A Theory of Behaviour Observation in Software Testing

Transcription

1 ehavour Observaton n Software Testng 20/08/99 Theory of ehavour Observaton n Software Testng Hong Zhu School of omutng and Mathematcs, Oxford rookes Unversty, Gsy Lane, Headngton, Oxford, OX3 0P, UK, Emal: hzhu@brookesacuk Xudong He School of omuter Scence, Florda Internatonal Unversty, Unversty Park, Mam, FL 33199, Emal: hex@csfuedu bstract Software testng s a rocess n whch a software system s dynamc behavours are observed and analysed so that the system s roertes can be nferred from the nformaton revealed by test executons Whle the exstng software testng theores mght be adequate n descrbng the testng of sequental systems, they are not caable to descrbe the testng of concurrent systems that can exhbt dfferent behavours on the same test case due to non-determnsm and concurrency Ths aer resents a theory of behavour observaton n software testng We frst ntroduce and formally defne the noton of observaton scheme that characterses systematc and consstent methods of behavour observatons and recordngs We roose a set of desrable roertes for observaton schemes and study the relatonshs among the roertes We rovde several constructons of observaton schemes that have drect mlcatons n current software testng ractce We then exlore the relatonshs between dfferent observaton schemes and examne these observaton schemes wth regard to the desrable roertes Fnally, we aly the theory to a concrete comutaton model for concurrent systems, hgh-level Petr nets, and demonstrate how to use observaton scheme to defne test adequacy crtera Keywords: software testng, behavour observaton, concurrent systems, Petr nets, comlete artally ordered set 1 Introducton Software testng s a rocess n whch a software system s dynamc behavours are observed, recorded and analysed so that roertes of the system can be nferred from the nformaton revealed by test executons The ast decades have seen a rad growth n the study of testng methods, see eg [ZHM97] for a survey The most well known testng methods nclude structural testng, functonal testng, and fault-based testng These testng methods not only dffer from each other n the way that test cases are selected or generated, but also n the way that the dynamc behavour of the software under test are observed and analysed For examle, n statement testng, executons of the statements n rogram source code are observed and recorded for analysng the ercentage of statements tested In ath testng methods, observatons are made and recorded to analyse varous tyes of aths executed durng testng In mutaton testng, what observed s the lveness of mutants, e whether or not a mutant roduces a result dfferent from that of the software under test These behavour observaton and recordng methods are usually mlemented through software nstrumentaton It s no doubt that how to observe system s behavour durng the testng rocess s a very mortant ssue of testng methods No matter how dynamc behavour s observed and recorded, t should be made consstently and systematcally Unfortunately, few exstng theores have addressed the mortant ssue of behavour

2 ehavour Observaton n Software Testng 20/08/99 observaton n software testng Instead, exstng theores of software testng assume exlctly or mlctly that a software system can only demonstrate one dynamc behavour on one test set and the test cases unquely determne the dynamc behavour of the rogram under test For examle, test crtera were defned as a functon or redcate on test sets, eg [GG75, u82, Wey86, hs87, DaW88 Wey88, PZ91, PZ93, FW93, ZH93] Ths assumton s sound for testng sequental software However, testng a concurrent system on the same test case twce may reveal dfferent behavours due to non-determnsm The unqueness assumton on software s dynamc behavour hence sgnfcantly lmts the alcaton of exstng software testng theores Ths aer addresses the comlex of non-determnsm and concurrency n software testng whle sequental software systems are treated as secal cases Ths aer rooses a theory that enables us to study the roertes of varous ways of behavour observatons and the relatonshs between them It s organsed as follows Secton 2 ntroduces the noton of observaton schemes and the relmnary concets and notatons used n the aer Secton 3 dscusses varous roertes of observaton schemes Secton 4 studes the extracton relaton between observaton schemes Secton 5 rooses a number of schemes constructons as abstractons of testng methods Secton 6 ales the theory to the testng of Petr net number of observaton schemes are defned and analysed Secton 7concludesthe aer wth a dscusson of related and future work 2 The Noton of Observaton Schemes The recorded observatons of system s dynamc behavour durng a testng can be (1) the set of executed statements, (2) the set of exercsed branches, (3) the set of traversed aths, (4) the sequences of communcatons occurred between rocesses, or (5) the set of dead mutants, and so on To make the behavour observaton and recordng meanngful, we requre that the observaton and recordng be systematc and consstent For examle, t s not accetable f we record executed statements at one tme and communcatons between rocesses at another tme durng a sngle testng executon We use the word scheme to denote a systematc and consstent way of observng and recordng dynamc behavours n software testng We requre that a scheme have a unverse of henomena on systems dynamc behavours that are observable from testng Ths unverse must have the structure of a comlete artally ordered set for the followng reasons (1) Partal orderng on observable henomena Suose that a software system s tested and an observaton s made y carryng out more tests on the same software system and use the same observaton scheme, we should be able to obtan more nformaton about the dynamc behavour of the system The fnal cumulatve observaton should contan more nformaton than the ntermedate observaton Therefore, there s an orderng between these two observatons However, observatons made durng two ndeendent tests are n general not ordered Therefore, all ossble observatons made durng testng a software system form a artal order (2) Exstence of a summaton oeraton summaton oeraton s needed to add u a number of ndvdual observatons made durng testng a software system The result of such a summaton s the least uer bound of the ndvdual observatons (3) Exstence of the least element No nformaton about the dynamc behavour of a system can be observed f the system s not executed To dentfy such a stuaton, the unverse of henomena s requred to contan the least element denotng no nformaton

3 ehavour Observaton n Software Testng 20/08/99 For examle, let an observaton of the dynamc behavour of a system be the set of executed statements durng testng, all such sets consttute a unverse of henomena and the artal orderng relaton on the unverse s the set ncluson The least element s the emty set The summaton oeraton s set unon efore we formally defne the noton of schemes and nvestgate ther roertes, we frst gve some mathematcal concets and notatons Readers are referred to [GS90] for detals about doman theory Let D be a non-emty set, be a bnary relaton on D, and S D be a subset of D Partal orderng: the bnary relaton s a artal orderng, f s reflexve, transtve and ant-symmetrc Uer bound: u s called an uer bound of S, f s u for all s S onsstent subset: S s a consstent subset f for all s 1, s 2 S, there s s D such that s 1 s and s 2 s We say that s 1 s consstent wth s 2 and wrte s 1 s 2 Drected subset: S s a drected subset, f for all s 1, s 2 S, there exsts s S such that s 1 s and s 2 s omlete artally ordered set (PO): <D, > s called a comlete artally ordered set, f (1) D has a least element, wrtten ; (2) for every drected subset S D, S has a least uer bound, wrtten as S, e for any other uer bound u, S u Let I be any ndex set, x be a varable that ranges over set X, We wrte x X Pred as a short hand for x 1 X 1 x 2 X 2 Pred x X Pred s defned smlarly bag(x) s used to denote the set of all multle sets on a set X The tradtonal set oeratons are used to denote ther multle set varants as well Y ' s the set obtaned by removng dulcated elements n a multle set Y Let ϕ be a mang from X to Y For all subsets X, we defne ϕ()={ϕ(a) a } We use the followng symbols n ths aer: and ts varants for a concurrent system, and P for the set of all concurrent systems; s and ts varants for a secfcaton, and S for the set of all secfcatons; D for the set of nut data for all concurrent systems, and D for the set of vald nuts of ; t and ts varants for a test case, e an nut datum for testng a concurrent system T and ts varants for a test set Generally, T s a multle set (or bag) so that multle executons on the same test case can be descrbed

4 ehavour Observaton n Software Testng 20/08/99 and ts varants for a henomenon observable durng testng a concurrent system, and Γ for a set of henomena We now formally defne a scheme as follows Defnton 1 (Observaton Scheme) scheme of behavour observaton and recordng, or smly an observaton scheme, s an ordered ar <Β, >, where ={<, > P}, and ={ P} s called the unverse of henomena For all concurrent systems P, <, > s a comlete artally ordered set s called the unverse of henomena on s called the recordng functon For all concurrent systems P, the recordng functon mas a test set T to a non-emty consstent subset of Informally, each element n s a henomenon observable from testng a concurrent system 1 2 means that henomenon 1 s a art of henomenon 2 The least element of denotes that nothng s observed (T) s the set of all ossble henomena observable by testng on test set T In other words, (T) means that s a henomenon that s observable by an executon of on test set T Examle 1 (Inut/Outut observaton scheme) Let IO = {<x,y> x D and y (x)}, where y (x) means that y s a ossble outut of concurrent system when executed on nut data x The unverse of observable henomena s defned to be the ower set of IO, and the artal orderng be set ncluson The recordng functon (T) s defned to be the collecton of sets of nut/outut ars observable from testng on T For nstance, assume that D = {0,1}, (1)= {1}, and (0) = {0, 1} due to non-determnsm Let test data t=0, and test set T 1 ={t}, then (T 1 )={{<0,0>}, {<0,1>}}, e one may observe ether {<0,0>} or {<0,1>} by executng on nut 0 once Let test set T 2 ={2t}, then (T 2 )={{<0,0>}, {<0,1>}, {<0,0>, <0,1>}}, e one of the followng three dfferent henomena can be observed by executng twce on the same nut 0: (1) {<0,0>} - outut 0 n two executons on nut 0; (2) {<0,1>} - outut 1 n two executons on nut 0; (3) {<0,0>, <0,1>} - outut 0 n one executon, and 1 n another executon on the same nut 0 Examle 2 (Dead mutant observaton scheme) onsder the observaton scheme for mutaton testng [DLS78, ud81, How82] Let Φ be a set of mutaton oeratons The alcaton of Φ to a rogram roduces a set of mutants of Let Φ() be the set of such mutants that are not equvalent to Defne the unverse of henomena to be the ower set of Φ() The artal orderng s defned to be the set ncluson relaton For all test sets T, the recordng functon ( T) s defned to be the collecton of sets of mutants Each element n ( T) s a set of mutants that can be klled by one testng of on

5 ehavour Observaton n Software Testng 20/08/99 T Examle 3 (Outut dversty observaton scheme) The observaton scheme n ths examle records the number of dfferent oututs on each nut data henomenon observable from testng a concurrent system on a set of test cases conssts of a set of records Each record has two arts <t, n>, where t s a vald nut, n s the number of dfferent oututs on the nut data observed from the testng Formally, an element of the unverse of henomena s a set n the form of {<t,n > t D, n >0, } The artal orderng relaton on henomena s defned as follows: ' < t, n > < t', n' > ' ( t = t' n n ' ) The least uer bound of and s, then, + = { < t, n > < t, n >= 1 2 % &K 'K < t,max( n, n ) >, f n, n > 0( < t, n > < t, n > ) < t, n >, f n > 0( < t, n > ) n > 0( < t, n > ) } < t, n >, f n > 0( < t, n > ) n > 0( < t, n > ) Proertes of Schemes The defnton of scheme rovded n the revous secton s stll nsuffcent to characterse the noton of consstent and systematc ways of behavour observaton For examle, gven a set of mutaton oerators, a set of mutants of a gven rogram can be generated by alyng the mutaton oerators to Suose that the henomena observed durng a mutaton testng of s the set of mutants stll alve after test Therefore, one mght defne the unverse of henomena to be the sets of lve mutants and the artal orderng on henomena to be the set ncluson relaton Our ntuton suggests that f an observaton 1 s made by a test executon, one should be able to have an observaton 2 such that 1 2 when carryng out more tests However, usng the above defnton t s just the ooste, because the more one tests a rogram, the less mutants may be alve, e 1 2 Therefore, we further requre that an observaton scheme satsfy addtonal roertes The frst roerty to be dscussed here s related to whether non-trval henomena can be observed from non-emty tests Defnton 2 (Observablty) n observaton scheme s sad to have observablty, f a concurrent system s tested on vald nuts, some henomenon of the system s behavour can always be observed, but nothng can be observed from a testng usng only nvald nut Formally, (1) P( T D (T ) ), and (2) P( T D = ( T ) = { } ) The followng are some roertes weaker than observablty ll of them state that the emty test set, whch means no testng has been done, s the weakest one n terms of the observable henomena

6 ehavour Observaton n Software Testng 20/08/99 Defnton 3 (Lower, Mddle and Uer Least Element) n observaton scheme s sad to have the lower least element roerty, f for all test set T, ( ) ' ( T)( ' ) It s sad to have the mddle least element roerty, f for all test sets T, (T) ( )( ) It s sad to have the uer least element roerty, f for all test set T, ( T) ' ( )( ' ) Lemma 1 (1) The mddle least element roerty mles the lower least element roerty (2) Observablty mles the lower least element roerty, the mddle least element and uer lower element roerty Proof Statement (1) s straghtforward by defnton Statement (2) can be roved by condton (2) of observablty nother roerty related to when a non-trval henomenon can be observed s the doman lmted roerty defned below Defnton 4 (Doman Lmted Proerty) n observaton scheme s sad to have doman lmted roerty, f only vald nuts effect the observaton Formally, P( ( T ) = ( T D ) ) The second grou of roertes about schemes s concerned wth the relatonsh between the henomena observable from testngs on dfferent test sets The consstency requres that a henomenon observable from one test set be consstent wth any henomenon observable from another test set f both are obtaned from testng the same rogram Defnton 5 (onsstency) set Γ of henomena s sad to be consstent wth a set Γ of henomena, wrtten as Γ Γ, f for all Γ and all Γ, n observaton scheme s sad to have consstency, f for all systems, all test sets T and T, the set of henomena observable from executng on T s consstent wth the set of henomena observable from executng on T Formally, P( (T) (T )) It should be notced that a concurrent system mght well roduce dfferent oututs on one test case n two test executons Whether such henomena are consdered as consstent deends on the defnton of the unverse of henomena For examle, the defnton of the nut/outut observaton scheme gven n Examle 1 consders such henomena as consstent However, they are consdered as nconsstent f the defnton of IO n Examle 1 s relaced wth IO' = { < t, f ( t) > t D} where f s a functon on D In fact, usng IO to defne the unverse of henomena excludes the ossblty of non-determnsm Hence, dfferent oututs on the same nut ndcate nconsstency roerty stronger than consstency s the comleteness roerty Defnton 6 (omleteness) n observaton scheme s sad to have comleteness, f every henomenon Ι observable

7 ehavour Observaton n Software Testng 20/08/99 from executng a concurrent system on a test set T Ι s contaned n the henomena observed from executng on T Formally, P( (T ) ( ( T ) ( ))) Lemma 2 Let T ={t } If an observaton scheme s comlete, then we have that P( ({t }) ( (T)( ))) Proof Let T ={t } for Ι n the defnton of comleteness Lemma 3 omleteness mles the mddle least element roerty That s, f an observaton scheme has comleteness, then for all test sets T, there s (T) such that, for all ( ), Proof For all test sets T, we have that T = T T, where T = The statement then follows the defnton of comleteness mmedately ( ) The thrd grou of roertes about schemes states the relatonsh between henomena observable from testng on a test set and ts subset Defnton 7 (Extendblty) n observaton scheme s sad to have extendblty, f every henomenon observable from executng a concurrent system on a test set T s a art of a henomenon observable from executng on a suerset T of T Formally, P( ( T ) T T ' ' ( T ' )( ' )) Lemma 4 Extendblty mles the lower least element roerty Proof Let T= n the defnton of extendblty Lemma 5 Extendblty mles consstency Proof Let T 1 and T 2 be any gven test sets Then, T = T 1 T 2 T 1 Smlarly, T T 2 y extendblty, for all 1 (T 1 ) there s (T) such that 1, and for all 2 (T 2 ) there s (T) such that 2 y the defnton of observaton schemes, (T) s a consstency set, therefore, there s * such that * and * y the transtvty of relaton, we have that * 1 and * 2 Hence, 1 2 The statement follows from the defnton of consstency Lemma 6 omleteness mles extendblty Proof ssume that T T' and (T ) Let T ' = T T" and " ( T" ) y the defnton of comleteness, + " ( T' ) The statement follows from the fact that + " Defnton 8 (Tractablty)

8 ehavour Observaton n Software Testng 20/08/99 n observaton scheme s sad to have tractablty, f every henomenon observable from executng a concurrent system on a test set T contans a henomenon observable from executng on a subset T of T Formally, P( ( T ) T T ' ' ( T ' )( ' )) Lemma 7 Tractablty mles the uer least element roerty, that s, for all test sets T, ( T) ' ( )( ') Proof Let be the T n the defnton of tractablty Defnton 9 (Reeatablty) n observaton scheme s sad to have reeatablty, f every henomenon observable from executng a concurrent system on a test set T can be observed from executng on the same test cases n the test set T for more tmes Formally, ' P( ( T ) T ' bag ( T ) T ' T ( T' ) ) The last grou of roertes about schemes states the relatonshs between the henomena observable from testng on a number of subsets and the henomena observable from the unon of the subsets These roertes are related to questons lke whether a testng task can be dvded nto a number of subtasks Defnton 10 (omosablty) n observaton scheme s sad to have comosablty, f the henomena observable by executng a concurrent system on a number of test sets T can be ut together to form a henomenon that s observable from executng on the unon of the test sets T Formally, P( (T ) ( ( T ))) omosablty means that the summaton oeraton s safe n the sense that the summaton of the henomena observable from a number of test sets s stll observable f the testng s not dvded nto subsets The decomosablty formally defned below states another knd of safety It means that dvdng a testng task nto small subtasks wll not loss the ossblty of observng a henomenon Unfortunately, some schemes that look sound do not have decomosablty as we wll see later Defnton 11 (Decomosablty) n observaton scheme s sad to have decomosablty, f for all test sets T, T = T mles that every henomenon observable from executng a concurrent system on the test set T can be decomosed nto the summaton of the henomena observable from executng on test sets T Formally, let T = T, P( (T) (T ) ( = ) )

9 ehavour Observaton n Software Testng 20/08/99 Lemma 8 (1) Decomosablty mles that P T={t }( (T) ({t }) ( = )), (*) (2) If an observaton scheme has comosablty, then (*) mles decomosablty Proof Statement (1) s obvous from the defnton of decomosablty To rove (2), let T = and T ={t,j j J } Let (T) y (*), we have that, j J,j ({t,j }) ( =, j, )) y comosablty, we have that for all, j ( T ),, Let = that = j J J j, j j J T Then, we have The followng lemma roves that comosablty and decomosablty are very strong roertes Lemma 9 (1) omosablty mles comleteness; (2) Decomosablty mles tractablty; (3) omosablty and decomosablty mly reeatablty Proof (1) The statement follows from the fact that for all, (2) ssume T T and ( T) Let T=T T y the defnton of decomosablty, 1 ( T' ) 2 ( T")( = ) The statement follows from the fact that ' (3) ssume that T={t }, T ' T and T ' bag( T ) Let (T ) y decomosablty, ' there are ({ t }) for all such that = Snce T ' T and T ' bag( T ), there s T* such that (a) T =T T*, and (b) for all t T*, there s such that t=t T Let t = y comosablty, + T 0 ' 5 The statement follows from the fact that + = + t t T * t T * t t t T *, and for all, + =

10 ehavour Observaton n Software Testng 20/08/99 The relatonshs between roertes of schemes are summarsed n Fgure 1 Uer least element roerty Mddle least element roerty Lower least element roerty onsstency Tractablty Decomosablty Observablty Reeatablty & Extendablty omleteness omosablty The followng theorem gves the structure of henomena observable from fnte test sets usng a scheme wth comosablty and decomosablty Theorem 1 (Structure Theorem) If a scheme =<, > has comosablty and decomosablty, we have (1) ({ nt}) = { + + ({ t}) 0 < + n}, for all n>0; k (2) ({ n1t 1, n2t2,, nktk}) = { + + = + + ({ t}) 0 < + n 1 k} = 1 Proof (1) y alyng nducton on n For n=1, the rght hand sde of equaton (1) s: { + ({ t}) + = 1} = { ({ t})} ({ t}) Therefore, the statement s true when n=1 + = ssume that the statement s true for n=n>0, e ({ Nt}) = { + + ({ t}) 0 < + N} (*) For n=n+1, by the defnton of comosablty, the followng equaton follows from (*) ({( N + 1) t}) { + ' ({ Nt}) ' ({ t})} Fgure 1 Relatonshs between roertes = { + + ({ t}) 0 < + N + 1}

11 ehavour Observaton n Software Testng 20/08/99 y the defnton of decomosablty, we have that N + 1 ({( N + 1) t}) { ({ t}) 1 N + 1} = 1 = { + + ({ t}) 0 < + N + 1} Therefore, the statement s also true for n=n+1 ccordng to the rncle of mathematcal nducton, the statement s true for all n>0 (2) The roof s smlar to the roof of statement (1) by alyng nducton on k Examle 4 (1) The nut/outut observaton scheme defned n Examle 1 satsfes all the roertes defned n defntons 2 ~ 11 (2) The dead mutant observaton scheme defned n Examle 2 has reeatablty, consstency, comleteness, comosablty, extendblty, tractablty, and decomosablty Its observablty and doman lmted roerty deends on the mutaton oeratons For some mutaton oeratons, a non-emty test set may kll no mutants It may also have no doman lmted roerty because an nvald nut for the orgnal rogram may be vald for a mutant, hence t klls the mutant (3) The outut dversty observaton scheme defned n Examle 3 has observablty, reeatablty, consstency, comleteness, comosablty, extendablty, tractablty and doman lmted roerty However, t does not have decomosablty Proof The roofs of the statements are straghtforward Detals are omtted for the sake of sace 4 Extracton Relaton etween Schemes Gven a henomenon observable from testng of a concurrent system, we can often extract nformaton from the observaton For examle, we can extract the set of executed statements from the set of executed aths Ths secton formally defnes such extracton relaton between schemes and studes ts roertes Let =<, > and =<, > be two schemes Defnton 12 (Extracton Relaton between Schemes) Scheme s an extracton of scheme, wrtten, f for all P, there s a homomorhsm ϕ from <,, > to <,, >, such that (1) ϕ ()= Α, f and only f = Β,, and (2) for all test sets T, ( T) = ϕ ( ( T)) Informally, scheme s an extracton of scheme means that scheme observes and records more detaled nformaton about dynamc behavours than scheme does The henomena that scheme observes can be extracted from the henomena that observes

12 ehavour Observaton n Software Testng 20/08/99 Defnton 13 (Equvalence Relaton on Schemes) Scheme and scheme are sad to be equvalent, wrtten ~, f there s an somorhsm ϕ from to such that for all test sets T, ϕ ( ( T)) = ( T) Obvously, we have that: Lemma 10 For all schemes and, ~ mles that, and Lemma 11 For all schemes, and, (a) ~; (b) ~ ~; (c) ~ ~ ~ It s also easy to see that the extracton relaton s a artal orderng on schemes when equvalent schemes are consdered as the same one Lemma 12 For all schemes,, and, we have that: (1) Reflexvty: ; (2) Transtvty: and mly that ; (3) nt-symmetry: and mly that ~ Proof Straghtforward, by the defntons The followng theorem roves that extracton mangs reserve the roertes of schemes dscussed n the revous secton Theorem 2 ssume that scheme =<, > s an extracton of scheme =<, > We have that: (1) f has observablty, so does ; (2) f has doman lmted roerty, so does ; (3) f has consstency, so does (4) f has comleteness, so does ; (5) f has extendblty, so does ; (6) f has tractablty, so does ; (7) f has reeatablty, so does ; (8) f has comosablty, so does ; (9) f has decomosablty, so does ; Proof Let ϕ, P, be the extracton mangs from to

13 ehavour Observaton n Software Testng 20/08/99 (1) Let T be any gven test set (a) ssume that T D y the observablty of,, ( T) Snce ( T) = ϕ ( ( T)), and ϕ ( ) =, =,, we have that, ( T) Therefore, T, ( T) D (b) Now assume that T D = y the observablty of, ( T) = { } y the defnton of extracton, ( T) = ϕ ( ( T)) =ϕ ({ Β, })={ Α, } That s, T D = ( T)={ Α, } Therefore, statement (1) s true (2) Let T be any gven test set We have that ( T) = ϕ ( ( T)) (by the defnton of extracton) = ϕ ( ( T D )) (by the doman lmted roerty of ) = ( T D ) (by the defnton of extracton) (3) Let T and T be any gven test sets Let ( T) and ' ( T ' ) y the defnton of extracton, there are ( T) and ' ( T' ) such that ϕ ( ) and, = ' = ϕ ( ' ) y the consstency of, we have that ( T) ( T' ) Hence, there exsts * such that * * and ' ecause ϕ s a homomorhsm, we have that * * ϕ ( ) ϕ ( ) and ϕ ( ) ϕ ( ' ) Therefore, ( T) ( T' ) (4) Let T I be a collecton of test sets and ( T ), I ecause ( T ) = ϕ ( ( T )), I, there are ( T ) such that = ϕ ( ) for I y the comleteness of, there exsts ( T ) such that for all I Snce ϕ s a homomorhsm, ϕ ( ) ϕ ( ) = y the defnton of extracton, ϕ ( ) ϕ ( ( T )) = ( ) T Therefore, s comlete (5) Let T, T be test sets and T T' Let ( T) y the defnton of extracton, there exsts ( T) such that ϕ ( ) y the extendblty of, there s ' ( T' ) such that = y the defnton of extracton, ϕ ( ' ) ( T' ) Snce ϕ s a ' homomorhsm, ϕ ( ' ) ϕ ( ) = Therefore, also has extendblty (6) Let T, T be test sets and T T' Let ( T) y the defnton of extracton, there exsts ( T) such that = ϕ ( ) y the tractablty of, there s ' ( T' ) such that ' Snce ϕ s a homomorhsm, ϕ ( ) ϕ ( ' ) Therefore, also has tractablty ' (7) Let T be any gven test set, and T bag( T ) and T T' Let ( T) y the defnton of extracton, there s ( T) such that ϕ ( ) y the defnton of =

14 ehavour Observaton n Software Testng 20/08/99 reeatablty, ( T' ) y the defnton of extracton, we have that ϕ ( ) ϕ ( ( T' )) = ( T' ) Therefore, also has reeatablty = (8) Let T I be a collecton of test sets and ( T ), I ecause ( T ) = ϕ ( ( T )), I, there are ( T ) such that = ϕ ( ) The comosablty of mles that ( T ) ecause ϕ s a homohormsm, ϕ ( ) = ϕ ( ) Therefore, ϕ ( ) ϕ ( ( T )) = ( T ) That s, also has comosablty (9) Let T I be a collecton of test sets Let ( T ) y the defnton of extracton, there s ( T ) such that = ϕ ( ) The decomosablty of mles that T there exst ( ) for I, such that = = Snce ϕ s a homomorhsm, we have that ϕ ( ) = ϕ ( ) Hence, ϕ ( ) y the defnton of extracton, T ϕ ( ) ( ) for all I Therefore, also has decomosablty 5 onstructons of Schemes Ths secton rovdes a number of constructons of observaton schemes and nvestgates ther roertes 51 Set onstructon In statement testng, software testers observe and record the subset of statements n the software source code that are executed, see eg [Mye79, e90] In ths observaton scheme, the executon of a statement s an atomc event to be observed, and the unverse of henomena conssts of all the sets of such events The artal orderng on henomena s just set ncluson Such a constructon of scheme s common to many testng methods The followng s a formal defnton of ths constructon Defnton 14 (Regular Set Scheme) Scheme =<, > s sad to be a regular set scheme (or smly regular scheme) wth base U P, f for all P, the elements n the PO <, > are subsets of U and the artal orderng s the set ncluson relaton Moreover, the followng condtons hold for the mang : (1) U = ( t}) ) t D (2) ( ) = { }, ({, (3) T D ( T),

15 ehavour Observaton n Software Testng 20/08/99 (4) ( T) = ( T D ), (5) ( T ) = { ( T ), I} Lemma 13 Let =<, > be an observaton scheme If s a regular scheme, then we have that (1) has observablty; (2) has doman lmted roerty; (3) has comosablty; (4) has decomosablty Proof (1) ecause s the least element n the unverse of henomena, condton (2) and (3) n the defnton of regular scheme mly the observablty; (2) ondton (4) s the doman lmted roerty; (3) omosablty and decomosablty follow from condton (5) mmedately Theorem 3 (Extracton Theorem for Regular Schemes) Let =<, Β > be a regular scheme Let =<, Α > ssume that for all P, there s a set U such that <, > s a PO on subsets of U wth set ncluson relaton If for all P, there s a surjecton f from U to U such that: (a) ( = { f ( x) x }), or n short =f ( ), and (b) for all test sets T, ( T) = < f( ) ( T), then we have that: (1) s a regular scheme wth base U, and (2) s an extracton of We say that s the regular scheme extracted from by the extracton mang f Proof (1) To rove statement (1), we frst rove that s a scheme, then rove that s a regular

16 ehavour Observaton n Software Testng 20/08/99 scheme wth base U (a) To rove that s a scheme, we only need to rove that for all test sets T, ( T) s consstent Ths s roved as follows Let and n ( T) Then, there s and n ( T) such that = f ( ) and ' = f ( ' ) The consstency of ( T) mles * that there s * such that * and ' y condton (a), we have that * * * f ( ) It s easy to see by set theory that f ( ) f ( ) and f ( ) f ( ' ) Therefore, ( T) s consstent (c) To rove that s a regular scheme wth base U, we check that satsfes condtons (1)~(5) n Defnton 14 It follows the defnton of and the regularty of Detals are omtted here for the sake of sace (2) It s easy to see that the mang ϕ ( ) = { f ( x) x } s a homomorhsm from <,, > to <,, > The statement follows drectly from the defntons 52 Partally Ordered Set onstructon Let X be a non-emty set and be a artal orderng on X subset S X s sad to be downward closed f for all x S, y x y S Let P Gven a artally ordered set (also called oset) <, >, we defne the unverse of henomena to be the set of downward closed subsets of The a bnary relaton, on henomena s defned as follows: 1, 2 x 1 y 2( x y) It s easy to rove that, s a artal orderng Moreover, f the oset <, > has a least element, the oset <,, > s a PO wth the least element { } The least uer bound of 1 and 2 s 1 2 Defnton 15 (Poset Scheme) n observaton scheme =<, > s sad to be a artally ordered set scheme (or oset scheme) wth base <, >, P, f ts unverse of henomena s defned as above and the recordng functon has the followng roertes: (1) ( ) ={{ }}, (2) T D { } ( T), (3) ( T) = ( T D ), (4) ( T ) = { ( T ), I} Lemma 14 oset scheme has observablty, doman lmted roerty, decomosablty, and comosablty

17 ehavour Observaton n Software Testng 20/08/99 Proof The observablty follows from condtons (1) and (2) mmedately The doman lmted roerty follows from condton (3) omosablty and decomosablty follow from the fact that the least uer bound of a drected subset of downward closed subsets s the unon of the subsets Detals are omtted from here Examle 5 (Observaton scheme for ath testng [How76, Mye79, e90]) Let be any gven rogram ath n s a sequence of statements n executed n order Let be the set of aths n, and the artal orderng be the sub-ath relaton Let s be a set of aths n The downward closure of s s the set of sub-aths covered by s, wrtten as s Let T be a test set Defne: T s T, ( ) = { s T, s a set of executon aths n that may be executed when testng on T} It s easy to see that the functon defned above satsfes the condtons (1)~(4) n the defnton of the oset scheme Therefore, by Lemma 14, t has observablty, doman lmted roerty, comosablty and decomosablty Smlar to Examle 5, we can defne observaton schemes that observe the sequences of a tye of events haened durng test executons of a system, such as the sequences of communcaton and synchronsaton events Such schemes have the same roerty as the scheme for ath testng 53 Product onstructon Gven two schemes and, we can defne a new scheme from them The followng defnes the roduct scheme of and Defnton 16 (Product onstructon) Let =<, > and =<, > The scheme =<, > s sad to be the roduct of and, wrtten =, f for all P, (1) =< { <, >, },, >, 2, 7 2, 7 2, 7; where <, > < ', ' > ' ' (2) for all test sets T, ( T) = ( T) ( T) Theorem 4 Let =<, > and =<, > be two schemes We have that: (1) f both and have observablty, so does ; (2) f both and have doman lmted roerty, so does ; (3) f both and have consstency, so does (4) f both and have comleteness, so does ;

18 ehavour Observaton n Software Testng 20/08/99 (5) f both and have extendblty, so does ; (6) f both and have tractablty, so does ; (7) f both and have reeatablty, so does ; (8) f both and have comosablty, so does ; (9) f both and have decomosablty, so does Proof Let scheme = (1) y the defnton of roduct scheme,, =<,,, > The followng roves that satsfes two condtons of observablty, resectvely (a) ssume that T D The observablty of and mles that, ( T) and, ( T) y the defnton of the roduct scheme, =<, > ( T) ( T) ;,,, (b) Now assume that T D = y the observablty of and, ( T)={ Α, } and ( T) = { } Therefore, ( T) ( T) = { <, > } = { }, (2) Let T be any gven test set Then,,,, ( T) = ( T) ( T) (by the defnton of the roduct scheme) = ( T D ) ( T D ) (by the doman lmted roerty of and ) = ( T D ) (by the defnton of roduct scheme) (3) Let T and T be any gven test sets Let ( T) and ' ( T' ) y the defnton of the roduct scheme, there are ( T), ' ( T' ), ( T) and ' ( T' ) such that =<, > and ' =< ', ' > y the consstency of and, ( T) ( T' ) and * ( T) ( T' ) Hence, there exsts such that * and * * ', and there s such that * and * * * * ' Let =<, > Then, * and * ' Therefore, ( T) ( T' ) (4) Let T I be a collecton of test sets and ( T ), I ecause ( T ) = ( T ) ( T ) for all I, there are ( T ) and ( T ) such that =<, > y the comleteness of and, there exsts ( T ) such that for all I, and there exsts ( T ) such that for all I Let =<, > Then, ( T ) ( T ) = ( T ) I Therefore, s comlete and for all

19 ehavour Observaton n Software Testng 20/08/99 (5) Let T, T be test sets and T T' Let ( T) y the defnton of the roduct T scheme, there exsts ( T) and ( T) such that =<, > y the ' extendblty of and, there s ' ( ' ) such that, and there s ' ( T' ) such that Let ' =< ', ' > y the defnton of the roduct ' scheme, ' = < ' ' > <, > = and ' ( T' ) Therefore, also has extendblty (6) Let T, T be test sets and T T' Let ( T) y the defnton of the roduct scheme, there exsts ( T) and ( T) such that =<, > y the tractablty of and, there s ' ( T' ) and ' ( T' ) such that ' and ' Let ' =< ', ' > y the defnton of the roduct scheme, ' ( T' ) and ' Therefore, also has tractablty ' (7) Let T be any gven test set, and T bag( T ) and T T' Let ( T) y the defnton of the roduct scheme, there are ( T) and ( T) such that =<, > y the defnton of reeatablty, ( T' ) and ( T' ) y the defnton of the roduct scheme, ( T' ) ( T' ) = ( T' ) Therefore, also has reeatablty (8) Let T I be a collecton of test sets and ( T ), I ecause ( T ) = ( T ) ( T ) for all I, there are ( T ) and ( T ) such that =<, > The comosablty of and mles that ( T ) and ( T ) Therefore, = <, > =<, > ( T ) ( T ) = ( T ) That s, also has comosablty (9) Let T I be a collecton of test sets Let ( T ) y the defnton of the roduct scheme, there are ( T ) and ( T ) such that =<, > The decomosablty of and mles that there exst ( T ) and ( T ), I, such that = and = ecause <, > =<, > and <, > ( ), also has decomosablty T Lemma 15 For all schemes, and, ( ) ~ ( ) Proof Straghtforward by the defnton Examle 6 (Tyed dead mutant observaton scheme) In Examle 2, an observaton scheme s defned for mutaton testng In software testng tools, mutaton oerators are often dvded nto a number of classes to generate dfferent tyes of mutants, see eg [KO91] Dead mutants of dfferent tyes are then recorded searately to rovde more detaled nformaton To defne the observaton scheme for ths, let Φ 1, Φ 2,,

20 ehavour Observaton n Software Testng 20/08/99 Φ n be sets of mutaton oerators For each Φ, =1, 2,, n, defne a dead mutant observaton scheme M as n Examle 2 Then, we defne the tyed dead mutant observaton scheme M Tyed = M 1 M 2 M n 54 Statstcal onstructons Let =<, Β > be an observaton scheme N be any gven set of numbers Then, <N, > s a totally ordered set under the less than or equal to relaton on numbers We can defne a scheme =<, Α > as follows Defnton 17 (Statstcal onstructon) scheme =<, Α > s sad to be a statstcal observaton scheme based on =<, Β >, f there exsts a set N of numbers and a collecton of mangs s P : N such that (1) For all P, =N, and, s the less than or equal to relaton on N; (2) For all P, the mang s from to the set N reserves the orders n, e, ' s ( ) s ( ' ); (3) For all test sets T, ( T) = { s ( ) ( T)} Examle 7 (Statement coverage) Let =(, Β ) be the regular scheme for statement testng Defne s () = / n, where n s the number of statements n rogram, s the sze of the set We thus defne a statstcal observaton scheme for statement coverage The henomena observed by the scheme are the ercentage of statements executed durng testng Examle 8 (Mutaton score) In mutaton testng, mutaton score s defned by the followng equaton and used as an adequacy degree of a test set [DLS78, ud81] numberofdeadmutants MutatonScore = numberofnon -equvalentmutants We defne the mutaton score as a statstcal observaton scheme based on the dead mutant observaton scheme wth the mang s ()= / m, where s the sze of the set and m s the number of non-equvalent mutants of generated by the set of mutaton oerators Notce that the statement coverage scheme defned above s not decomosable, although the observaton scheme for statement testng s regular, whch has decomosablty accordng to Lemma 13 Smlarly, the mutaton score scheme does not have decomosablty whle the dead mutaton scheme has decomosablty The examles show that the sace of statstcal nformaton observable from testng searately on several smaller test sets may be smaller than the sace observable from a large test set In software testng, statstcs can be also made on the henomena observed from testng on each test case The followng defnes the general constructon of such schemes Defnton 18 (ase-wse Statstcal onstructon)

21 ehavour Observaton n Software Testng 20/08/99 scheme =<, Α > s sad to be a case-wse statstcal observaton scheme based on =<, Β >, f there exsts a set N of numbers and a collecton of mangs s P : N such that (1) For all P, =D N, where D N s the set of artal functons from D to N, and, s defned by the followng equaton: 1, 2 t D ( 1( t) = undefned 1( t) 2( t)), where s the less than or equal to relaton on N; (2) For all P, the mang s from to the set N reserves the order n, e, ' s ( ) s ( ' ); (3) For all test sets T = { nt I} where j t t j, ( T) ff (a) I ({ n t })( ( t ) = s ( )), and (b) t T ( t) = undefned Notce that f the base scheme of a case-wse statstcal scheme has comosablty and decomosablty, the case-wse statstcal nformaton can be derved from the henomena observed by usng the base scheme However, the case-wse observaton scheme may have dfferent roertes from ts base scheme The followng s such an examle Examle 9 The outut dversty observaton scheme defned n Examle 3 s the case-wse statstcal observaton scheme based on the nut/outut observaton scheme wth the mang s beng the set sze functon 6 ehavour Observatons on Petr Nets In ths secton, we aly the theory of behavour observaton to Petr nets - a well-known model of concurrent and dstrbuted systems [Mur89] 61 asc Noton of Petr Nets In ths subsecton, we rovde an overvew of a class of hgh-level Petr nets called redcate transton nets (PrT nets n the sequel) [GL81] formal defnton of redcate transton nets can be found n [He96] PrT net s a tule (Nt, Sec, Ins), where (1) Nt = (Pl, Tr, Fl) s the net structure, n whch () Pl and Tr are non-emty fnte sets satsfyng Pl Tr = Pl and Tr are the sets of laces and transtons of Nt resectvely; () Fl (Pl Tr) (Tr Pl) s a flow relaton, called the arcs of Nt; (2) Sec s the underlyng secfcaton, whch defnes the tyes, tokens, labels, and constrants of Nt; (3) Ins = (ϕ, L, R, M 0 ) s a net nscrton that assocates a net element n Nt wth ts denotaton n Sec: () ϕ s a mang that assocates each lace l n Pl wth a vald tye defned n Sec, () L s a mang that mas each arc n Fl to a vald label defned n Sec, () R s a mang that assocates each transton tr n Tr wth a frst order logc formula defned n Sec, (v) M 0 s a set of ntal markngs Each ntal markng assgns a multle set of tokens

22 ehavour Observaton n Software Testng 20/08/99 defned n Sec to each lace l n Pl markng s a dstrbuton of tokens n laces transton s enabled f ts re-set laces contan enough tokens and ts constrant s satsfed wth an occurrence mode, whch s a substtuton of relevant label varables wth tokens n enabled transton can fre The frng of an enabled transton consumes the tokens n the re-set laces and roduces tokens n the ost-set laces We denote the frng of transton tr wth occurrence mode α n markng M by M transtons, ncludng the same transton wth two dfferent occurrence modes, are n conflct f the frng of one of them dsables the other Two transtons not n conflct can fre concurrently onflcts are resolved non-determnstcally n executon ste of a PrT net tr/ α M, where M s the resultng markng We call the ar tr/α a frng Two conssts of the smultaneous frngs of non-conflct enabled transtons, and s denoted by M TrO M, where TrO s a set of frngs n executon of a PrT net s a sequence of consecutve executon stes startng from an ntal markng The dynamc semantcs of a Petr net s the set of all ossble maxmal executon sequences startng from ntal markngs Notce that, frstly, we have used an nterleaved-set semantc model here and thus true concurrency can be studed lternatvely, other semantc models such as branch structure and artal order can be used [Re85] Secondly, we have consdered the dynamc semantcs of a PrT net from a set of ntal markngs, nstead of a sngle ntal markng The followng PrT net secfes the well-known dnng hlosohers roblem h Pcku Thnkng Eatng Putdown h ch1, ch2 <h,ch1,ch2> hostck <h,ch1,ch2> ch1,ch2 Fgure 2 PrT Net Secfcaton of Dnng Phlosohers Problem Fgure 2 shows two hlosoher states denoted by laces Thnkng and Eatng resectvely, two transtons Pcku and Putdown, and the avalable chostck state defned by lace hostck The net nscrton (ϕ, L, R, M 0 ) s as follows (1) Place Tyes: ϕ(thnkng) = ϕ(eatng) = 2 PHIL, ϕ(hostck) = 2 HOP, where tyes PHIL and HOP are nduced from ntegers and defned n Sec (2) rc Labels: L(Thnkng, Pcku) = h, and the rest are obvous from Fgure 2 (3) Transton onstrants: R(Pcku) = (h = ch1) (ch2 = h 1), R(Putdown) = true, where s modulus k addton (4) Intal Markng: M 0 = {m k k=2, 3,,}, where m k s defned as follows: m k (Thnkng) = {1, 2,, k}, m k (Eatng) =, m k (hostck) = {1, 2,, k} The above secfcaton allows concurrent executons such as multle non-conflctng (nonneghborng) hlosohers ckng u chostcks smultaneously, and some hlosohers ckng u chostcks whle others uttng down chostcks The constrants assocated wth transtons Pcku and Putdown also ensure that a hlosoher can only use two desgnated

23 ehavour Observaton n Software Testng 20/08/99 chostcks defned by the mlct adjacent relatonshs Table 1 below gves the detals of a artal executon of the PrT net n Fgure 2 Table 1 n Executon of the PrT Net n Fgure 2 Markngs m Frngs f Thnkng Eatng hostck Fred Transton Occurrence Mode {1,2,3,4,5} { } {1,2,3,4,5} Pcku {h 1, ch1 1, ch2 2} {2,3,4,5} {<1,1,2>} {3,4,5} Putdown {h 1, ch1 1, ch2 2} {1,2,3,4,5} { } {1,2,3,4,5} Pcku {h 2, ch1 2, ch2 3} {1,3,4,5} {<2,2,3>} {1,4,5} Pcku {h 4, ch1 4, ch2 5} {1, 3, 5} {<2,2,3>, <4,4,5>} {1} Putdown {h 2, ch1 2, ch2 3} {1, 2, 3, 5} {<4, 4, 5>} {1,2,3} Putdown {h 4, ch1 4, ch2 5} {1,2,3,4,5} { } {1,2,3,4,5} Pcku {h 5, ch1 5, ch2 1} {1,2,3,4} {<5,5,1>} {2,3,4} Pcku {h 3, ch1 3, ch2 4} {1,2,4} {<5,5,1>, <3,3,4>} {2} Putdown {h 3, ch1 3, ch2 4} {1,2,3,4} {<5,5,1>} {2,3,4} Putdown {h 5, ch1 5, ch2 1} {1,2,3,4,5} { } {1,2,3,4,5} 62 ehavour Observaton Schemes on Petr Nets Let be a PrT net and M 0 be the set of ntal markngs of Thus, M 0 can be vewed as the doman of vald nut for From the defnton of PrT nets, an executon of on a test case m 0 s a maxmum sequence of executon stes startng from m 0, and s denoted by TrO0 TrO1 TrOk : m0 m1 m2m k where m 0 M 0, m, =1,2,, are markngs such that each m s obtaned from m 1 by frngs TrO 1 For many concurrent systems, a maxmum sequence of markngs can be nfnte, e the executon does not termnate However, n software testng ractce, we cannot observe and record an nfnte executon wthn a fnte erod of tme Therefore, we sto executon manually and observe and record a artal executon We use n to denote the artal executon (or refx) consstng of the frst n executon stes of, e n TrO0 TrO1 TrOn : m m m m m We denote the set of all fnte artal n 1 n + + executons from an ntal markng m 0 of as Σ m, and Σ 0 as the set of fnte artal executons from the set of ntal markngs M 0, resectvely We call the frng sequence TrO 0 TrO 1 TrO n 1 extracted from a artal executon n a trace and denote t by Trace ( n ) We use Frng ( n ) to denote the set of frngs extracted from the trace Trace ( n ) Furthermore, we denote the transton sequence Tr 0 Tr 1 Tr n 1 extracted from a trace by drong all occurrence bndngs as Trans ( n ), where each Tr s a multle set It s worth notng that artal executons can have dfferent numbers of executon stes, whch are determned by the tester, hence form an addtonal dmenson of non-determnsm of testng Wthout the loss of generalty, we call both comlete executons and artal executons as test executons n the sequel In the followng sectons, we defne several concrete behavour observaton schemes based on the transton coverage crtera roosed n testng ER nets (a tye of hgh-level Petr nets) n [MP90] Sx transton coverage crtera roosed n [MP90] are: (1) Frng Sequence dequacy: an adequate testng must nclude all feasble frng sequences from the ntal markng set; (2) Frng dequacy: an adequate testng must cover all feasble frngs;

24 ehavour Observaton n Software Testng 20/08/99 (3) Transton Sequence dequacy: an adequate testng must cover all feasble transton sequences; (4) Transton dequacy: n an adequate testng, every feasble transton must be fred at least once; (5) N-Tmes dequacy: an N-Tmes adequate testng must cover those feasble transton sequences that contan any one transton no more than N tmes; (6) N-Notable dequacy: an N-Notable adequate testng must cover those feasble transton sequences that contan any one transton no more than N tmes and no more than one notable sub-sequence, where a sub-sequence s sad notable, f all sequences of the same length contanng t result n the same fnal markng In [MP90], nterleavng semantcs was used such that each executon ste nvolved only one frng of a transton, whch s a secal case of our executon ste that can have smultaneous frngs of multle transtons The subsumng relatonshs among the above adequacy crtera are shown n Fgure Defnton 19 (Frng Sequence Scheme) For all P, the unverse of observaton on under the frng sequence scheme FS > s defned as follows: (1) FS = 2 Trace, where Trace s the set of all traces from the set of ntal markngs, (2) The artal orderng FS on FS s the set ncluson; Ω = < FS, FS (3) For all m M0, ({ m}) = {{ } Tracem, }, where Trace,m s the set of all traces from the ntal markng m; FS FS (4) For all test sets T, ( T )= ( T) The doman Dof a PrT net s the set of ntal markngs M 0 From the defnton of traces, FS t s easy to see the followng roertes of () FS ( ) = { }, FS () T M0 ( T), FS FS () ( T) = ( T M0) Thus the Frng Sequence Scheme s regular 6 Fgure 3 The Subsumng Relatonshs FS

25 ehavour Observaton n Software Testng 20/08/99 Defnton 20 (Frng Sequence overage) Let E be a collecton of test executons of Petr net E s sad to satsfy the frng sequence coverage crteron ff Trace (E) = Trace The coverage measurement of E s defned by the formula FS(, E) = Trace ( E) Trace It s worth notng that most concurrent systems n ractcal use contan an nfnte number of frng sequences To satsfy such an adequacy crteron, we may need an nfnte amount of comutaton resource Defnton 21 (Frng Scheme) For all P, the unverse of observaton on under the frng scheme defned as follows (1) FT Frng = 2, where Frng s the set of all feasble frngs n, (2) The artal orderng (3) For all test sets T, FT on FT s the set ncluson; FT FS (T)= Frng( x) u ( T) x u Ω = < FT, FT FT > s It s easy to see that the Frng Scheme s an extracton of the Frng Sequence Scheme such that the orders between frngs are gnored The followng defnes the frng coverage crteron Defnton 22 (Frng overage rteron) Let β be an observaton under the frng scheme on Petr net durng a testng The testng s sad to be adequate accordng to the frng coverage crteron ff β = Frngs Moreover, for all β FT, the adequacy measurement s FT(, β) = β Frng For examle, the executon of the Petr net gven n Table 1 satsfes the frng coverage crteron ll the transtons (e Pcku and Putdown) and ossble occurrences have aeared Defnton 23 (Transton Sequence Scheme) For all P, the unverse of observaton on under the transton sequence scheme < TS, TS > s defned as follows TS Ω = (1) TS Trans = 2, where Trans s the set of transton sequences extracted from Trace by gnorng occurrence modes n executon stes; (2) The artal orderng TS on s the set ncluson; TS TS FT FT (3) For all test sets T, (T)= Trans( x) u ( T), where ( T) s the recordng x u functon defned n the Frng Sequence Scheme It s easy to see that the Transton Sequence Scheme s an extracton of the Frng Sequence Scheme such that the bndngs of occurrence modes to transtons are gnored