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¹ - Calculus

¹ - Calculus. Based on: “Model Checking”, E. Clarke and O. Grumberg (ch. 6, 7) “Symbolic Model Checking: 10^20 States and Beyond”, Burch, Clark, et al “Introduction to Modal Mu-Calculi”, J. Bradfield and C. Stirling. Agenda. Review Some fixpoint theory Syntax and semantics of ¹ -Calculus

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¹ - Calculus

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  1. ¹-Calculus Based on: “Model Checking”, E. Clarke and O. Grumberg (ch. 6, 7) “Symbolic Model Checking: 10^20 States and Beyond”, Burch, Clark, et al “Introduction to Modal Mu-Calculi”, J. Bradfield and C. Stirling

  2. Agenda • Review • Some fixpoint theory • Syntax and semantics of ¹-Calculus • Examples • Symbolic Model Checking • Applications

  3. Reminder: Kripke Structure • M=(S,R,L ) p q p,q AP={p,q}

  4. Reminder: CTL* (I) • State formulae: • p2AP • If f and g are state formulae, so are: fÆg:ffÇg • If f is a path formula, the following are state formulae: Af Ef

  5. Reminder: CTL* (II) • Path formulae: • If f is a state formula, it is also a path formula • If f and g are path formula, so are: fÆg:ffÇg • If f and g are path formula, so are: XfGf Ff fUg fWg f f f f f … f … g f f f g f f f f f f f …

  6. Agenda • Review Some fixpoint theory • Syntax and semantics of ¹-Calculus • Examples • Symbolic Model Checking • Applications 

  7. { 1 , 2 , 3 } { 1 , 2 } { 1 , 3 } { 2 , 3 } { 1 } { 2 } { 3 } ; Fixpoints: definitions (I) • The power-set lattice • Defined over P(S) for some finite set S • Partial order: µ • Example:

  8. Fixpoints: definitions (II) • Predicate transformer: ¿: P(S)!P(S) asdf • F2P(S)is a fixpoint of ¿ iff ¿(F) = F S S ¿

  9. Fixpoints: definitions (III) • F2P(S) is a least fixpoint of ¿ iff • F is a fixpoint of ¿, and • If G is a fixpoint of ¿, then FµG Notation: ¹X . ¿(X) • F2P(S) is a greatest fixpoint of ¿ iff • F is a fixpoint of ¿, and • If G is a fixpoint of ¿, then GµF Notation: ºX . ¿(X) G F

  10. Fixpoint properties (I) • Is there always a fixpoint? • No, e.g.: S{1}P(S) = { ;, { 1 } } ¿(;)  { 1 } ¿({ 1 })  ;

  11. Fixpoint properties (II) • If there is a fixpoint, is there always a least fixpoint? • No, e.g.: S {1, 2} ¿({ 2})  { 2} ¿({ 1})  { 1} ¿(;)  { 1}

  12. Monotonous functions • ¿ is monotonic iff for all FµG : ¿(F)µ¿(G) ¿(G) ¿ G ¿(F) F

  13. Fixpoint properties (IV) • Theorem (Knaster-Tarski): If ¿ is monotonous and S is finite, ¿ has a unique least fixpoint and a unique greatest fixpoint. • Proof: constructive.

  14. Computing least fixpoints Qold:=; Qnew=¿(Qold) while Qold Qnew do Qold:= Qnew Qnew:=¿(Qold) end while return Qnew • Need to show: • Termination • Result is a least fixpoint • Result is unique

  15. Correctness (I) • Qi : the value of Qnew in the i-th iteration Qold := ; Qnew = ¿(Qold) while Qold Qnew do Qold := Qnew Qnew := ¿(Qold) end while return Qnew ¿ ¿ ¿ ¿ … = ; Q0 Q1 Q2 Qn Qn+1 = Q! ¿(;) ¿2(;) ¿n(;) ¿n+1(;)

  16. Correctness (II) • Lemma: QiµQi+1 for all i • Proof by induction: • Base: i=0 Qold := ; Qnew = ¿(Qold) while Qold Qnew do Qold := Qnew Qnew := ¿(Qold) end while return Qnew ¿ ; µ Q0 Q1 (;)

  17. Correctness (III) • Lemma: QiµQi+1 for all i • Proof by induction: • Step: Qold := ; Qnew = ¿(Qold) while Qold Qnew do Qold := Qnew Qnew := ¿(Qold) end while return Qnew ¿ ¿  ? µ µ Qi-1 µQi Qi-1 Qi Qi+1  ¿is monotonic Qi= ¿(Qi-1) µ ¿(Qi) = Qi+1 Induction hypothesis

  18. Correctness (IV) Lemma: QiµQi+1 for all i • Termination: S is finite Qold := ; Qnew = ¿(Qold) while Qold Qnew do Qold := Qnew Qnew := ¿(Qold) end while return Qnew ¿ ¿ ¿ ¿ … = ; µ µ µ µ Q0 Q1 Q2 Qn Qn+1 • Need to show: • ) Termination • Result is a least fixpoint • Result is unique ¿2(;) ¿n(;) ¿n+1(;) ¿(;) 

  19. Correctness (V) • Q! is a least fixpoint: • Let G be some fixpoint. • Need to show: Q !µG • We will show: QiµG for all i • Base: Q0=;µG • Step: Assume QiµG Qi+1= ¿(Qi)µ¿(G) = G Qold := ; Qnew = ¿(Qold) while Qold Qnew do Qold := Qnew Qnew := ¿(Qold) end while return Qnew • Need to show: •  Termination • ) Result is a least fixpoint • Result is unique  

  20. Correctness (VI) • The least fixpoint is unique: • Let F and G be least fixpoints • FµGandGµF )F=G  

  21. The Initial Estimate • We used Q0=; • Can start with any “conservative” estimate • Iµleast fixpoint

  22. Computing greatest fixpoints Qold:=S Qnew=¿(Qold) while Qold Qnew do Qold:= Qnew Qnew:=¿(Qold) end while return Qnew

  23. Agenda • Review • Some fixpoint theory Syntax and semantics of ¹-Calculus • Examples • Symbolic Model Checking • Applications 

  24. ¹-Calculus (I) • Let AP be a set of atomic propositions • LetVAR={Y1, Y2, …} be a set of relational variables • The formulas of ¹-Calculus: • p2AP • Y2VAR • If f and g are formulas, so are fÇg, fÆg, f

  25. ¹-Calculus (II) • The formulas of ¹-Calculus (cont’d): • If f is a formula, so are ¤f and }f • If Y is a relational variable and f is a formula, the following are formulas: • ¹Y . f • ºY . f AX EX x. P(x) ¹Y . f(Y) bind Y A formula is closed if all its fixpoint variables are bound

  26. ¹-Calculus Semantics (I) • For Y2VAR, Y is a formula. • But what does it mean? • e:VAR!2S is an environment • Define: e[QÃW]is e with W substituted for Q • (e[QÃW])(Q) = W • The environment is not needed for closed formulas

  27. ¹-Calculus Semantics (II) • A formula f is interpreted as a set of states in which f is true • Notation: «f¬Me • «p¬Me={s2S|p2L(s)} • «Y¬Me=e(Y) • «:f¬Me=Sn«f¬Me • «fÆg¬Me=«f¬MeÅ«g¬Me • «fÇg¬Me=«f¬Me[«g¬Me M,s ⊨ f s «f¬M

  28. ¹-Calculus Semantics (II) • «}f ¬Me={s|9t:R(s,t)Æt2«f¬Me} • «¤f ¬Me={s|8t:R(s,t)!t2«f¬Me} • «¹Y.f¬Me is the least fixpoint of: ¿(W)=«f¬Me[YÃW] • «ºY.f¬Me is the greatest fixpoint s s «f¬ «f¬

  29. Restrictions on ¹-Calculus • Are all formulae monotonic? • fÆg, fÇg • :f ) fixpoint variables must be under an even number of negations ¹Y.:Y ºY.:(YÇp) ¹Y.:(:YÇp)   ¹Y.:(:YÇp) ¹Y.(::YÆ:p) ¹Y.(YÆ:p) ¹-Calculus is closed under negation  ¿(;)  { 1 } ¿({ 1 })  ;  : 

  30. Agenda • Review • Some fixpoint theory • Syntax and semantics of ¹-Calculus Examples • Symbolic Model Checking • Applications 

  31. Why are fixpoints interesting? • Recall from Logic I: • I( A, P ) : the smallest set W such that • AµW, and • If x2W and f2P then x2W. • I( A, P ) = ¹Y. AÇP( Y ) P A

  32. Intuition for least fixpoints • x2«¹Y .¿(Y)¬ • “Finite iteration” • Example: • EF'=¹Y . 'Ç}Y … ; x

  33. Intuition for greatest fixpoints • x2«ºY .¿(Y)¬ • “Invariant” • Example: • EG'=ºY.'Æ}Y S= … = x x x x x

  34. ¹-Calculus aerobic (I) • ¹Y.qÇ(pƤY) = ? A[p U q] • ºY.qÇ(pƤY) = ? A[p W q] q p p … Y0 Y1 Y2

  35. ¹-Calculus aerobic (II) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Can pass through Y a finite number of times • Each time p holds • Can pass through Z infinitely • Each time p doesn’t hold ) “p is true only finitely often on all paths”

  36. ¹-Calculus aerobic (III) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Inner computation 1: Y0=;, Z00=S • Z!0 = ºZ. :p Æ ¤Z=AG:p Notation: Yi : ith estimate for Y Zij : ith estimate for Z, using the jth estimate for Y ! denotes the last iteration p S :p AG:p p p :p :p :p :p …

  37. ¹-Calculus aerobic (IV) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Outer iteration 1: • Y1 = (pƤY 0)Ç(:pƤZ!0) AG:p AG:p :p :p :p :p …

  38. AG:p :p :p :p :p … AG:p p :p :p :p … AG:p :p :p p :p … ¹-Calculus aerobic (V) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Inner computation 2: • Z!1 = ºZ.(pƤY1)Ç(:pƤZ) A[:pW( pƤY1 )] : p p AG:p

  39. AG:p :p :p :p :p … AG:p p :p :p :p … AG:p :p :p p :p … ¹-Calculus aerobic (VI) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Outer iteration 2: • Y2 = (pƤY1)Ç(:pƤZ! 2) :p p AG:p p AG:p Y1 Z! 2

  40. ¹-Calculus aerobic (VI) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • Every inner computation: A[:pW( pƤYn )] • Add a “layer” of :p (with infinite behaviors) • Every outer iteration: (pƤYn)Ç(:pƤZm) • Add a single p

  41. ¹-Calculus aerobic (VII) • ¹Y.ºZ.(pƤY)Ç(:pƤZ) = ? • p can appear a finite number of times finite no. AG:p p p :p :p p :p :p p p :p …

  42. Agenda • Review • Some fixpoint theory • Syntax and semantics of ¹-Calculus • Examples Symbolic Model Checking • Applications 

  43. Symbolic Model Checking eval(f, e) f states that satisfy f M, e

  44. Model Checking Algorithm (I) • if f=p : return {s|p2L(s)} • if f=Q: return e(Q) • if f = g1Æg2: return eval(g1,e)Åeval(g2 , e ) • if f = g1Çg2: return eval(g1,e)[eval(g2 , e )

  45. Model Checking Algorithm (II) • if f=} g : return {s|9t[R(s,t)Æt2eval(g,e)] } • if f=¤g: return {s|8t[R(s,t)!eval(g,e)(t)] }

  46. Model Checking Algorithm (III) • if f=¹Y.g(Y): Qnew = ; repeat Qold = Qnew Qnew = eval( g,e[YÃQnew]) until Qnew = Qold return Qnew

  47. Model Checking Algorithm (III) • if f=ºY.g(Y): Qnew = S repeat Qold = Qnew Qnew = eval( g,e[YÃQnew]) until Qnew = Qold return Qnew

  48. Model Checking Complexity (I) if f=p : return {s|p2L(s)} if f=Q: return e(Q) if f = g1Æg2: return eval(g1,e)Åeval(g2 , e ) if f = g1Çg2: return eval(g1,e)[eval(g2 , e ) if f=} g : return {s|9t[R(s,t)Æt2eval(g,e)] } if f=¤g: return {s|8t[R(s,t)!eval(g,e)(t)] } O( |M| )

  49. Model Checking Complexity (II) • if f=¹Y.g(Y): Qnew = ; repeat Qold = Qnew Qnew = eval( g,e[YÃQnew]) until Qnew = Qold return Qnew O( |S| ) Repeat entire computation of eval(g) Overall complexity: nesting depth O( |M| ¢ |f| ¢ |S|k)

  50. Improved Model Checking (I) • Example: ¹Y.g(Y,¹Z.h(Y, Z)) ¹Y ¹Z ; ; Before: Now: Y1 =g(;,Z! 0) = = Z! 0= ¹Z.h(;, Z) |S| iterations |S| iterations = Z! 1= ¹Z.h(Y1, Z) Y2 =g(Y1,Z!1) = |S| iterations O(|S|2))O(|S| + |S|)

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