Comparing Refinement Settings

Comparing Refinement SettingsHeiko SchmidtChristian-Albrechts-Universit¨a t zu Kiel,Germanyhsc@informatik.uni-kiel.de1IntroductionThe process of specializing a specification is called refinement.This concept is,e.g.,frequently used in model-driven software development,where the process of development starts with an abstract model,which is refined in later design phases.If thefinal system should be runnable on a computer,this refinement process is continued until a concrete system,also called im-plementation,is reached.Abstraction,a technique to tackle the state explosion problem that occurs in model checking,is dual to refinement.Here,the model of a program is abstracted by combining program states and reducing their absolute number.We compare settings that can be used for refinement and abstraction(shortly called refine-ment settings)with respect to their relative expressive power.We present several approaches that can be taken,discussing their possible applications.We also present some results of com-parisons between existing refinement settings.We concentrate on a reactive setting with labeled transition systems as implementations and consider various refinement settings based on them,including(disjunctive)modal and mixed transition systems[2,3,7,8]as well asµ-and modal automata(without fairness)[6,4].Work presented here and the mentioned future work are planned to constitute part of my PhD thesis. 2Comparing Refinement SettingsA refinement setting consists of a set of models,a subset of models to be considered imple-mentations,and a refinement preorder,defining whether one models refines another.In the following,we present possible approaches to the comparison of expressiveness:Elementwise comparison.If the two compared abstraction settings are based on the same class of models,e.g.,labeled transition systems,the two preorders can be compared elementwise,i.e., one setting is as least as expressive as another setting,if each pair in the preorder relation of the second setting is also a pair in the preorder relation of thefirst setting.This kind of comparison is performed in[11]and[5],where preorders of the linear time–branching time spectrum are compared1.However,since we primarily want to compare settings based on different models, we do not follow this approach.Implementation-based comparison.This comparison approach can be applied to refinement settings based on different models.It identifies an abstract system primarily with the set of its refining implementations and considers one setting to be at least as expressive as another setting,if every set of implementations expressible by an abstraction in the second setting is also expressible by an abstraction in thefirst setting.Formally,an implementation-based embedding must exist,which is defined as follows:This work is in partfinancially supported by the DFG project Refism(FE942/1-1).1Those preorders,however,mostly do notfit into our understanding of a refinement setting,because the implementations are not distinguished,and mostly not even satisfactorily distinguishable.Modal transition systemsMixed transition systems Disjunctive modal transition systems Disjunctive mixed transition systems µ-automata(without fairness) modal automata(without fairness) LTS with ready simulationLTS with ready trace inclusionLTS with ready pair inclusionLTS with failuretrace inclusion LTS with failure pair inclusionFig.1.Hierachy of refinement/abstraction settings with respect to deterministic transition systems as implementations and the implementation-based comparison approach.Settings above are more ex-pressive than settings below.Ready(resp.failure)pair inclusion is the preorder of the readiness(resp. failures)semantics,LTS stands for labelled transition systems.Definition1(Implementation-based embedding).A function f from one setting A to another setting B is an implementation-based embedding,if it preserves the sets of concrete refinements,i.e.,the set of concrete refinements of each abstraction a in A equals the set of concrete refinements of f(a)in B.Such transformations are satisfactory in many cases.For example,if there is an algorithm for generalized model checking[1]for one setting,satisfaction in the other setting can be checked as well,byfirst applying the transformation.This works for generalized model checking,because in this technique an abstraction satisfies a given formula if and only if each implementation satisfies the formula.Using the implementation-based comparison approach,we compared several abstraction settings with respect to deterministic transition systems as implementations.This resulted in the hierarchy illustrated in Figure1.This work will soon be submitted for presentation. Preorder-based comparison.The previously presented approach does not consider the actual structure of the refinement preorder,because it only compares sets of implementations.However, the relations between abstractions may be of interest,e.g.,if a compositional satisfaction relation is given that approximates the validity of formulae in abstractions(then an abstraction might not satisfy a formula,although all its implementations do).As before,settings are compared using transformations(or their non-existence),but now making use of order homomorphisms or embeddings on the refinement equivalence classes,where two models are refinement equivalent if and only if they refine each other in both directions:Definition2(Preorder-based homomorphism/embedding).A function f from one set-ting A to another setting B is an preorder-based homomorphism(resp.embedding),if1.f induces an order homomorphism(resp.embedding)on the refinement equivalence classes,and22.for all concrete systems a of A,we have that a and f(a)are refinement equivalent.Using the preorder-based comparison approach,we compared disjunctive modal transition sys-tems(DMTS)with a variant,introduced by us,called1-selecting modal transition systems (1MTS)[10,9],which interprets hypertransitions exclusively(exclusive-or choice of exactly one witness).We got the result that1MTS and DMTS have the same expressive power when com-pared using the implementation-based approach,but1MTS are more expressive when compared using the preorder-based embedding approach.3Future WorkWe are seeking applications that give insight on the usefulness of the different comparison approaches and their corresponding transformations.In which cases is it desired to switch refinement settings and is the achievement worth the price of increased complexity of the model that will be introduced by most of the transformations?Furthermore,many comparisons are still to be performed.By now,we have an overview over the hierarchy of refinement settings with respect to deterministic transition systems as implementations and the implementation-based comparison approach(Figure1).Results with respect to the preorder-based approach are still missing.Furthermore,the settings are not com-pared yet with respect to transition systems as implementations.Weak notions of the refinement settings,i.e.,those that consider internal actions that cannot be observed from the outside,also remain to be examined.References[1]G.Bruns and P.Godefroid.Generalized model checking:Reasoning about partial state spaces.InC.Palamidessi,editor,CONCUR2000,volume1877of LNCS,pages168–182.Springer-Verlag,2000.[2] D.Dams.Abstract interpretation and partition refinement for model checking.PhD thesis,Tech-nische Universiteit Eindhoven,The Netherlands,1996.[3] D.Dams,R.Gerth,and O.Grumberg.Abstract interpretation of reactive systems.ACM Trans-actions on Programming Languages and Systems,19(2):253–291,1997.[4] D.Dams and K.S.Namjoshi.Automata as abstractions.In R.Cousot,editor,Verification,Model Checking,and Abstract Interpretation,volume3385of LNCS,pages216–232.Springer-Verlag,2005.[5]R.Eshuis and paring refinements for failure and bisimulation semantics.Fundam.Inf.,52(4):297–321,2002.[6] D.Janin and I.Walukiewicz.Automata for the modalµ-calculus and related results.In J.Wie-dermann and P.H´a jek,editors,Mathematical Foundations of Computer Science,volume969of LNCS,pages552–562.Springer-Verlag,1995.[7]rsen and B.Thomsen.A modal process logic.In LICS’88,pages203–210.IEEE ComputerSociety Press,1988.[8]rsen and L.Xinxin.Equation solving using modal transition systems.In LICS’90,pages108–117.IEEE Computer Society Press,1990.[9]paring disjunctive modal transition systems with their one-selecting variant,2006.Diploma thesis.Available at rmatik.uni-kiel.de/~hsc/diplomarbeit.pdf.[10]H.Schmidt and paring disjunctive modal transition systems with an one-selecting variant.In Nordic Workshop on Programming Theory,2006.Available at http: //rmatik.uni-kiel.de/~hsc/nwpt06.pdf.[11]R.J.van Glabbeek.The linear time–branching time spectrum I;the semantics of concrete,sequential processes.In J.Bergstra,A.Ponse,and S.Smolka,editors,Handbook of Process Algebra,chapter1,pages3–99.Elsevier,2001.3。