Preliminries Closure properties Model Checking Results Conclusion Seite 17
Theoretische Informtik MTL-MC(, dep, ) is P NP[1] hrd A generl P NP[1] mchine cn e reduced to: input x f SAT 1 0 cc rej f SAT input x f SAT 1 0 rej cc f SAT Preliminries Closure properties Model Checking Results Conclusion Seite 17
Theoretische Informtik MTL-MC(, dep, ) is P NP[1] hrd A generl P NP[1] mchine cn e reduced to: input x f SAT 1 0 cc rej f SAT input x f SAT 1 0 rej cc f SAT input x f SAT cc 1 cc 0 Accept lwys Preliminries Closure properties Model Checking Results Conclusion Seite 17
Theoretische Informtik MTL-MC(, dep, ) is P NP[1] hrd A generl P NP[1] mchine cn e reduced to: input x input x input x input x f SAT 1 0 cc rej f SAT 1 0 rej cc f SAT cc 1 cc 0 f SAT 1 0 rej rej f SAT f SAT Accept lwys Reject lwys Preliminries Closure properties Model Checking Results Conclusion Seite 17
Theoretische Informtik MTL-MC(, dep, ) is P NP[1] hrd A generl P NP[1] mchine cn e reduced to: input x input x input x input x f SAT 1 0 cc rej f SAT 1 0 rej cc f SAT cc 1 cc 0 f SAT 1 0 rej rej f SAT f SAT Accept lwys Reject lwys Since MDL-MC(, dep) is NP-complete, MTL-MC(, dep) cn simulte the SAT orcle questions. Preliminries Closure properties Model Checking Results Conclusion Seite 17
Theoretische Informtik Conclusion Summry Filure of downwrds closure is not the reson for the complexity lep. Restricting the clssicl negtion is equl to restrict possile quntor lterntions. Open Questions Complexity of MTL-MC(, ). Mye PSPACE-complete. Complexity of MTL stisfiility prolem. Definition nd clssifiction of dependence temporl logic. Preliminries Closure properties Model Checking Results Conclusion Seite 18