Branching Bisimilarity on Normed BPA Is EXPTIME-complete

0.41 MB·English

by  Chaodong He

Branching Bisimilarity on Normed BPA Is
EXPTIME-complete
Chaodong He Mingzhang Huang
BASICS, Department of Computer Science and Engineering, Shanghai Jiao Tong University
Abstract—We put forward an exponential-time algorithm for large bisimulation base for branching bisimilarity on normed
5 deciding branching bisimilarity on normed BPA (Bacis Process BPA,andbyguessingthebase,theyshowthatthecomplexity
1 Algebra)systems.Thedecidabilityofbranching(orweak)bisim- of this problem is in NEXPTIME [25]. The current best
0 ilarity on normed BPA was once a long standing open problem
lowerbound for weak bisimilarity is the EXPTIME-hardness
2 whichwas closed byYuxi Fuin[1].TheEXPTIME-hardnessis
an inference of a slight modification of the reduction presented establishedbyMayr[2],whoseproofcanbeslightlymodified
r
a by Richard Mayr [2]. Our result claims that this problem is to show the EXPTIME-hardness for branching bisimilarity
M EXPTIME-complete. as well. As to the general BPA, decidability of branching
bisimilarity is still unknown.
7 I. INTRODUCTION
Inthispaper,weconfirmthatanexponentialtimealgorithm
1 Basic process algebra (BPA) [3] is a fundamentalmodelof
exists for checking branching bisimilarity on normed BPA.

] infinitestatesystems,withitsfamouscounterpartinthetheory Comparing with the known EXPTIME-hardness result, we
O of formal languages: context free grammars in Greibach nor- get the result of EXPTIME-completeness. Thus the com-
L mal forms, which generate the entire context free languages. plexity class of branching bisimilarity on normed BPA is
In 1987, Baeten, Bergstra and Klop [4] proved a surprising
. completely determined.
s
result at the time that strong bisimilarity on normed BPA
c Basically, we introduce a family of relative bisimilarities
[ is decidable. This result is in sharp contrast to the classical
parameterizedbythe referencesets, which canbe represented
fact that language equivalence is undecidable for context free
2 by a decompositionbase defined in this paper. The branching
grammar[5].Afterthisremarkablediscovery,decidabilityand
v bisimilarity is exactly the relative bisimilarity whose refer-
8 complexity issues of bisimilarity checking on infinite state
ence set is the empty set. We show that this base can be
4 systems have been intensively investigated. See [6], [7], [8],
approximated. The approximation procedure starts from an
7 [9], [10] for a number of surveys.
4 initial base, which is relatively trivial, and is carried on by
As regards strong bisimilarity on normed BPA, Hu¨ttel and
0 repeatedly refining the current base. In order to define the
Stirling [11] improved the result of Baeten, Bergstra and
. approximation procedure and to ensure that the family of
1
Klop using a more simplified proof by relating the strong
0 relative bisimilarities is achieved at last, a lot of technical
bisimilarityoftwonormedBPAprocessestotheexistenceofa
5 difficulties need overcoming. Some of them are listed here:
1 successfultableausystem.Later,HuynhandTian[12]showed
v: that the problem is in ΣP2, the second level of the polynomial • Despite the seeming resemblance, the relative bisimilar-
hierarchy.Beforelong,anothersignificantdiscoverywasmade ities (Section III) defined in this paper is significantly
i
X by Hirshfeld, Jerrum and Moller [13] who showed that the superior to the corresponding concepts in [25]. The
r problem can even be decided in polynomial time. Improve- relative bisimilarities in this paper is suffix independent.
a
ments on running time was made later in [14], [15], [16]. Thispropertyisextremelycrucialforour algorithm.The
ThedecidabilityofstrongbisimilarityongeneralBPAisaf- correctness of definition is characterized in Theorem 2.
firmedbyChristensen,Hu¨ttelandStirling [17]. 2-EXPTIME • We show that a generalized unique decomposition prop-
is claimed to be an upper bound by Burkart, Caucal and erty holds for the family of relative bisimilarities (The-
Steffen [18] and is explicitly proven recently by Jancˇar [19]. orem 3). In the decompositions, bisimilarities with dif-
As to the lower bound, Kiefer [20] achieves EXPTIME- ferent reference sets depend and impact on each other.
hardness,whichisanimprovementofthepreviousPSPACE- The notion of decomposition bases (Section V) provides
hardness obtained by Srba [21]. an effective representation of an arbitrary family of pro-
In the presence of silent actions, however, the picture is cess equivalencesthat satisfies the uniquedecomposition
less clear. The decidability for both weak bisimilarity and property.
branchingbisimilarityonnormedBPAwasoncelongstanding • Inaniterationofrefinementoperation,anewdecomposi-
openproblems.Forweakbisimilarity[22],theproblemisstill tionbaseisconstructedfromtheold(SectionVI).Thatis,
open,whileforbranchingbisimilarity[23],[24],aremarkable anewfamilyofequivalencesisobtainedfromtheoldone.
discoveryismadebyFu[1]recentlythattheproblemisdecid- Besides,comparingwithallthepreviousalgorithms[26],
able. Very recently, using the key property developed in [1], [15], [27] which take partition refinement approach, our
Czerwin´skiandJancˇarshowsthatthereexistsanexponentially refinement procedure possesses several hallmarks:
– The new base is constructed via a globally greedy If ≍ is an equivalence relation on processes, then we will
strategy, which means that all the relevant equiva- useα−≍→α′ todenotethefactα−τ→α′ andα≍α′,anduse
lences with differentreference sets are dealt with as α −6≍→ α′ to denote the fact α −τ→ α′ and α 6≍ α′. We write
a whole. =⇒ for the reflexive transitive closure of −τ→, and ⇐⇒ for
– The refinement operation in previous works heavily thesymmetricclosureof=⇒(i.e.⇐⇒d=ef =⇒∪=⇒−1).Ac-
depends on predefined notions of norms and de- cordingly,=≍⇒isunderstoodasthereflexivetransitiveclosure
creasingtransitions.Thesenotionscanbedetermined of −≍→. That is, α=≍⇒α′ if and only if α−≍→·...·−≍→α′.
from the normed BPA definition immediately. Such
Remark 1. α=≍⇒α′ is slightly different from α=⇒α′ ≍α.
amethoddoesnotworkatpresent.Oursolutionisto
IfComputationLemma(Lemma1)holdsfor≍,thenα=≍⇒α′
definenormsinasemanticway(SectionIV).Norms,
if and only if α=⇒α′ ≍α.
relyingontherelevantequivalencerelations,together
with decreasing transitions, can change dynamically A process α is normed if α −ℓ→1 ·...· −ℓ→n ǫ for some
in every iteration. When we start to construct a new ℓ1,...,ℓn. A BPA system Γ = (C,A,∆) is normed if all
base, no information on norms is available. Thus at the processes defined in Γ are normed. In other words, Γ is
thistimewecannotdeterminewhetheratransitionis normed if X is normed for every X ∈ C. In the rest of the
decreasing.Oursolutionistoincorporatethetaskof paper, we will invariably use Γ = (C,A,∆) to indicate the
computing normsinto the globaliteration procedure concerned normed BPA system. A BPA system Γ is called
via the greedy strategy. realtime if for every (X −→ℓ α)∈∆, we have ℓ6=τ.
– Inpreviousworkstheorderofprocessconstantscan A process α is called a ground process if α=⇒ǫ. The set
be determined in advance. Every time a new base is of ground constants is denoted by C . Apparently C ⊆ C
G G
constructed from the old, the constants are treated and α is ground if and only if α∈C∗.
G
in the same order.There is no such predefinedorder Remark 2. A BPA system Γ = (C,A,∆) is totally normed
in our algorithm. The treating order is dynamically if and only if rules of the form X −τ→ ǫ are forbidden. Γ =
determined in every iteration. (C,A,∆) is totally normed if and only if C =∅.
G
EquivalencecheckingonnormedBPAissignificantlyharder
B. Bisimulation and Bisimilarity
than the related problem on totally normed BPA. For to-
In the presence of silent actions, branching bisimilarity of
tally normed BPA, branching bisimilarity is recently shown
van Glabbeek and Weijland [23], [24] is well-known.
polynomial-timedecidable[27].Whatisobtainedinthispaper
is significantly stronger than previous results [28], [29], [27]. Definition 1. Let ≍ be an equivalence relation on processes.
≍ is called a branchingbisimulation, if the following bisimu-
II. PRELIMINARIES lation property hold: whenever α≍β,
A. Normed Basic Process Algebra • If α −a→ α′, then β =≍⇒ · −a→ β′ for some β′ such that
A basic process algebra (BPA) system Γ is a triple α′ ≍β′.
(C,A,∆), whereC isa finiteset ofprocessconstantsranged • If α −6≍→ α′, then β =≍⇒ · −6≍→ β′ for some β′ such that
over by X,Y,Z,U,V,W, A is a finite set of actions, and ∆ α′ ≍β′.
is a finite set of transition rules. The processes, ranged over The branchingbisimilarity ≃ is the largestbranchingbisimu-
by α,β,γ,δ,ζ,η, are generated by the following grammar: lation.
α ::= ǫ | X | α .α . Remark 3. In this paper, branching bisimulations in Defini-
1 2
tion1andotherbisimulation-likerelationsinlaterchaptersare
The syntactic equality is denoted by =. We assume that forced to be equivalence relations. This technical convention
the sequential composition α1.α2 is associative up to = and does not affect the notion of branching bisimilarity.
ǫ.α = α.ǫ = α. Sometimes α.β is shortened as αβ. The set
The branching bisimilarity is a congruence relation, and it
of processes is exactly C∗, the finite strings over C. There is
satisfies the following famous lemma.
a special symbol τ in A for silent transition. ℓ is invariably
used to denotean arbitraryaction, while a is used to denotea Lemma1(ComputationLemma[23]). Ifα=⇒α′ =⇒α′′ ≃
visible(i.e.non-silent)action.Thetransitionrulesin∆ areof α, then α′ ≃α.
ℓ
theformX −→α.Theoperationalsemanticsoftheprocesses IfΓisrealtime,thebranchingbisimilarityisthesameasthe
are defined by the following labelled transition rules. strong bisimilarity. In this paper, branching bisimilarity will
be abbreviated as bisimilarity. For realtime systems, the term
(X −→ℓ α)∈∆ α−→ℓ α′
bisimilarity will also be used to indicate strong bisimilarity.
ℓ ℓ
X −→α α.β −→α′.β III. RELATIVIZEDBISIMILARITIES ONNORMED BPA
A centraldot‘·’ is oftenused to indicate an arbitraryprocess. A. Retrospection
Forexample,wewriteα−ℓ→1 ·−ℓ→2 β,orevenα−ℓ→1 −ℓ→2 β,to In [1], Yuxi Fu creates the notion of redundant processes,
meanthatthereexistssomeγ suchthatα−ℓ→1 γ andγ −ℓ→2 β. and discover the following Proposition 3, which is crucial to
the proof of decidability of bisimilarity for normed BPA. Two processes are R-equalif they differ only in suffixesin
R∗.R-equalityisanequivalencerelation.Eliminatingasuffix
Definition2. Aprocessαisa≃-redundantoverγ ifαγ ≃γ.
in R∗ from a process does not change the = -class.
R
We use Rd(γ) = {X|Xγ ≃ γ} to indicate the set of all
Lemma 4. 1) α= αγ if and only if γ ∈R∗.
constants that is ≃-redundantover γ. Clearly, Rd(γ)⊆C . R
G 2) α= ǫ if and only if α∈R∗.
R
Thefollowinglemmaconfirmsthattheredundantprocesses
overγ are completelydeterminedby the redundantconstants. Definition 4. Let R⊆CG. α is in R-normal-form (R-nf) if
1) either α=ǫ,
Lemma 2. αγ ≃γ if and only if α∈(Rd(γ))∗.
2) or there exist α′ and X such that α=α′X and X 6∈R.
The crucial observation in [1] is the following fact. If α= α′ and α′ is in R-nf, then α′ is called an R-nf of α.
R
Proposition 3. Assume that Rd(γ ) = Rd(γ ), then αγ ≃ The (unique) R-nf of α is denoted by αR.
1 2 1
βγ1 if and only if αγ2 ≃βγ2. From Definition 4, taking the R-nf of α is nothing but
removing any suffix of α in R. R-equality is the syntactic
Proposition 3 inspires us to define a relativized version of
equality on R-nf’s. In particular, ∅-equality is exactly the
bisimilarity ≃ for a given suitable reference set R, which
R ordinary syntactic equality.
will satisfy the following theorem.
Lemma 5. α = β if and only if α = β . In particular,
Theorem 1. Let γ be a process satisfying Rd(γ) =R. Then R R R
α= β if and only if α=β.
α≃ β if and only if αγ ≃βγ. ∅
R
The transition relations can be relativized as follows.
Proposition 3 confirms that ≃ does not depend on the
R
special choice of γ under the assumption of the existence of Definition 5. The R-transition relations between R-nf’s are
γ such thatR=Rd(γ). However,itismuchwisernotto take defined as follows: We write ζ −→ℓ η if there exists α and
R
Theorem1asthedefinitionof≃R fromacomputationalpoint β such that ζ =αR, η =βR, and α−→ℓ β.
of view. Here are the reasons.
ℓ
AccordingtoDefinition5,therelation−→ isdefinedonly
R
• We cannot tell beforehand (except when we can decide ℓ
on the set of processes in R-nf. When we write α−→ β, α
≃) whether, for a given R, there exists γ such that R= R
and β are implicitly supposed to be R-nf’s.
Rd(γ), nor can we tell whether R=Rd(γ) even if both
R and γ are given. Lemma 6. α −→ℓ β if and only if α= ·−→ℓ ·= β.
R R R R R
• The algorithm developed in this paper takes the refine-
ℓ ℓ
mentapproach.Imaginethat≍isanapproximationof≃, Let α −→R β. Intuitively, if α 6= ǫ, then α −→R β is
we can define, for example, the ≍-redundant constants induced by α; if α = ǫ, then α −→ℓ β is induced by one
R
Rd≍(γ) accordingly. It is quite possible to run into the of the constantsin R. This importantfact is formalizedin the
situation where, for a specific R, there is no δ such that following lemma.
R=Rd≍(δ) even if R=Rd(γ) for some γ.
ℓ
Lemma 7. α −→ β if and only if
R R R
Therefore it is advisable to make ≃ well-defined for every
R satisfying R ⊆ CG. ImportantlyR, ≃R should be defined 1) either αR = ǫ and X −→ℓ β′ for some β′ and X ∈ R
without the knowledge of the existence of γ. such that β′ =R β.
2) or α 6=ǫ and α−→ℓ β′ for some β′ such that β′ = β.
Remark 4. In [25], Czerwin´ski and Jancˇar also define a R R
relativizedversionofbisimilarities.Thedifferenceisthatthey As usual, we write =⇒ for the reflexivetransitive closure
R
directly take Theorem 1 as the definition. After that, they of −τ→ , and ⇐⇒ for the symmetric closure of =⇒
R R R
establish a weaker version of unique decomposition property. (i.e. ⇐⇒ d=ef =⇒ ∪ =⇒ −1). Accordingly, α −≍→ α′
R R R R
In[25],≃R isdefinedonlyforthoseR’ssuchthatR=Rd(γ) is understood as α −τ→ α′ ≍ α, and =≍⇒ is the reflexive
R R
forsomeγ.UsingTheorem1,apropertyof≃ canbeproved ≍
R transitive closure of −→ .
R
byatranslationofapropertyof≃.Thoughseeminglysimilar,
The ground processes are robust under relativization.
thepropertieswhichwillbedevelopedinthissectionaremuch
stronger than those properties in [25]. Lemma 8. α=⇒ǫ if and only if αR =⇒R ǫ.
Now it is time for defining R-bisimilarity.
B. Definition of R-Bisimilarities
Definition 6. Let R ⊆ C and let ≍ be an equivalence
G
Now we elaborate on the definition of ≃R. Some auxiliary relation such that =R ⊆ ≍. We say ≍ is an R-bisimulation,
notations are introduced to make things clear. if the following conditions are satisfied whenever α≍β:
Definition 3. Let R ⊆ C . Two processes α and β are R- 1) ground preservation: If α=⇒ǫ, then β =⇒ǫ.
G
equal, denoted by α = β if there exist ζ,α′,β′ such that 2) If α −6≍→α′, then β =≍⇒ · −6≍→ β′ for some β′ such
R R R R
α=ζα′, β =ζβ′, and α′,β′ ∈R∗. that α′ ≍β′.
3) If α −a→ α′, then β =≍⇒ · −a→ β′ for some β′ such C. R-identities and Admissible Reference Sets
R R R
that α′ ≍β′.
Clearly, R-bisimilarity has the following basic property.
The R-bisimilarity ≃ is the largest R-bisimulation.
R Lemma 12. Let R⊆C . If X ∈R, then X ≃ ǫ.
G R
Remark 5. R-bisimilarity ≃ is well-defined, based on the
R Be aware that the converse of Lemma 12 does not hold in
following observations:
general.Thatis,ifX ≃ ǫ,thereisnoguaranteethatX ∈R.
R
• =R is an R-bisimulation. This basic observation leads to further discussion.
• If ≍1 and ≍2 are both R-bisimilations, then (≍1∪≍2)∗ Definition 7. Let R ⊆ C . A process α is called a ≃ -
is an R-bisimulation. G R
identity if α≃ ǫ. We use Id to denote {X |X ≃ ǫ}.
R R R
If R=∅, then ≃ is exactly the ordinary bisimilarity ≃.
∅
R-bisimulations can actually be understood as the bisimu- ByLemma12andDefinition6,R⊆IdR ⊆CG.Moreover,
lations on R-nf’s under R-transitions, as is stated below. Lemma 13. α≃ ǫ if and only if α∈(Id )∗.
R R
Proposition 9. Let R ⊆ CG and let ≍ be an equivalence Below we will demonstrate that, as a reference set, IdR
relation such that =R ⊆ ≍. Then ≍ is an R-bisimulation if plays an important role. At first we state a useful proposition
and only if whenever α≍β, for relative bisimilarities. It says that ≃ is monotone.
R
1) if αR =⇒R ǫ, then βR =⇒R ǫ; Proposition 14. Let R ⊆ R ⊆ C . If α ≃ β, then
2) sifucαhRth−a6≍→t αR′ ≍α′Rβ,′t;hen βR =≍⇒R · −6≍→R βR′ for some β′ α≃R2 β. 1 2 G R1
3) if αR −a→R α′R, then βR =≍⇒R · −a→R βR′ for some β′ Corollary 15. Let R1 ⊆R2 ⊆CG. Then, IdR1 ⊆IdR2.
such that α′ ≍β′. Intuitively, ≃ is the relative bisimilarity which is induced
R
byregardingtheconstantsinRasǫpurposely.Itisreasonable
Comparing with the definition of bisimulation (Defini-
tion1),Definition6andProposition9containsanextraground to expectthatX ≃IdR ǫ if andonlyif X ∈IdR. Thisintuition
is confirmed by Proposition 16 and its corollaries.
preservationconditionwhichguaranteesthatagroundprocess
cannotberelatedtoanon-groundprocessinanR-bisimilation. Proposition 16. α≃R β if and only if α≃IdR β.
Inthedefinitionofbisimulation,thisconditionisalsosatisfied,
Corollary 17. Let R ⊆C and R ⊆C . If Id =Id ,
for it can be derived from other bisimulation conditions. As 1 G 2 G R1 R2
then α≃ β if and only if α≃ β.
to R-bisimulation, this is not always the case, as is illustrated R1 R2
in the following example. Corollary 18. Let R,S ⊆ C such that R ⊆S ⊆Id , then
G R
Id =Id .
Example 1. Consider the following normed BPA (C,A,∆): S R
• C={A0,A1}; A direct inference of Corollary 18 is the following fact.
•• A∆=is{thae,τs}et;caontaining the foallowing rules:τ LIdeIdmRm=aI1d9R..X ≃IdR ǫifandonlyifX ∈IdR.Inotherwords,
A −→A , A −→A , A −→ǫ
0 1 1 1 1 The above discussions lead to the following definition.
Let R = {A }, and let ≍ be the equivalence relation which
1
relates every processes defined in Γ to ǫ. Clearly A0 6=⇒R ǫ. Definition 8. An R⊆CG is called admissible if R=IdR.
However, we can show that (A ,ǫ) satisfies R-bisimulation
0 ThesignificanceofProposition14,Proposition16,andtheir
conditions except for the ground preserving condition:
corollariesistherevelationofthefollowingfact:Theset{≃
R
maCtcohnesdidbeyrinAg0,thitatretmheainRs-ttroanshsiotwiontshaotfǫǫcacnanmabtechtrtaihveiaRlly- b}Ry⊆thCoGseo≃f aRl’lsrienlawtivheicbhisRimiislaarditmieisssiisblceo.mpletely determined
transitionsofA .TheuniqueR-transitionofA isA −→ ǫ,
which can be m0atched by ǫ −a→R ǫ since A01 −a→0 A1 Rand Lemma 20. For everyR⊆CG, IdR is admissible. IdR is the
(A ) =ǫ. smallest admissible set which contains R.
1 R
Therelativebisimilarity≃ isnota congruenceingeneral. D. R-redundant Constants
R
For example, we may not have αγ ≃R βγ even if α ≃R β. The properties of ≃-redundant processes (Definition 2 and
However, we have the following result. Proposition3)inSectionIII-Acannowbegeneralizedforthe
relative bisimilarity ≃ .
Lemma 10. If γ ≃ δ and α ≃ β, then αγ ≃ βδ. In R
R R
particular, If γ ≃ δ, then αγ ≃ αδ. Definition9. LetR⊆C .Aprocessαis≃ -redundantover
R R G R
γ if αγ ≃ γ. We use Rd (γ) to denote {X |Xγ ≃ γ}.
The computation lemma also holds for ≃ . R R R
R
NotethatRd(γ)definedinSectionIII-AisexactlyRd (γ).
Lemma 11 (Computation Lemma for ≃R). If α =⇒R Also note that Id is the same as Rd (ǫ). ∅
α′ =⇒ α′′ ≃ α then α′ ≃ α. R R
R R R
Lemma 21. If γ ≃R δ, then RdR(γ)=RdR(δ). • X is a ≃R-composite if X ≃R αX′ for some X′ and α
such that X′ 6∈Id and α6∈Rd (X′).
Lemma 22. 1) α≃ ǫ if and only if α∈(Id )∗. R R
2) αγ ≃ γ if and oRnly if α∈(Rd (γ))∗. R • X is a ≃R-prime if X is neither a ≃R-identity nor a
R R ≃ -composite.
R
Lemma 21 is a direct inference of Lemma 10. Lemma 22
According to Definition 10, a constant X ∈C must act as
is the strengthened version of Lemma 2.
one of the three different roles: ≃ -identity, ≃ -composite,
R R
Now we can state the fundamental theorem for ≃R. or ≃ -prime. We will use Pr and Cm to indicate the set
R R R
Theorem 2. Let R′ = RdR(γ), then α ≃R′ β if and only if of ≃R-primesand ≃R-composites, respectively.Accordingto
αγ ≃R βγ. Proposition 16, PrR =PrIdR and CmR =CmIdR.
Proposition 23. Assume that RdR(γ1) = RdR(γ2), then Definition 11. We call Pr.Pr−1.....P1 a ≃R-prime decom-
αγ1 ≃R βγ1 if and only if αγ2 ≃R βγ2. position of α, if α ≃R Pr.Pr−1.....P1, and Pi is a ≃Ri-
prime for 1 ≤ i ≤ r, if R = Id and R = Rd (P ) for
Proposition24. Supposethatγ ≃ δ andletR′ =Rd (γ)= 1 R i+1 Ri i
R R 1≤i≤r−1.
RdR(δ). Then αγ ≃R βδ if and only if α≃R′ β.
Note that according to Lemma 27 every R for 1 ≤ i ≤ r
i
Theorem2andProposition23arethestrengthenedversions
is admissible.
ofTheorem1andProposition3.Proposition24isaninference
Thefollowing‘relativizedprimeprocessproperty’iscrucial
of Lemma 10 and Theorem 2. Theorem 2 and Proposition 24
to the unique decomposition property (Theorem 3).
act as the relativized version of the congruence property and
the cancellation law. Lemma 28. Suppose that X, Y are ≃R-primes and αX ≃R
The following lemma is an inference of Theorem 2. βY. Then X ≃R Y.
Lemma 25. RdRdR(δ)(γ)=RdR(γδ). Theorem3(UniqueDecompositionPropertyforR-bisimilari-
ties). LetP .P .....P andQ .Q .....Q be≃ -prime
r r−1 1 s s−1 1 R
In the following we discuss the significance of the admis- decompositions. Let R ,S = Id and let R = Rd (P )
sible reference sets. First it is easy to see the following fact for 1 ≤ i < r and S 1 =1 Rd R(Q ) for 1i≤+1j < s.RTihein,
according to Proposition 16. j+1 Sj j
r =s, R =S and P ≃ Q for 1≤i≤r.
i i i Ri i
Lemma 26. RdR(γ)=RdIdR(γ) for every γ and R. IV. NORMS AND DECREASING BISIMULATIONS
The following lemma ensures that the admissible set is A. Syntactic Norms vs. Semantic Norms
preserved under the ‘redundant’ operation.
When Γ is realtime, a natural number called norm is
Lemma27. IfR=RdR′(γ)forsomeγ,thenRisadmissible. assigned to every process. The norm of α is the least number
k such that α−a→1 ·−a→2 ...−a→k ǫ for some a ,a ,...,a .
Remark 6. Even if R is admissible, it is not guaranteed that 1 2 k
Thenormforrealtimesystemsisbothsyntactic (static)and
R=RdR′(γ)forsomeγ andR′.Thisfactindicatesthat,even semantic (dynamic).It is syntactic because its definition does
if ≃ is onlyattractive foronlyadmissible R’s, ournotionof
R notrelyonbisimilarity,anditcanbeefficientlycalculatedvia
R-bisimilarities strictly generalizes the ones in [25].
greedystrategymerelywiththeknowledgeofrulesin ∆.Itis
semantic,becausethenormofarealtimeprocessαistheleast
E. Unique Decomposition Property for R-bisimilarities
numberk suchthatα≃·−a→1 ·≃·−a→2 ·≃...≃·−a→k ·≃ǫ
When Γ is realtime, the set C of process constants can be
for some a ,a ,...,a .
divided into two disjoint sets: primes Pr and composites Cm. 1 2 k
Therefore,wegetthecoincidenceofthesyntacticnormand
Every process α is bisimilar to a sequential composition of
semanticnormforrealtimesystems.Fornon-realtimesystems,
prime constants P .....P , and moreover, the prime decom-
1 r however, the syntactic norms and the semantic ones do not
positionisunique(uptobisimilarity).Thatis,ifP .....P ≃
1 r coincide any more. They must be studied separately.
Q .....Q , then r = s and P ≃ Q for every 1 ≤ i ≤ r.
1 s i i
This property is called unique decomposition property, which B. Strong Norms and Weak Norms
isfirstestablishedbyHirshfeldetal.in[13].WhenΓistotally
We define two syntactic norms for non-realtime systems.
normed, the unique decomposition property still holds [27].
The strong norm takes silent actions into account while the
If Γ is not totally normed, the unique decomposition prop-
weak norm neglects the contribution of silent actions.
erty in the above sense does not hold due to the existence of
redundant processes. However, we expound that, apart from Definition 12. The strong norm of α, denotedby |α|st, is the
theexistenceofredundantconstants,therelativebisimilarities least number k such that α −ℓ→1 · −ℓ→2 ... −ℓ→k ǫ for some
{≃ } enjoys a ‘weakened’ version of unique decomposition ℓ ,ℓ ,...,ℓ .
R 1 2 k
property (Theorem 3), which is still called unique decompo- Theweak normofα, denotedby|α| , is theleastnumber
wk
sition property in this paper. k such that α=a⇒1 ·=a⇒2 ...=a⇒k ǫ for some a ,a ,...,a .
1 2 k
Definition 10. Let R⊆C , and X ∈C. Lemma 29. 1) |ǫ| =|ǫ| =0;
G st wk
2) |αβ| =|α| +|β| ; |αβ| =|α| +|β| . Proposition 34. ≃ is a decreasing bisimulation for every
st st st wk wk wk R
R⊆C .
Lemma 30. 1) |α| =0 if and only if α=ǫ. G
st
2) |α| =0 if and only if α∈C∗ (i.e. α=⇒ǫ). Thereisnoneedtodefinetheso-calledR-decreasingbisim-
wk G
ulation. The following lemma confirms that, for decreasing
Lemma 31. If α≃ β, then |α| =|β| .
R wk wk ℓ ℓ
transitions, −→ and −→ are essentially the same.
R
Lemma 32. |X| is exponentially bounded for every X.
st
Lemma 35. If α −→ℓ β is ≃ -decreasing, then α −→ℓ β′
R R
C. The Semantic Norms
for some β′ such that β′ =β.
R
Thesemanticnormsplayanimportantroleinouralgorithm.
Thefollowinglemmaprovidesaboundforsemanticnorms.
They depend on the involved semantic equivalence. Let ≍ be
a process equivalence. A transition α −→ℓ α′ is called ≍- Lemma 36. If ≍ is a decreasing bisimulation, then kαk ≤
≍
preserving if α′ ≍α. |α| for every α.
st
Definition 13. Let ≍ be a process equivalence. The ≍-norm Remark 7. The labelled transition graph defined by a normed
of α, denoted by kαk≍, is the least number k, such that BPA Γ can be perceived as a directed graph G (with infinite
number of nodes) whose nodes are the processes and whose
α≍·−ℓ→1 ·≍·−ℓ→2 ·≍...≍·−ℓ→k ·≍ǫ.
edges are the labelled transitions. G can be extended to
for some ℓ1,ℓ2,...,ℓk. If kαk≍ = k, then any transition a weighted direct graph in different ways. Let Gst be the
sequence of the form weightedextensionof G in which everyedgeof G hasweight
one. Let G be the one which is the same as G except
α=≍⇒·−ℓ→1 ·=≍⇒·−ℓ→2 ·=≍⇒...=≍⇒·−ℓ→k ·=≍⇒ǫ (1) that the wewigkht of every silent transition is zero. Thste strong
(resp. weak) norm of a process α is the length of the shortest
is called a witness path of ≍-norm for α. The length of the
path from α to ǫ in G (resp. G ). Let ≍ be an equivalence
witness path is k. st wk
relation on processes. We can define the graph G which is
≍
Clearly, the ≍-norms have the following basic fact: the same as G exceptthat the weight of γ −→ℓ γ′ is set zero
st
Lemma 33. If α≍β, then kαk =kβk . if γ ≍ γ′. A witness path of ≍-norm of α corresponds to a
≍ ≍
shortest path from α to ǫ in G .
≍
If ≍ is an arbitrary equivalence relation, the witness path
If ≃ is known for every R, then ≃ -norm of a BPA
≍ R R
doesnotalwaysexist,becauseitis notalwaysthecase α=⇒
process(orconstant)canbecalculatedviathegreedystrategy
β whenever α ≍ β. This is one of the motivations of the
in an efficient way. It depends on the following property:
forthcomingnotionofdecreasingbisimulation(Definition15).
1) kαk =0 if and only if α≃ ǫ.
For the moment, we introduce the ≍-decreasing transitions. ≃R R
2) kαβk =kαk +kβk in which R′ =Rd≃(β).
Definition 14. A transition α −→ℓ α′ is ≍-decreasing if ≃R ≃R′ ≃R R
It works like a generalization of Breadth-first search, or a
kα′k <kαk .
≍ ≍ variantofDijkstra’salgorithm.Thedetailofthecalculationis
AccordingtoDefinition13,kα′k =kαk −1ifα−→ℓ α′ omitted, but this idea will be used to calculate the semantic
≍ ≍
is a ≍-decreasing transition. In witness path (1), every transi- norms k·kB later (Section VI) in the refinement procedure
tion −ℓ→i must be ≍-decreasing for 1≤i≤k. when const=ruRcting new base from the old.
Definition 15. A process equivalence ≍ is a decreasing D. Decreasing Bisimulation with R-Expansion of ≏
bisimulation, if the following conditions are satisfied:
Based on decreasing transitions, we can define a special
1) If α≍ǫ, then α=⇒ǫ.
2) If α≍β and α−→ℓ α′ is a ≍-decreasingtransition,then notion called decreasing bisimulation with R-expansion of
≏, which will be taken as the refinement operation in our
there exist β′′ and β′ such that β =≍⇒ β′′ −→ℓ β′ and
algorithm. This notion is crucial to the correctness of the
α′ ≍β′.
refinement operation. The readers are suggested to review
Decreasingbisimulationisaweakerversionofbisimulation. Definition 6 and Definition 15 before going on.
The difference lies in that only decreasing transitions need
Definition16. Let≍and≏betwoequivalencesonprocesses
to be matched. Be aware that the transition β′′ −→ℓ β′ in such that = ⊆≍⊆≏. We say that ≍ is an R-expansion of
R
Definition 15 is forced to be ≍-decreasing. ≏ if the following conditions hold whenever α≍β:
Let≍beadecreasingbisimulation.Thenany≍-decreasing
1) α=⇒ǫ if and only if β =⇒ǫ.
transitionofαcanbeextendedtoawitnesspathof≍-normof
α. The norm kαk≍ is equalto the least numberof decreasing 2) If α−6≏→α′, then either βR =≍⇒R ·−6≏→R β′ for some β′
transitions from α to ǫ. such that α′ ≏β′.
Nearlyallequivalencesappearinginthis paperare decreas- 3) If α −a→α′, then β =≍⇒ · −a→ β′ for some β′ such
R R R
ing bisimulation. For example: that α′ ≏β′.
We say that ≍ is a decreasing bisimulation with R-expansion constants are R-bisimilar to each other, thus they can be
of≏if≍isbothadecreasingbisimulationandanR-expansion contracted into a single one. In the work [27] for totally
of ≏. normed BPA, we prevent the occurrence of X ⇐⇒ Y
via a preprocess in which mutually associate constants are
The following lemma provides another characterization of
contracted into a singe one. For normed BPA discussed in
the decreasing bisimulation with R-expansion of ≏.
this paper, we take the same idea but the difficulty is that the
Lemma 37. Assume that = ⊆ ≍ ⊆ ≏. ≍ is an decreasing contracting operation cannot be performed uniformly, for it
R
bisimulation with R-expansion of ≏ if and only if following depends on the reference set R. The only way we can take
conditions hold whenever α≍β and α,β are in R-nf: is to introduce the R-association and to contract R-associate
1) if α=⇒ ǫ, then β =⇒ ǫ; constants into R-blocks for individual R’s. The members in
R R
2) if α −6≍→ α′, being ≍-decreasing, then β =≍⇒ · −6≍→ an R-block are interchangeable.
R R R
β′ for some β′ such that α′ ≍β′; Be aware that it is possible that X ≃ ǫ even if X 6∈ R.
R
3) if α −6≏→R α′, not being ≍-decreasing, then β =≍⇒R In this case we must have [X]R ⊆ IdR by the Computation
·−6≏→R β′ for some β′ such that α′ ≏β′; Lemma(Lemma11).AlsonotethatitispossiblethatX ⇐⇒R
4) if α −a→ α′, being ≍-decreasing, then β =≍⇒ · −a→ ǫ for some R. But these kinds of R is uninteresting because
R R R
β′ for some β′ such that α′ ≍β′. byputtingsuchX intoR wecangetalargerreferencesetR′
5) if α −a→R α′, not being ≍-decreasing, then β =≍⇒R such that ≃R′ =≃R.
·−a→R β′ for some β′ such that α′ ≏β′. We call a reference set R qualified if X ⇐⇒R ǫ cannot
happen for every X 6∈ R. The unqualified R’s can be pre-
Remark 8. Thestyleofthedefinitionof‘decreasingbisimula-
determined. They are useless from now to the end of this
tionwithexpansion’alsoappearsin[27].Themaindifference
paper. From now on we assume that every reference set R is
is that in this paper, the ‘decreasing transitions’ are semantic, qualified. For example when we write ‘for every R ⊆ C ’,
G
while in [27], the ‘decreasing transitions’ are syntactic. we refer to every qualified R which is a subset of C . In
G
Thenotionof‘decreasingbisimulationwithexpansion’isa
particular, every admissible set is qualified.
better understandingof the previous refinement operationson
totallynormedBPAandBPP[15],[26].Moreover,thisnotion Lemma 38. All constants in a block [X] are R-bisimilar.
R
is crucial to the development of a polynomial time algorithm
Lemma39. If[X] 6=[Y] andX =⇒ Y,thenY 6=⇒ X.
for branching bisimilarity on totally normed BPA [27]. R R R R
The behaviours of [X] can be more than the total be-
V. DECOMPOSITION BASES R
haviours of its member constants. All the processes asso-
In this section, we define a way for finitely representing
ciate to X should be taken into account. It is possible that
a family of equivalences which satisfies unique decomposi-
X ⇐⇒ ζX′ for some ground process ζ. For instance we
R
tion property in the sense of Theorem 3. Such family of
can have X =⇒ Z =⇒ ζY =⇒ Y =⇒ X. In this
R R R R
equivalencesinclude {≃ } and allthe intermediatefamilies
R R example, X,Y,Z,ζX,ζY,ζZ are mutually R-associate. Thus
of equivalences constructed during the iterations. This finite
the behaviour of ζ should also be taken into account.
representation is named decomposition base.
Definition 18. Y is an R-propagating of X (or of [X] ) if
A. R-blocks and R-orders R
X ⇐⇒ YζX′ for some ζ and X′. (In this case we must
R
To make our algorithm easy to formulate, we need some have X′ ⇐⇒ X, and Yζ is ground.)
R
technical preparations. The reason will be clear later.
Lemma 40. Y ∈Rd (X) if Y is an R-propagatingof X.
Definition 17. Let R ⊆ C . We call that α is R-associate R
G
to β if α ⇐⇒R β. Let X ∈ C\R. The R-block related to Lemma 41. Suppose X ⇐⇒R ζX′ −→ℓ R ζ′X′ such that
X, denoted by [X]R is the set of all the constants which is ζ′X′ 6=⇒R X. Then X′ ∈ [X]R, and ζ = Yγ for some Y
R-associate to X. Namely, [X] d=ef {Y | X ⇐⇒ Y}. We and γ such that
R R
use the term block to specify any R-block for R⊆CG. • Y is an R-propagating of [X]R.
Clearly,twoR-blockscoincidewhentheyoverlap.ThusR- • Y −→ℓ α and ζ′ =αγ.
blocks composea partition of C\R. The partition is denoted • X ≃R γX. (i.e. γ ∈(RdR(X))∗)
by CR d=ef {[X]R |X ∈C\R}. • Y.X −→ℓ R αX with Y.X ≃R ζX′ ≃R X and αX ≃R
We will use the convention that the members of [X]R for ζ′X ≃R ζ′X′.
different R’s are taken from different copy of C. In other
Lemma 41 shows that the behaviours of [X] are com-
words,ifR 6=R ,then[X] and[X] arealwaysdisjoint, R
1 2 R1 R2 pletely determined by the associate constants and the propa-
andtheyareregardedasdifferentobjects,eveniftheyindicate
gatingconstantsofX,whichleadstothefollowingdefinition.
the same set.
ℓ
Remark 9. The intuition of R-blocks is obvious. According Definition 19. The R-derived transition 7−→ is defined as
R
to the Computation Lemma (Lemma 11), The R-associate follows:
ℓ
1) Let X ∈[X] and X −→ α. If either ℓ6=τ, or ℓ=τ activatedafterYζ isconsumedcompletely.Thisniceproperty
R R
and α6=⇒ X, then [X] 7−→ℓ α. of normed BPA simplifies the situation greatly.
b R b R R
ℓ
2) Let Y be an R-propagating of [X]R and Y −→R α. If B. Decomposition Bases
ℓ
αei.tXhe.r ℓ 6= τ, or ℓ = τ and α 6=⇒R ǫ, then [X]R 7−→R eveArydeBcomispoasiqtuioinntbuapslee(BIdisBa,PfarmBil,yComf{BB,RD}cRB⊆,CRGdiBn)w.hich
R R R R R R
Lemma 42. Suppose X ⇐⇒R · −→ℓ R α. If either ℓ 6= τ, or • IdBR is a subsetof groundconstantscalled BR-identities.
ℓ=τ and α6=⇒R X, then [X]R 7−→ℓ R ·≃R α. • CmBR specifies the set of BR-composites. A BR-
composite is an IdB-block.
R
ItistechnicallyconvenienttotreattheR-blocksasthebasic • PrBR specifies the set of BR-primes. A BR-prime is an
objects in the algorithm, because of the following lemma. IdB-block.
R
Lemma 43. If [X]R 7−τ→R ·=⇒R Y, then [Y]R 6=[X]R. • RdBR isa functionwhosedomainisPrBR. Let[X]IdBR be
Finally we can define an order on R-blocks based on a BR-prime. The value RdBR([X]IdBR) is a set of ground
constants which are called BR-redundant over [X]IdB.
wLehmenmevae3r9[.XF]or<ever[yYR] ,,wweefihxavaelXine6=a⇒r ordYer.<R such that • DcBR is a function whose domain is CmBR. Let [X]RIdB
R R R R R
Lemma 44. If [X]R 7−τ→R ·=⇒R Y, then [Y]R <R [X]R. btheeaBBRR-d-eccoommppoossiittei.onThoef v[Xalu]IedBD,cwBRh(i[cXh]IidsBRa) sistricnaglleodf
R
Example2. Theexampleillustrateswhywehavetointroduce blocks [Xr]Rr[Xr−1]Rr−1...[X2]R2[X1]R1 with r ≥ 1,
possibly different orders <R for different R’s. R1 = IdBR, [Xi]Ri ∈ PrBRi and Ri+1 = RdBRi([Xi]Ri)
In a normed BPA system Γ = (C,A,∆), we can have for every 1≤i<r.
the following fragment of definition: Let A ,A ,B ,B be To make a decompositionbase B work properly,we need the
1 2 1 2
constants in C . We have transition rules: following constraints:
G
A −τ→A B , A −τ→A B . 1) R⊆IdBR ⊆CG.
1 2 1 2 1 2 2) If R ⊆ S, then IdB ⊆ IdB. If R ⊆ S ⊆ IdB, then
R S R
Therecanbeothertransitionsrelatedtothereconstantswhich IdB =IdB. In particular, IdB =IdB.
R S IdB R
is •ofInnothimepcoarsteanocfeR. N1o=w,{tBak1e},nwoteicheaovef tAhe1f−oτ→lloRw1inAg2f.aTchtsu:s 3) BBR-ad=miBssIidbBRle.foBr eisvecroymRp.leWtelhyeRndeRter=minIdedBR,byRthisoscealBleRd
• Iwnethhaevcea[sAe2o]fRR1 2<=R1{[BA21}]R,1w.eOhr,avinesAh2or−t,τ→AR22<AR11. ATh1u.s 4) iInfRwhisicBh-RadmisisBs-ibadlem,tihsseinblCe.mBR andPrBR area partition
we have [A1]R2 <R2 [A2]R2. Or, in short, A1 <R2 A2. of R-blocks:CmBR∪PrBR =CR and CmBR∩PrBR =∅.
These two orders <R1 and <R2 are clearly not consistent. 5) RdBR([X]IdBR) is B-admissible provided that [X]IdBR is a
ThisfeaturereflectsabigdifferencebetweennormedBPAand B -prime. Thus DcB is well-defined.
R R
totally normed BPA. AdecompositionbaseB definesafamilyofstringrewriting
B
Remark 10. There is also a big difference between normed system{→R}R⊆CG.ThefamilyofBR-reductionrelationsare
BPA and normed BPP. In the case of normed BPP [30], [31], defined according to the following structural rules.
tlheteuospsearyattohraotfXpagreanleleraltceosmYpoifsiXtio⇐n.⇒ThYus,kifXX,ingewnheirlaete‘ks’Yis, X ∈IdBR X 6∈IdBR DcBR([X]IdBR)=α
then X ⇐⇒ Yn k X hence X ≃ Yn k X for every n ∈ N. X →BR ǫ X →BR [X]IdB [X]IdB →BR α
a R R
Suppose that X −→ǫ, we have a [X]IdBR ∈PrBR α→BRdBR([X]IdBR) α′ β →BR β′
X ⇐⇒Y k...kY kX −→Y k...kY α.[X]IdB →BR α′.[X]IdB α.β →BR α.β′
ntimes ntimes R R
B -reductionrelationsaredeterministic.Thusforanyprocess
for every n∈N. N|ow,{izf all}the | {z } R
α, the B -normal-form (in the sense of string rewriting
R
Y k...kY kX systems) is unique, and it is called the BR-decomposition
of α. We use the notation dcmpB(α) to indicate the B -
ntimes R R
decomposition of α. Processes α and β are B -equivalent,
foreveryn∈N areco|ntrac{tezdint}oablock[X],thenwehave B R
notation α= β, if they have the same B -decomposition.
R R
ℓ
[X]7−→Y k...kY Lemma 45. α=B β if and only if dcmpB(α)=dcmpB(β).
R R R
ntimes
According to B -reduction rules, we have the following
for everyn∈N, as is donein|the{szame}way as Definition19. R
characterization of dcmpB(α).
This example shows that the behaviour of [X] is infinite R
branching.NotethatinthecaseofnormedBPA, thissituation Lemma 46. If R is B-admissible, then
isnotpossible.IfX ⇐⇒YζX,thenactionsinX canonlybe • dcmpBR(ǫ)=ǫ.
• If X ∈R, then dcmpBR(γX)=dcmpBR(γ). • In the third step, for every non-B-admissible R, BIdR
• If X 6∈R, then dcmpBR(γX)=dcmpBR(γ.[X]R). is assigned to BR. That is, PrRb:= PrIdR, CmRb :=
• If [X]R ∈CmR, then CmIdR, and sobon.
dcmpBR(γ.[X]R)=dcmpBR(γ.DcR([X]R)). Pay special attention to the descriptions of PrR and CmR.
• If [X]R ∈PrR, then TheyhaveslightlydifferentfromPrR andCmR.Semantically,
If R is notdcBm-padBRm(γis.[sXib]lRe,)th=en(ddccmmppBRBd(BRα()[X=]Rd)(cγm)p)B.[X(]Rα.). idfesXcri∈ptiPonrRofanPdrRXa≃ndRCYm,Rth,ewneYne∈edPtrhRe.BIRn-tphreimseysnttaoctbice
R IdB absolutely unique,which is accomplishedvia < . The orders
R b R
We list some basic facts. <R take effects in double means: Let R be admissible, then
Lemma 47. α =B ǫ if and only if α ∈ IdB. When R is 1) Among the R-blocks of ≃R-primes, there is exactly one
R R distinguished R-block that is qualified as a B -prime,
B R
B-admissible, α= ǫ if and only if α∈R.
R which is the < -minimum one in the related ≃ -class.
R R
b b
Lemma 48. If X ,X ∈ [X] , then X =B X and 2) Let[X] beaB -prime.If[X] 7−τ→ α,thenX =6B α.
1 2 R 1 R 2 R R R R R
b
kX k =kX k . B
1 =BR 2 =BR If X =⇒R Y 6=b⇒R X, then X 6=R Y.
Every decomposition base constructed during the refinement
We can write k[X] k =kXk for any X ∈[X] .
R B B R
=R =R procedure in our algorithm will satisfy these two properties.
Lemma 49. kXk ≥1 if R isbB-admissible abnd X 6∈R.
B Remark 11. For realtime normed BPA (or BPP), there is an
=R
even strong property. There exists a uniform order ‘<’ on all
B
Lemma 50. If =R is a decreasing bisimulation, then the size the constants such that, whenever X −→ℓ α is syntactically
of DcB([X] ) is exponentially bounded.
R R (also semantically) decreasing or ≍-preserving (for some
In the following the superscript B will often be omitted if appropriate≍),all the constantsin α will be strictly less than
B is clear from the context. For example sometimes we write X in order ‘<’. In history, this property plays a significant
Pr for PrB. role in the previousfast bisimilarity decision algorithms [26],
R R
[15].
C. Representing ≃ via Decomposition Base
R For totally normed BPA, we have an adaptation of this
We definea decompositionbaseB whichcan represent≃R strong property, in which the condition becomes X −→ℓ α
b
B
foreveryR.Thatis,α=R βifandobnlyifα≃R β.Theorem3 is syntactically (no longer semantically) decreasing, or weak-
is crucial. Moreover, there are other subtleties which deserve norm-preserving(no longer ≍-preserving) [27].
to be mentioned. For non-realtime normed BPA systems, the above require-
mentisdefinitelytoostrongtobesatisfied,sothatthedecision
Lemma 51. All constants in a block [X]Id are R-bisimilar.
R algorithm must be developed in some other ways. This is the
ThedescriptionofB ={(IdR,PrR,CmR,DcR,RdR)}R origination of putting semantic norms into the algorithm.
relies on the family of orders{< } definedin Section V-A.
b R R Ultimately we have the following coincidence result.
It contains three steps:
b
• In the first step, we determine IdR for every R: IdR = Proposition 52. α≃R β if and only if α=BR β.
Id . According to Proposition 14, Proposition 16 and
R
their corollaries, Id satisfies constraints 1–3 in Sec- VI. DESCRIPTION OF THEALGORITHM
R
tion V-B. In particular, R is admissible if and only if Our algorithm takes the partition refinement approach. The
R is B-admissible. purpose is to figure out the B defined in Section V-C. The
• In thebsecond step, we determine other constituents of strategy is to start with a spebcial initial base B0 satisfying
B for every B-admissible R: B ⊆B and iteratively refine it. We will use notation B ⊆D
R 0
– Pr = {[X] | X ∈ Pr and X 6≃ tomeanthat=B ⊆=D foreveryR.Therefinementoperation
b R b R R R b R R
Y for every Y < X}. will be denoted by Ref. By taking B = Ref(B ), we have
R i+1 i
– Cm = {[X] | X ∈ Cm , or X ∈ a sequence of decomposition bases
R R R
Pr and X ≃ Y < X for some Y}.
R R R
– If [X] ∈ Pr , then Rd ([X] ) = {Y | YX ≃ B0,B1,B2,...
R R R R R
X}. Be aware that RdR([X]R) is admissible (also such that
B-admissible) according to Lemma 27. B ⊇B ⊇B ⊇....
0 1 2
– If [X] ∈ Cm , then Dc ([X] ) =
R R R R
b
[X ] [X ] ...[X ] [X ] , in which The correctness of the refinement operation adopted in this
r Rr r−1 Rr−1 2 R2 1 R1
X ≃ X .X .....X , R = R, [X ] ∈ Pr paper depends on the following requirements, which will be
R r r−1 1 1 i Ri Ri
and R = Rd ([X ] ) for every 1 ≤ i < r. proved gradually:
i+1 Ri i Ri
Thanks to the B-admissibility of Rd ([X ] ) for 1) B ⊆B .
Ri i Ri 0
1≤i≤r, Dc ([X] ) is well-defined. 2) Ref(B)=B.
R R
b b
b b
3) If B (B, then B ⊆Ref(B)(B. C. Expansion Conditions
Accordibngtotheabovbethreerequirements,oncethesequence We start to define new base B from the old base D. This
{Bi}i∈ω becomes stable, say Bi = Bi+1 for some i, we can is the core of our algorithm. The newly constructed =BR is
affirm that B =Bi. made to be a decreasing bisimulation with R-expansion of
On the whole, our algorithm is an iteration: =D . Referring to Lemma 37, we have =B ⊆ =D , and for
b R R R
every α,β in R-nf, the following conditions hold whenever
1) Compute the initial base B0 and let D:=B0. B
2) Compute the new base B from the old base D. α=R β:
3) If B equals D then halt and return B. 1) if α=⇒ ǫ, then β =⇒ ǫ;
R R
4) D:=B and go to step 2. 2) Whenever α−τ→ α′,
R
Apparently, the algorithm relies on the initial base and the a) if α −τ→ α′ is =B -decreasing, then β ==B⇒R · −τ→
refinement step which computes B =Ref(D) from D. β′ for soRme β′ sucRh that α′ =B β′; R R
R
A. Relationships between Old and New Bases b) if α−τ→R α′ is not =BR-decreasingand α6=DR α′, then
B
Beforedescribingthealgorithmindetails,weinvestigatethe β ==⇒RR ·−τ→R β′ for some β′ such that α′ =DR β′;
relationship between two bases B and D assume that B ⊆D. 3) Whenever α−a→ α′,
R
Lemma 53. If B ⊆D, then IdBR ⊆IdDR for every R. a) if α −a→R α′ is =BR-decreasing, then β ==B⇒RR · −a→R
Remark 12. An interesting consequence according to β′ for some β′ such that α′ =BR β′.
Lemma 53 is that, if R = IdDR, then R = IdBR must hold b) if α −a→R α′ is not =BR-decreasing, then β ==B⇒RR
because R⊆IdBR ⊆IdDR. This confirms the fact that, during ·−a→R β′ for some β′ such that α′ =DR β′.
the iteration of refinement, once a reference set R becomes
The above conditions will be called expansion conditions in
B-admissible, it preserves the admissibility in the future.
the following. Our task is to construct B from D and validate
Lemma 54. If B ⊆D and IdB =IdD, then PrD ⊆PrB. theseexpansionconditions.Fromexpansionconditionswecan
R R R R
B D B
Lemma 55. If B ⊆ D, IdB = IdD, and PrB = PrD, then see that, in case =R = =R, =R must be an R-bisimulation.
R R R R D B D
RdB ⊆RdD. Thus when ≃R (=R, we must have =R (=R.
R R
Basically, the construction contains three steps:
Lemma 56. If B ⊆ D, and moreover IdBR = IdDR, PrBR = 1) DetermineIdB foreveryqualifiedR.Afterthat,weknow
PrD, and RdB =RdD for every R, then B =D. R
R R R whether a given R is B-admissible. Note that some R’s
The purpose of Lemma 53 to Lemma 56 is to get the which are not D-admissible can be B-admissible.
following fact. 2) Determine other constituents of BR for every B-
admissible R.
Proposition57. Thetotalnumberofiterations(i.e.refinement
operations) in our algorithm is exponentially bounded. 3) For non-B-admissible R’s, BIdR is copied to BR.
The third step is relatively trivial. Its correctness depends
on the following lemma.
B. The Initial Base
B B
The initial base B0 ={B0,R}R is defined as follows: Lemma 60. If ≃IdB ⊆=IdB, then ≃R ⊆=R.
R R
• IdR :=CG ={X ∈C||X|wk =0} for every R. The first and secondsteps of the constructionare described
Thus C is the only B -admissible set. in Section VI-D and Section VI-E.
G 0
• PrR := {[P]CG} where [P]CG is the <CG-minimum D. Determining IdB
C -blocksatisfying|P| =1.Pr :=∅ incaseC = R
G wk R G
C. First of all, we must determine what IdB is. This problem
R
• CmR :=CCG \PrR. asksunderwhatcircumstancewecanbelievethatX =BR ǫfor
• DcR([X]CG):=[P]CG...[P]CG if [X]CG ∈CmR. X ∈CG.BeawarethatLemma53confirmsthatIdBR ⊆IdDR.
The basic idea is to make use of the expansion conditions.
|X|wktimes
• RdR([P]IdR):=|CG, if{PzrR =}{[P]IdR}. Definition 20. Let S be a set that makes R ⊆ S ⊆ IdDR.
Now B0,R is defined as (IdR,PrR,CmR,DcR,RdR). No- We call S an IdBR-candidate if the following conditions are
tice that B0,R is the same for every R⊆CG. satisfied whenever X ∈S\R:
Lemma 58. αB=0 β if and only if |α| =|β| . 1) If X −τ→R α and α 6∈ (IdDR)∗, then ǫ −τ→R β for some
R wk wk β such that dcmpD(α)=dcmpD(β).
R R
B a a
Lemma 59. B ⊆B . Namely, ≃ ⊆= for every R. 2) If X −→ α, then ǫ −→ β for some β such that
0 R R R R
dcmpD(α)=dcmpD(β).
One can chbeck that all the five constraints described in R R
Section V-B are satisfied by B . According to Definition 20,
0