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

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

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

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)

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))

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)))

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:

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 :

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

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,

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

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

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

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)

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,

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 :

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,

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

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

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

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

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

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