1. Asymmetry and 3-Valued Symmetry Reduction Course Project of CSC 2108H, 2003 Ou Wei Yong Yuan Department of Computer Science, University of Toronto, 2004

2. Presentation Outline Introduction Background Component Symmetry Symmetry Reduction in 3-Valued Models Related Work

3. PqPq S1S1  P  q S3S3  p q S5S5  P  q S4S4 PqPq S2S2 P qP q S0S0 TM MT P=T q=M S0S0 P=M q=T S5S5 P=F q=F S4S4 S3S3 P=T q=T S1S1 S2S2 M M T T Introduction Extended symmetry reduction to handle asymmetric system –Defined component symmetry with weaker constraints Developed symmetry reduction to 3-valued models

4. S5S5 S4S4 S2S2 S3S3 A Permutation σon a set S –bijectionσ: S  S –Cyclic notation. e.g., (1 5 2)(4 3) denotes 1  5, 5  2, 2  1, 4  3, 3  4 Automorphism of a Kripke structure M = (S, s 0, R, L) –The permutation that preserves all the transition relations –Formally, –Example: Background S1S1 S2S2 S3S3 S4S4 S5S5 S1S1 σ = (s2, s3)(s4, s5)

5. Orbit –Orbit of a state s: –Intuitively, the set of the states that can be reached from the state s by applying a permutation –A representative state can be chosen from orbit, denoted by rep(θ(s)) A permutation σ of Kripke structure M = (S, s 0, R, L) is an invariance permutation for an atomic proposition p iff (  s  S)( p  L(s)  p  L(σ(s))

6. Quotient Structure –Given a Kripke structure M = (S, s 0, R, L), and a permutation group G on S, the Quotient Structure M G = (S G, S 0 G, R G, L G ) S G = {θ(s) | s  S } S 0 G = θ(s 0 ) R G  S G X S G and  s  S,  t  S,(θ(s), θ(t))  R G iff (  s’  θ(s))(  t’  θ(t)) (s’, t’)  R L G (θ(s)) = L(rep(θ(s)))

7. Inspiration To Component Symmetry The existential abstraction and the universal abstraction on the set of orbits M G is bisimilar to M if the following condition holds:

8. Component Symmetry Definition –A permutation σ of a Kripke structure M = (S, s o, R, L) is called a component symmetry of M if and only if the following condition holds :

9. P=T q=M S0S0 P  q S0S0 PqPq S0S0 P=M q=T S5S5  p q S5S5 pqpq S5S5 P=F q=F S3S3  P  q S3S3 S3S3 Example of Component Symmetry σ = (s 1, s 2 )(s 3, s 4 ), G = = {σ, I} P=F q=F S4S4 P=T q=T S1S1 S2S2 PqPq S1S1 PqPq S2S2  P  q S4S4 1.(σ(s 1 ), σ 2 (s 4 )) = (s 2, s 4 ) 2.(σ(s 2 ), σ 2 (s 4 )) = (s 2, s 4 ) 3.Orbits: {s 1, s 2 }, {s 3, s 4 }, {s 0 }, {s 5 } PqPq S1S1

10. Properties of Component Symmetry Given a Model M = (S, s 0, R, L) with AP as a set of atomic propositions, G is a component symmetry group of M If G is an invariance group for all the propositions in AP, M G, ComSym is bisimilar to M If G is an invariance group for a temporal logic formula φ, then we can verify φ in M G, ComSym. Formally,

11. Component Symmetry Generated Group G is component symmetry generated group of a Kripke structure M = (S, s 0, R, L) if and only if G =, where σ i (1 <= i <= k) is a component symmetry of M. The composition σ = σ m σ n may not be a component symmetry permutation. However, the reduced structure M G, ComSymGen is still bisimilar to M

12. Symmetry Reduction on 3-Valued Models Automorphism-based 3-valued symmetry reduction Component-symmetry-based 3-valued symmetry reduction Relation of symmetry reduction and 3-valued model reduction

13. Automorphism-based 3-Value Symmetry Reduction Invariance group for 3-valued model Automorphism

14. Quotient Structure For 3-Valued Model Quotient Structure –S G = {θ(s) | s  S } –S 0 G = θ(s 0 ) –R G : S G X S G  { , M,  } and  s  S,  t  S, R G (θ(s), θ(t))   iff (  s’  θ(s))(  t’  θ(t)) R(s’, t’) =  (1) R G (θ(s), θ(t))  M iff (1) is false and (  s’  θ(s))(  t’  θ(t)) R(s’, t’) = M (2) R G (θ(s), θ(t))   iff both(1) and (2) are false –L G : S G x AP  { , M,  } is defined as : (  s  S)(  p  AP) L G (θ(s), p)  L(rep(θ(s)), p)

15. Properties of 3-valued Symmetry Reduction Given a 3-valued model M = (S, s 0, R, L) with AP as a set of atomic propositions, G is an automorphism group of M If G is an invariance group for all the propositions in AP, M and M G refine each other. That is, M  ref M G and M G  ref M If G is an invariance group for a temporal logic formula φ, then we can verify φ in M G. Formally,

16. Component Symmetry on 3-Valued Model Definition –A permutation σ of a 3-valued Kripke structure M = (S, s o, R, L) is called a component symmetry of M if and only if the following condition holds :

17. Properties of 3-valued Symmetry Reduction Given a 3-valued model M = (S, s 0, R, L) with AP as a set of atomic propositions, G is a component symmetry of M If G is an invariance group for all the propositions in AP, M and M G refine each other. That is, M  ref M G and M G  ref M If G is an invariance group for a temporal logic formula φ, then we can verify φ in M G. Formally,

18. Example P=T q=M S0S0 P=M q=T S5S5 P=F q=F S4S4 S3S3 P=T q=T S1S1 S2S2 M M M M T T T T P=T q=M a0a0 P=M q=T a3a3 P=F q=F a2a2 P=T q=T a1a1 M T TT σ = (s1, s2)(s3, s4), G = = {σ, I } M MGMG

19. Symmetry and 3-Valued Model Reduction P=T q=F S0S0 P=F q=T S5S5 P=F q=F S4S4 S3S3 P=T q=T S1S1 S2S2 T T T T S0S0 S5S5 P=F q=F S4S4 S3S3 P=T q=T S1S1 S2S2 T T T T T T T T MM MMMM

20. P=T q=F a0a0 P=F q=T a3a3 P=F q=F a2a2 P=T q=T a1a1 T TT a0a0 P=M q=T a3a3 P=F q=F a2a2 P=T q=T a1a1 T T TT MGMG MGMMGM P=T q=M a0a0 P=M q=T a3a3 P=F q=F a2a2 P=T q=T a1a1 M T TT Combination of M G  and M G  M

21. Let M = (S, s 0, R, L) be a 3-valued Kripke structure with AP as a set of atomic propositions. Let G be an component symmetry group of M, and M G be the quotient structure induced by G. If G is an invariance group for all atomic propositions in AP, then M G is the combination of the quotient structures induced by G on M  and M  M. Relation of Symmetry Reduction and 3- Valued Model Reduction

22. Related Work Virtual Symmetry (Emerson et al) –Let M = (S, R) be a structure, and G be a group acting on S. M G = (S G, R G ) is the symmetrization of M by G where S G = S, and R G = {(σ(s), σ(t)) | σ  G and (s, t)  R} –M is virtually symmetric w.r.t G if (  (s, t)  R G )(  σ  G)(s, σ(t))  R

23. Thank you!