A Survey of Methods for Generating a Test Sequence for Conformance Testing of Finite State Machine

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1 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), A Suvey of Methods fo Geneting Test Sequence fo Confomnce Testing of Finite Stte Mchine Zin Eid King Adul Aziz Univesity Jeddh, Sudi Ai zineid@hotmil.com Hnn Ndeem Reem Al Gmdi King Adul Aziz Univesity King Adul Aziz Univesity Jeddh, Sudi Ai Jeddh, Sudi Ai hnnndeem@hotmil.com Reem382@hotmil.com Astct Finite stte mchines testing polems hs een studied due to its pplictions to confomnce testing of communiction potocol; we will focus on the pimy polems of confomnce testing (o fult detection polem) sed on knowing whee we e duing the epeiment, using sttus messge, eset nd distinguishing. Given specifiction mchine we hve its stte tnsition nd output functions in the fom of tnsition digm o "lck o" which we cn only oseve its input/output ehviou, nd we will design test to detemine whethe implementtion confoms to the specifiction y geneting nd using test clled checking. We discuss thee methods fo geneting checking : Sttus Messge nd Reset, Distinguishing Sequence Method nd Using Septing Sequences Insted of Sttus Mssges. Finlly, we will eview smll compison etween those methods out how we cn choose etween them.. Intoduction An FSM (finite stte mchine) is mchine of finite nume of sttes poduces outputs on stte tnsitions fte eceiving inputs. FSM e widely used to model systems in divese es including sequentil cicuits, pticul types of pogms (epesented y leicl nlysis, ptten mtching etc.), nd moe ecently communiction potocols. An FSM cn e descied y quintuple M = (Q,,, q,f) whee: Q is set of finite intenl sttes. is set of finite input lphet :QS Q is totl functions clled the tnsition function which epesent tht the mchine M is moving fom stte q to the stte (q,) when eceiving n input. q in Q is the stt stte, F Q; elements of F e clled ccept o finl sttes.[] An FSM cn e lso epesented y stte tnsition digm diected gph whose vetices coespond to the sttes of the mchine nd whose edges coespond to the stte tnsitions nd lelled with the input nd output of tht tnsition s shown in Fig.. Testing FSM polem hs een studied in diffeent es nd t vious times nd the e seemed to hve mostly died down until few yes go when the testing polem ws esuected nd is now eing studied new due to its pplictions to confomnce testing of communiction potocol.it is vey wide fields of testing hdwe nd softwe includes with n etensive litetue which we cnnot hope to include hee we will focus on the sic polems of confomnce testing (o fult detection polem). Figue. Tnsition digm of finite stte mchine Thee e two types of FSM (finite stte mchines): Mely nd Mooe mchines. The theoy of oth types is vey simil, we minly conside Mely mchines ecuse its model finite stte systems moe popely. And lso we hve two types of testing polems: IJCTA Mch-Apil 24 Aville online@ 435

2 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), in the fist one we hve mchine M with complete desciption nd we hve the tnsition digm of it ut we lck some infomtion out in which stte it is Polem: detemining the finl stte fte the test. Polem2: stte identifiction polem to identify the initil stte nd the test tht solve this polem is clled distinguishing. Polem3: stte veifiction polem which veify tht the mchine in the specified stte nd the test tht solve this polem is clled unique Input/output UIO..2. In the second type of testing the mchine M tht is eing tested is eithe lck o i.e. we don t know its stte digm (the tnsition nd output function) nd only we cn ecognize its I/O ehvio o we hve nd know the mchine tnsition digm, nd we hve to test whethe the implementtion confoms to the specifiction which clled the confomnce testing o fult detection polem nd the test tht solve this polem is clled checking. In ou ppe, we will discuss 3 methods fo constucting checking fo the confomnce testing polem. In section II, we will discuss the concept of confomnce testing polem, in section III we descie the Sttus Messge nd Reset messge, In section IV the Distinguishing Sequence Method is descied, nd in section V Using Septing Sequences Insted of Sttus Mssges including(w method, Wp method, UIO method, Distinguishing Sequence Method), In section VI we will descie the Using Identifying Sequences Insted of Distinguishing Sequences method. 2. Confomnce Testing Given n FSM M S which cts s specifiction nd fo which we know its tnsition digm, nd nothe finite stte mchine M I, which we cn only oseve its ehvio,; we wnt to test whethe M I coectly implements o confoms to M S. Confomnce is fomlly defined s equivlence o isomophism whee M I confoms to its specifiction M S if nd only if thei initil sttes e equivlent, i.e. they will poduce the sme output fo evey input, nd in genel two sttes s i nd s j e equivlent (iff) if nd only if fo evey input the mchine genete the sme output[2]. To pove this equivlence of M I nd M S we hve to look fo set of input s to pply it to the mchine M I to pove tht it is equl to its specifiction nd tht input is clled checking. We e inteested to find if M I fils to implement M S o not so the polem of confomnce testing is clled fult detection, The method of testing nd detection the fult in the M I is s follows:. Genete fom the mchine M S set of input s. 2. Applying ech input to M S, poduce the equied output s. 3. Ech pi of the input nd pospective output is test nd the set of ll tests is test suite. 4. Apply ech of the input to the mchine M I nd oseve the output. 5. Compe ctul output with the epected one, nd if they diffe, then the fult is detected [3]. This pocedue of testing s it pesented so f, cn only e used to show the pesence of ugs, ut neve to show thei sence[3]. Geneting checking (set of s tht conctented ct s unique checking ), is diffe fo thei cost to poduce test s, fo the totl size of the test suite (i.e. the totl length of the checking ), nd fo thei fult detection cpility. In fct, test suites should e the shot to e pplicle in pctice, nd it should cove the implementtion s much s possile nd detect s mny fults s possile, we will use diffeent methods to chieve these two opposite gols in which the min diffeence mong those methods e in the ssumptions they mke out the mchines M S nd M I. Some methods e vey efficient in poducing checking ut cn e esticted unde stong ssumptions while Othe methods poduce eponentilly long s, ut pefom the test unde moe genel ssumptions, we cn mke decision of choosing etween methods depending on the fcts we hve o ssumed out the two mchines M S nd M I, nd these ssumptions e of high impotnce ecuse testing mchine without ny ssumption is impossile nd we hve to intoduce some ssumptions out the mchines we wnt to veify nd these ssumptions is s follows:. M S is educed o miniml: ecuse equivlent mchines hve the sme I/O pefomnce, nd it is impossile to distinguish them y oseving the outputs; we cn minimize the mchine M S nd otin n equivlent educed mchine nd In tht miniml mchine thee e no equivlent sttes. Fo evey sttes s nd t, it should eist, clled septing which distinguish s fom t ecuse the set of outputs geneted fom pplying to s e diffeing. 2. M S is completely specified: oth of stte tnsition function δ nd output function λ e defined fo evey stte in Q nd evey input in. IJCTA Mch-Apil 24 Aville online@ 436

3 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), M S is stongly connected: which men evey stte is echle fom evey othe vi one o moe stte tnsitions. 4. M I does not chnge duing testing, in cse it hs sme inputs nd outputs s M S. which men M I cn ccept nd espond to ll input symols in. 5. Initil stte: the mchines M I is in its initil stte efoe we conduct confomnce test epeiment. If M I is not in its initil stte, we cn pply homing. 6. Sme nume of sttes: M I nd M S hs the sme nume of sttes. Depending on this ssumption the possile fults in M I could e one of two kinds: output fults, occus ecuse tnsition in the implementtion poduces wong output, nd tnsfe fults, in cse of implementtion goes to the wong stte. Fig. 2 shows two fulty implementtions of the specifiction mchine M S given in Fig.. 7. eset messge: fo ll s, δ(s,eset)=s nd the output function λ(s,eset)= which men tht the mchine hve pticul input eset tht fom ny stte s cuses tnsitions ends in the initil stte s with no output poduced. 8. sttus messge: if we lel the sttes s,s 2,...,s n, we ssume tht the sttus outputs the inde i when pplied to stte si, nd the mchines do not chnge stte. Fomlly si S, λ(s i,sttus)=i δ(s i,sttus)=s i so the sttus messge genetes n output tht uniquely identifies thei cuent stte. 9. set messge: when pticul set of inputs set(s j ) messge is eceived in the initil system stte, the mchines move to stte s j without poducing ny output. Fomlly fo ll t, δ(eset,set(t)) = t nd λ(s,set(t)) = [3][4]. The fist fou popeties listed ove e equiements nd without them confomnce test of the type to e discussed is not possile, the est of the ssumptions e convenient ut not essentil. Confomnce test hs mny methods tht genete the checking using those ssumptions nd some of these methods sed on knowing whee we e duing the epeiment using the pesent o sent of set, eset, nd sttus messges which we will discuss in ou ppe in net sections. Figue.2 two fulty implementtion of Ms 3. STATUSE MESSAGE AND RESET A sttus messge tells us the cuent stte of the mchine. Conceptully, we cn imgine tht thee is specil input sttus, nd upon eceiving this input, the mchine outputs its cuent stte nd stys thee. With sttus messge, the mchine is highly osevle t ny moment. We sy tht the sttus messge is elile if it is gunteed to wok elily in the implementtion mchine M i ; i.e., it outputs the cuent stte without chnging it. Suppose the sttus messge is elile. Then checking cn e esily otined y simply constucting coveing pth of the tnsition digm of the specifiction mchine M s nd pplying the sttus messge t ech stte visited. Becuse sttus messge checked ech stte, we veify whethe M i is simil to M s o not. Futhemoe; evey tnsition is tested ecuse its output is oseved eplicitly, nd its stt nd end stte e veified y thei sttus messges, nd If tht messge is not elile, then we still cn get checking y pplying the sttus messge twice in ow fo ech stte s i t some point duing the epeiment when the coveing pth visits s i ; we just need this doule ppliction of the sttus messge once fo ech stte. The doule ppliction of the sttus messge ensues tht it, woks popely fo evey stte. Fo emple, conside the specifiction mchine M s in Fig., stting t stte S. We hve coveing pth fom input =. Let s denote the sttus messge. If it is effective, then we chieve the checking "sssssss". If it is unelile, then we hve the "ssssssssss". We sy tht the mchine M s hve eset cpility if thee is n initil stte S nd n input symol tht tkes the mchine fom ny stte ck to S, i.e., M s (s i,) = S fo ll sttes s i ; We sy tht the eset is elile if it is gunteed to wok popely in the implementtion mchine M i, i.e. M i (s i, ) = S fo ll s i ; else it is unelile. We hve to know tht fo mchines with elile eset, thee is IJCTA Mch-Apil 24 Aville online@ 437

4 ISSN: polynomil time lgoithm fo constucting checking []. 4. distinguishing methods Distinguishing cn e used s unelile sttus messge ecuse it gives diffeent output fo ech stte; it is like sttus messge, ecept tht it moves the mchine to nothe stte when pplied. This method hs two phses to check whethe the implementtion mchine M I is equivlent to the specifiction Checking Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), τ(t,s2) τ(t2,s3) τ(t3,s) output implementtion mchine M I is simil to the specifiction mchine M S. Now it s the time fo estlishing evey stte tnsition in phse2: we wnt to check the tnsition fom stte s i to s j with input/output /o when the mchine is cuently in stte t k. We would fist tke the mchine fom t k to s i, pply input, oseve the output o, nd veify the ending stte s j. We cnnot simply use τ(t k, s i ) to tke the mchine to stte s i, since fults my lte the ending stte. Insted, we pply the following input : τ(t k, s i )τ(t i, s i )[]. Fist tnsfe is ssumed to tke the mchine to stte s i, which is veified y its esponse to, nd s hs een veified y (), τ(t i, s i ) definitely tkes the mchine to stte s i. We then stt testing the tnsition using input nd veify the ending stte y. Theefoe, the following tests fo tnsition fom s i to s j :τ(t k,s i )τ(t i,s i ).(2) mchine M S ; It fist uilds n input tht visits ech stte using tnsfe s insted of eset nd then pplies its distinguishing to test whethe M I is simil to M S. It then uilds n input to test ech tnsition to guntee tht M I confoms to M S. Definition(Tnsfe Sequence): A tnsfe τ(s i,s j ) is tht tkes the mchine fom stte s i to s j ; And since the mchine is stongly connected, so tht is lwys eists fo ny two sttes y the ssumption, Phse : y pplying the distinguishing stting fom stte s i nd let t i e the finl stte which denoted y t i = δ(s i,) nd τ(t i,s i +) is the tnsfe fom t i to s i +. The input tht checks the esponse Afte pefoming this, the mchine will e in stte t j. We epet the sme pocess fo ech stte tnsition nd otin checking. Note tht the checking length is polynomil in the size of the mchine M S nd the length of the distinguishing []. Emple: Fo mchine in Fig., the distinguishing is = nd the esponses fom stte s, s 2, nd s 3 e,, nd espectively. When we pply the distinguishing in sttes s, s 2, nd s 3 it will tke the mchine to t = s 2, t 2 = s 3 nd t 3 = s. And the tnsfe s cn epesented y τ(t,s 2 )=τ(t 2,s 3 )=τ(t 3,s )=. The () ecomes: of the mchine to the distinguishing in ech stte stting fom the initil stte s is: τ(t,s 2 ) τ(t 2,s 3 )...τ(t n,s ) () nd we cn epesent it s follows: Checking τ(t,s2) τ(t2,s3) τ(t3,s) output Stting in stte s, tkes the mchine to stte t nd then τ(t, s 2 ) tnsfes it to stte s 2 fo its esponse to. In the end the mchine esponds to τ(t n, s ). If it opetes coectly, it will e in stte s, nd this is veified y its esponse to the finl. Duing the test, we should ecognize n diffeent esponses to the distinguishing fom n diffeent sttes, nd this veifies tht the We cn see tht this input ends in stte t = s 2, The input s (2) cn e conctented to otin: tns. to test / S- S3 / S2- S2 / S3- S3 / S- S / S2- S3 / S- S2 IJCTA Mch-Apil 24 Aville online@ 438

5 ISSN: input τ(t,s 3) τ(t,s 2) τ(t2,s 3) τ(t3,s 2) τ(t,s 2) τ(t3,s ) end stte output The checking length is 27. Insted of using unique pesent distinguishing fo ll the sttes, we cn use n dptive distinguishing ; (ADS) which is decision tee specifies how to choose the net input dptively depending on the spotted output to identify the initil stte[4]. Figue.3 Adptive Distinguishing Sequence of mchine in figue. Fo ech stte s i, we emine the decision tee nd tke the input i fom the oot to the lef node s i []. Emple: An dptive distinguishing fo the mchine in Figue. is depicted in Figue.3, We pply the input nd if we get the output we know tht the mchine ws in the stte s 2. If we get the output, we should pply, nd if we get the output then we know tht the mchine ws in s othewise we get nd the mchine wee in s 3. We hve =, 2 =, 3 =, nd τ = y using dptive distinguishing (ADS) in ou emple, nd the () ecomes input sequenc e Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), τ(t,s3 ) 2 τ(t,s3 ) 3 τ(t,s3 ) Length nd Cost of ADV: it hs O(n 2 ) length with tnsfe cnnot e longe thn n. The () is long O(n 3 ). Becuse thee e pn tnsitions, nd evey (2) hs the length O(n 2 ), nd the cost is gin O(pn 3 ) to find the complete checking. The dvnce of this method is tht it does not need eset messge. 5. Using Septing Sequences Insted of Sttus Mssges Unfotuntely, sttus messge is seldom ville. In the net section, we pesent how not to ely on sttus messge to detemine the cuent stte duing the test. We ssume now tht the mchines hve no sttus messge (ut they still hve eset messge), nd we wish to test whethe M S is equivlent to M I only oseving the etenl ehviou. In the following, we pesent some methods tht cn e unified s poposed y Lee nd Ynnkkis [2]. All these methods she the sme technique nd used to identify the stte y eplcing the use of the sttus messge with mny kinds of s clled in genelly the septing s [2] used to identify the stte which they pplied to. Rememe tht y ssuming M S is miniml, it fo sue does not contin two equivlent sttes,in nothe wod, fo evey pi s i, s j thee must eist n input tht we cll septing, nd tht input will distinguish them ecuse it will poduce diffeent outputs, i.e. λ(s i, ) λ(s j, ). 5.. W Method: This method use specific septing s tht is clled chcteizing set nd nothe set clled tnsition cove set which visit ech tnsition in the mchine. Definition 2. tnsition cove set of M S (o P set fo shot) is set P of input s such tht fo ech stte s S nd ech input I thee eist n input P stting fom s the initil stte nd ending y pplying to stte s. Fomlly s S nd I y P : = y. nd δ(s, y) = s. [3][4]. A P set foces the mchine to pefom evey tnsition nd then stop. A P set cn e uilt y using stndd edth-fist visit of the tnsition digm of the mchine M S. Note tht P set is closed unde the opetion of selecting pefi: if elongs to P, then ny pefi of is in P too. One wy of constucting P[3] is to uild fist testing tee T of M S s eplined in Algoithm nd then get the input geneted fom the ptil pths of T. Whee ptil pth is of consecutive nches, stting fom the oot nd ending in teminl o non-teminl node. Becuse the nches in T e leled y n input symol, so the geneted fom ptil pth q is the input symols on q. In ddition to tht, the empty input is pt of ny P set. Note tht IJCTA Mch-Apil 24 Aville online@ 439

6 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), Algoithm temintes ecuse the nume of sttes is finite. Algoithm ( Building test tee). Denote the oot of the tee with the initil stte s,and this is the level 2. Becuse we ledy uilt the tee up to level k, so we hve to continue with the k + st level. ) With ech node t t level k fom left to ight ) if the node t is equl to nothe one t level j, with j k, then t is teminted nd consideed lef of T c) othewise, fo evey input, if the mchine moves fom stte s i to s j, we ttch nch with lel nd node with lel s j Emple: A test tee fo M S of Fig. is shown in Fig. 4. Fom this test tee we otin P = {,,,,,, }. Figue 4. A test tee fo MS of Figue To test evey tnsition of M I the W method uses P set nd uses nothe set insted of the sttus messge, clled chcteizing set o W set, which veify tht we hve the one epected end stte of ech tnsition. Definition 3. chcteizing set is set W of input s such tht fo evey pi of distinct sttes s nd t in S, thee eists n input in W such tht λ(s, ) λ(t, ) [4]. The chcteizing set is iefly clled W set o sometimes septing set. The input s in the W set e lso clled septing s. A W set eists fo evey mchine tht is miniml (Assumption ). The choice of W set is not unique, nd the fewe e the elements in the W set, the longe e the septing s in the W set. An lgoithm fo uilding W set s follows: Ptition the set of sttes S into locks B i with i =,...,. Initilly W is, B = S nd =. Until evey B i is singleton, tke two distinct sttes s nd t in B i (tht includes t lest two sttes) nd uild thei septing. Add to W nd ptition the sttes s ik in evey B j into smlle locks B j,..., B jh sed on thei diffeent output λ(s ik, ). Repet the pocess until ech B i ecomes singleton nd ecomes n. Fo evey pi of sttes s i nd s j, the esulting W set includes n input tht septes s i fom s j. Note tht thee e no moe thn n ptition, nd theefoe W set hs no moe thn n septing s. The W method consists in using the entie W set insted of the sttus messge to test tht the end stte of ech tnsition is the epected one.we hve to ce out W set, ecuse it my contin mny s, we hve to visit fo sevel times the sme end stte of evey tnsition to pply ll the septing s in W nd fo this gol we cn use eset messge nd the s in P set. The set of input is simply otined conctenting evey input in the P set with evey input of W then pply them in ode fte eset messge to tke the mchine ck to the initil stte. In this wy, ech input p ij is the conctention of the i th of P set(to test the i th tnsition) with the j th of W set, with n initil eset input. Fo emple nd fomlly if we hve two sets of input X nd Y, so we cn denote with X.Y the set of input s tht geneted y conctenting ll the input of X with ll the input of Y. The input s poduced y the W method is equl to {eset}.p.w. If we do not oseve ny fult, the implementtion is poved to e coect [3][4]. By pplying the of P we cn simply detecte fulty-output, while ny tnsfe fult is detected y the ppliction of W. Emple[4]: the chcteizing set W of the mchine in Fig. is {,}: Fo stte s, tnsitions / /, Fo stte s 2, tnsitions / /, Fo stte s 3, tnsitions / /. A distinguishes s fom s 2 nd s 3 fom s 2. distinguishes s fom s 3. P = {,,,,,, } the set of test s P.W is epoted in the following tle, whee we indicte with the eset messge.[4] P Tle. test P.W indicte with the eset messge IJCTA Mch-Apil 24 Aville online@ 44

7 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), PW Tn-to-test output s / S-S / S- S2 / S2-S2 / S2-S3 / S3-S3 / S3-S nd s j gve the sme output fo ech in W. Mchine M I is simil to M S if fo evey stte s i of M S, the mchine M I hs simil stte s s i, hee we hve n sttes (Assumption 6) then thee eists one-to-one coespondence etween simil sttes of the two mchines M S nd M I. If the fult is uncove y using the input withen the fist phse, we cn pct tht evey stte in M S hs simil stte in the implementtion, nd fo sue M I nd M S e simil. But this is not enough to confim tht thy e equivlent. The poof of the equivlence will e withen phse2. The totl length of the checking is 52. The fult in mchine M s of Fig. 2 is detected y the input, while the tnsfe fults in mchine M s2 e detected y the input tht tests the end stte of the tnsition, fo emple tnsition fom s with input eoneously moves the mchine to s 2 is detected y the input nd Wp Method W o Wp method [4] hs the min dvntge of educing the length of the test suite with espect to the W method. This is the fist method we pesent tht splits the confomnce test in two phses. Duing the fist phse we test tht evey stte defined in M S lso eists in M I, while duing the second phse we hve to check ll tnsitions (not ledy checked duing the fist phse) e coectly implemented. Fo the fist phse, the Wp dosen't use tnsition cove set insted of tht it use stte cove set. Definition 4.[5] stte cove set is set Q of input s such tht fo ech s S, thee eists n input Q tht tkes the mchine to s, i.e. δ(s, ) = s. Using Q set, we cn tke the mchine to evey stte. A Q set cn e geneted y edth-fist of the tnsition gph of M S. On the second phse, the Wp use n identifiction set W i fo stte s i insted of W set (the unique chcteizing set) fo ll sttes whee W i is suset of W. Definition 5.[5] n identifiction set of stte s i is set W i of input s such tht fo ech stte sj in S (with i j) thee eists n input of W i such tht λ(s i, ) λ(s j,) nd n suset of W i hs this popety. Note tht the union of ll the identifiction sets W i is chcteizing set W. Wp Method Phse The input s fo phse one consist in the conctention of Q set with chcteizing set (W set) fte eset. Fomlly, the set of input s is {eset}.q.w. In this wy, evey stte is checked in the implementtion with W set. We sy tht stte q i in M I is simil to stte s i if it poduces the sme outputs on ll the s in W set. A stte q i in M I cn e simil to t most one stte of M S, ecuse if we suppose tht q i is simil to sttes s i nd s j then s i Phse 2: The second phse tests ll the tnsitions. To this im, Wp method uses the identifiction sets. Fo evey tnsition fom stte s j to stte s i on input, we pply (fte eset) which let the mchine go to the stte s j long tnsitions ledy veified, then we pply the input, which let the mchine go to s i, nd then pply one identifiction of W i. finly, epet this test fo evey identifiction in W i, nd if these tests do not uncove ny fult, we hve veified tht the tnsition in the mchine M I fom stte tht is simil to s j on input poduces the ight output (thee is no fulty-output) nd moves to simil stte to s i (unde no tnsfe fult). By pplying these tests to evey tnsition, we cn pove tht M I confoms to its specifiction. The set of input tht coves evey tnsition (nd tht is closed unde the opetion of selecting pefi) is P set. Theefoe, the input of phse2 contin the s of P ending in stte s i tht e not contined in the Q set used y phse, conctented with ll s contined in the identifiction set W i. Fomlly if R = P Q nd i in R ends in s i, the pplied s duing the second phse is {eset}.r. W i. A complete foml poof of coectness fo the Wp method is given in the ppe[4]. Emple.[5] Fig. mchine hs stte cove set Q={,, }. Duing the fist phse we genete the following test s: stte to test Q.Q.W output 2 3 Duing the second phse, we fist compute the identifiction sets. W = {,} ll the s in W e needed to identify s, W 2 = {} distinguishes the stte s 2 fom ll othe sttes, W 3 = {} distinguishes the stte s 3 fom ll othe sttes, R = P Q ={,,, } IJCTA Mch-Apil 24 Aville online@ 44

8 ISSN: R stt stte.r.wi output end stte Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), Hee we hve totl length of the checking is 44 (note tht Wp method yields smlle test suite thn the W method). The output fult in mchine M I of Fig. 2 is detected duing the fist phse gin y the input. Some tnsfe fults in M I2 cn e detected on the fist phse, while othes, like the tnsfe fult fom stte s 3 with input is detected only y the input s duing phse UIO Methods If W i set contins only one, this is clled stte signtue [2] o unique input/output (UIO) [5], tht is unique fo the stte s i. By pplying UIO, we cn distinguish stte s i fom ny othe stte, ecuse the output poduced pplying UIO is specific to s i. In this wy, UIO cn detemine the stte of the mchine efoe its ppliction. A UIO hs the opposite ole of homing o synchonizing ; it identifies the fist stte in the insted of the lst one. Note tht not evey stte of the FSM hs UIOs. If UIO eists fo evey stte s i, then UIOs cn e used to identify ech stte in the mchine; in this cse the UIO cts s sttus messge, ecept it moves the mchine to nothe stte. The oiginl UIO method [5] uilds set of input s tht visit evey tnsition fom s i to s j y pplying tnsition cove set P nd then check the end stte s j y pplying its UIO. In this cse, UIO is used insted of sttus messge. Although used in pctice, the UIO method does not guntee to detect evey fult in the implementtion [6] ecuse the uniqueness of the UIO s my not hold in fulty implementtion which my contin stte s' with sme UIO s nothe stte s, nd fulty tnsition ending in s' insted of s my e tested s coect. Note tht, fo this eson, the Wp method uses the W i sets only in the second phse, while, in phse it pplies the complete W insted. A modified vesion of the UIO method, clled UIOv, genetes coect checking s [6]. The UIOv method uilds the test suite in thee phses: () Uv pocess: fo evey stte s in M S pply n input tht egins with eset nd eches s nd then pply the UIO of s. To ech ech stte 2 use Q set. The set of input s consist of (2) Uv pocess: visit evey stte s nd pplies the input pt of the UIO s of ll othe sttes nd check tht the otined output diffes fom the output pt of the UIO pplied. Skip UIO s tht hve the input pt equl to pefi α of the input pt of the UIO of s. Indeed, in this cse, we hve ledy pplied α duing the Uv pocess, nd we know tht the output diffes ecuse two sttes cnnot hve the sme input nd output pt of thei UIO s. At the end of Uv nd Uv pocess we hve veified tht M I is simil to M S. (3) Tnsition test phse: check tht evey tnsition not ledy veified in nd 2 poduces the ight output nd ends in the ight stte y pplying its UIO.[5] Note tht the UIOv method cn e consideed s specil cse of Wp method, whee the W set is the union of ll the UIO s nd phse of the Wp method includes oth Uv pocess nd Uv pocess nd phse 2 is the tnsition test phse. Emple. Fo the mchine in Fig. the UIO s e: UIO = / distinguishes the stte s fom ll othe sttes, UIO2 = / distinguishes the stte s 2 fom ll othe sttes, UIO3 = / distinguishes the stte s 3 fom ll othe sttes. Uv pocess Q stte to test 2 3.Q.UIO output 2. Uv pocess stte to 2 3 test.q. UIO output 3. Tnsition test phse: tnsition to test input / S- S / S2- S2 / S3-S / S3- S3 output The output fult of mchine M I of Fig, 2 is detected duing the Uv pocess gin y the input. Some tnsfe fults in mchine M I2 e detected duing the fist phses, while othes, like the tnsfe fult fom stte s 3 with input is detected only y the input s duing the tnsition test phse. IJCTA Mch-Apil 24 Aville online@ 442

9 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), Distinguishing Sequence Method In cse we cn find one tht cn e used s UIO fo evey stte, we cll such distinguishing (DS). In this sitution, we cn pply the DS method using the eset messge [7]. Note tht this DS method cn e viewed s pticul cse of the W method when the chcteizing set W contins only peset distinguishing. The test s e simply otined comining P set with. Emple:Fo the mchine in Fig. we cn tke the = s peset distinguishing. In fct λ M S (s, ) =, λ M S (s 2, ) =, nd λ M S (s 3, ) = P.P. Figue. 5. Using two septing s to identify the stte As usul, we fist check tht M I is simil to M S. We disply fo ech stte s i the esponses to ll the septing s in chcteizing set W (Definition 2). Suppose tht W hs two septing s w nd w 2. We wnt to pply the steps shown (in sque oes) in Fig. 5 () : tke M I to s i, pply w (step ), tke the mchine ck gin to s i (step 2) nd then pply w 2 (step 3). If we oseve the ight output, we cn sy tht the mchine M I hs stte q i simil to s i. We cn stt fom i = nd poceed to veify ll the sttes without using neithe eset no distinguishing. Ou polem Cost nd Length: All the methods pesented in Section 3 she the sme considetions out the length of the checking nd the cost of poducing it. Fo the W method, the cost to compute W set is O(pn2) nd W set contins no moe thn n s of length no moe thn n. The cost to uild tee T set using the Algoithm is O(pn), nd its mimum level is n. The genetion of P set, y visiting T, tkes time O(pn2) nd poduces up to pn s with mimum length n. Since we hve to conctente ech tnsition fom in P set with ech tnsition in W set, we otin up to pn2 s of length n + n, fo totl length of O(pn3) nd totl cost of O(pn3). The Wp method hs the sme totl cost O(pn3) nd sme length O(pn3). Epeimentl esults [4] show tht checking s poduced y the Wp method e genelly shote thn the checking s poduced y the W method. The UIO method nd the method using peset distinguishing e moe epensive, ecuse detemining if stte hs UIO s o peset distinguishing ws poved to e PSPACE hd[8]. Thee e specifiction mchines with eset messge tht equie checking s of length Ω(pn3) [9]. 6. Using Identifying Sequences Insted of Distinguishing Sequences Not evey finite stte mchine hs distinguishing s. In cse the mchine hs no eset messge, no sttus messge, no UIO s, nd no distinguishing s, we cnnot pply the methods poposed so f. We still cn use Assumption nd utilize the eistence of septing s which cn distinguishing the sttes. So confomnce testing is still possile [], nd the esulting checking s could e eponentilly long. tns. to test output s / S- S / S- S2 / S2- S2 / S2- S3 hee pesented ecuse we do not know how to ing M I ck to s i in veifile wy, ecuse in fulty mchine, s shown in Fig. 5 (), tnsfe τ(t i, s i ) (step 2) cn mke the mchine moves to nothe stte s' i whee we cn look fo the epected output pplying the w 2, without needing fo veifying tht s' i is s i nd without le to pply gin w. We use now the Assumption 6, nmely tht M I with only n sttes. Let to e the input nd n is n intege, is the conctention n times of. Theoem. Let s e stte of M I, is n input, o the epected output poduced pplying to s, i.e. o = λ(s, ), τ tnsfe fom t = δ(s, ) ck to s, nd o' the epected output poduced pplying τ to t. By pplying the input ( τ to stte s in M I, if we oseve the output (o o', then the mchine ends in stte whee pplying gin we oseve the sme output o. Figue 6. Applying n times nd τ / S3-S3 / S3-S Poof. The scenio of the theoem is shown in Fig. 6. Suppose tht M I is initilly in stte s. Applying τ the mchine should come ck to s. Howeve, due to some fults, the mchine M I my go to nothe stte q even if the output we oseve is the one epected, i.e. o o'. Assume tht pplying n times τ, we oseve evey time the sme output o o'. Let q e the stte of M I fte the ppliction of ( τ). IJCTA Mch-Apil 24 Aville online@ 443

10 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), Note tht even if the n pplictions of τ poduce n times the sme coect output o o', we e not sue tht s, q,..., q n e the sme stte yet. Howeve the n+ sttes s, q,..., q n cnnot e ll distinct, ecuse M I hs n sttes. Hence q n is equl to some q with < n nd, theefoe, it would poduce the sme output o if we pply [8]. Emple: Conside the mchine in Fig. nd tke ny lleged implementtion M I. Apply the input (in this cse τ = ) to the initil stte s of M I nd check tht the output is. We e not sue tht M I is now in stte s, s well. We cn pply gin nd oseve the output nd so on. When we hve pplied nd oseved the output, M I my hve tvesed sttes s, q, q 2, nd the finl stte q3. Ou mchine M I hs only thee sttes, so fo sue q3 is equl to one of s, q, o q 2 nd cetinly if we pply we should oseve. We use Theoem s follows. Assume tht M S hs the chcteizing set W = { w, w 2 } nd let s i e the stte we e going to veify. Let τ e the tnsfe tht tkes M S ck to s i fom t i = δ(w, s i ). We fist pply (wτ) n to s i. If we oseve wong output we hve poved tht M I does not confom to M S. Othewise we cn pply theoem with = w nd we e sue tht M I ends in stte tht would poduce the sme output s if we pplied w. We pply w 2 insted. If we oseve the specified output we cn conclude tht s i hs simil stte in M I. We cn genelize this method when the chcteizing set W contins m septing s. Suppose tht the chcteizing set is W = { w,..., w m }. Let τj e the tnsfe tht tkes the mchine ck to s fte the ppliction of wj, i.e. τj = τ(δ(s, wj ), s). We cn define inductively the s β s follows: identifying then moving to the net stte. Let Ii denote the identifying of stte s i nd τ(t i, s i+ ) to e the tnsfe fom ti = δ(s i, Ii) to s i+, y pplying the following we cn emphsize the simility etween M I nd M S. I τ(t, s 2 ) I2τ(t2, s 3 )... I (2) Once we hve poved tht M I is simil to M S we hve to veify the tnsitions. To do this, we cn use ny Ii s elile eset. Fo emple, we cn tke I s eset to the stte t = δi (s, w m ) nd use t s the initil stte to test evey tnsition. Indeed, we e sue tht if we do not oseve ny fult, I tkes the mchine to t. If we wnt to eset the mchine fom the stte s k to t we pply τ(s k, s )I nd even if τ(s k, s ) fils to tke the mchine to s, we e sue tht I will tke it to t. Now we poceed s eplined in Section VI. To test tnsition etween s i nd s j we cn pply eset I to t, then tnsfe to s i, then we pply the input, nd get the output, finlly pply the identifying Ij to mke sue of the end stte is s j. Emple. Conside the mchine M S in Fig.. W = {, }. Fo s, τ =, I = (w τ) 3 w 2 =, Fo s, τ =, I2 = (w τ) 3 w 2 = nd Fo s, τ =, I3 = (w τ) 3 w 2 =. The (2) ecomes: input sequen ce I τ(t,s 2) I2 τ(t2,s 3) I3 τ(t3, s) I β = w β = (β τ ) n w () By induction, one cn pove tht pplying β fte pplying (β τ ) n would poduce the sme output. Consideing how βi e defined, this mens tht pplying w,..., w - would poduce the sme output. Fo this eson we pply w fte (β τ ) n. Theefoe, one cn pove tht βm is n identifying of s i, in the following sense: if the implementtion mchine M I pplying βm poduces the sme output s tht poduced y the specifiction mchine stting fom s i, then M I hs stte tht is simil to s i nd such stte is the stte ight efoe the ppliction of the lst w m (egdless of which stte M I stted fom). We indicte the identifying fo stte s i with Ii. Once we hve computed the identifying fo evey stte, we cn pply method simil to tht eplined in fome section to visit ech stte, to confim the esponse of the mchine to the Length nd Cost: Identifying lengths gows eponentilly with the nume of septing s nd with n the nume of the sttes. Indeed, y eqution 2, evey βi is n times longe thn βi, the identifying I is equl to βm, the nume of septing s is epesented y m, nd tht cn e up to n. The esulting checking is eponentilly long. We cn optimize the IS method y using diffeent septing fmily Z i fo evey stte s i [2][4]. RESULTS The W, Wp, UIOv, nd DS methods detect fults of ny kind, while the UIO method my miss some fults. Methods W, Wp, nd DS with n dptive distinguishing poduce minimum cost while the othes hve gete cost. IJCTA Mch-Apil 24 Aville online@ 444

11 ISSN: If the sttus messge is not ville (which is mostly occus in softwe lck o testing), then we should use some et inputs to veify the sttes nd tht inputs should e unique, like in Wp, UIO nd DS. Asent messge Sttus messge Reset messge Not even Distinguishin g No UIOs Tle2. summey fo choosing est method Futue wok Pesent messge Reset messge Distinguishin g Method to use W, Wp, UIO(Unique Input Output), DS (Distinguishin g ) DS method uses tnsfe insted of eset Identifying IS (not coveed) checking Length nd cost Length O(pn3) Cost O(pn3) (the minimum Cost) Length O(pn3) Cost O(pn3) when used in conjunction with dptive distinguishin g It poduces eponentill y long checking The min wekness of some methods is tht it needs the set messge, which my not e ville. To void the use of set nd to possily shoten the test suite, we cn genete tht tveses the mchine nd thow evey stte nd evey tnsition t lest once without needing to estt nd get ck to the initil stte fte ech test nd without using set messge. Such is clled tnsition tou[3] which cn e used nd comped with ou eplined methods. Conclusions Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), Two mchines M s nd M i e equivlent if they will poduce the sme output fo evey input, we looking fo checking tht pove tht the two mchines e equl. Thee e mny methods fo geneting checking nd some of them equie only tht ll sttes e echle fom the initil one, pemit the mchines with dedlocks o sttes without ny eiting tnsition. Anywise, these methods must equie eset messge tht tke ck the mchine to its initil stte, futhemoe dedlock my stop the test epeiment. Ou methods depend on the pesent o sent of set, eset, nd sttus messges s i [2][3]. Refeences [] Lee, D.; Ynnkkis, Mihlis, "Pinciples nd methods of testing finite stte mchines- suvey," Poceedings of the IEEE, vol.84, no.8, pp.9,23, Aug 996 doi:.9/ [2] Lee, D.; Ynnkkis, Mihlis, "Testing finitestte mchines: stte identifiction nd veifiction," Computes, IEEE Tnsctions on, vol.43, no.3, pp.36,32, M 994 doi:.9/ [3] Chow, T.S., "Testing Softwe Design Modeled y Finite-Stte Mchines," Softwe Engineeing, IEEE Tnsctions on, vol.se-4, no.3, pp.78,87, My 978 doi:.9/tse [4] Fujiw, S.; v.bochmnn, G.; Khendek, F.; Amlou, M.; Ghedmsi, A., "Test selection sed on finite stte models," Softwe Engineeing, IEEE Tnsctions on, vol.7, no.6, pp.59,63, Jun 99 doi:.9/ [5] Kishn Snni nd Anton Dhu A new technique fo geneting potocol test. SIGCOMM Comput. Commun. Rev. 5, 4 (Septeme 985), [6] Yu, S.S.; Liu, M.T., "A new potocol test genetion method sed on UIOS," INFOCOM '92. Eleventh Annul Joint Confeence of the IEEE Compute nd Communictions Societies, IEEE, vol., no., pp.268,277 vol.3, 4-8 My 992 [7] Sidhu, D.; Leung, T.-K., "Foml methods fo potocol testing: detiled study," Softwe Engineeing, IEEE Tnsctions on, vol.5, no.4, pp.43,426, Ap 989 doi:.9/ [8] Dn Angluin Lening egul sets fom queies nd counteemples. Inf. Comput. 75, 2 (Noveme 987), [9] Roth, J.P., "Dignosis of Automt Filues: A Clculus nd Method," IBM Jounl of Resech nd Development, vol., no.4, pp.278,29, July 966 doi:.47/d [] Hennine, F. C., "Fult detecting epeiments fo sequentil cicuits," Switching Cicuit Theoy nd Logicl Design, 964 Poceedings of the Fifth Annul Symposium on, vol., no., pp.95,, -3 Nov. 964 IJCTA Mch-Apil 24 Aville online@ 445

12 ISSN: Hnn Ndeem et l, Int.J.Compute Technology & Applictions,Vol 5 (2), [] "Chpte 2 Deteministic Finite Automt DFA include Lectue 3..." Inset Nme of Site in Itlics. N.p., n.d. We. 22 Fe. 24 < pte-2-deteministic-finite-automt-dfainclude-lectue-3-nd-4-finite-automt-ndegul-sets-lnguges-sttes-nd-tnsitions>. [2] "Finite Stte Automt nd Mophogmmtics.n." Inset Nme of Site in Itlics. N.p., n.d. We. 22 Fe. 24 < MophoFSM/Finite%2Stte%2Mch ines%2nd%2mophogmmtics. html_>. [3] "4ConfomnceTesting - unig.it." Inset Nme of Site in Itlics. N.p., n.d. We. 22 Fe. 24 < ch/ppes/motes5.pdf_>. [4] "4ConfomnceTesting - unig.it." Inset Nme of Site in Itlics. N.p., n.d. We. 26 Fe. 24 < ch/ppes/motes5.pdf_>. [5] "Confomnce Testing. - ResechGte." Inset Nme of Site in Itlics. N.p., n.d. We. M. 24 < tion/ _confomnce_testi ng_> IJCTA Mch-Apil 24 Aville online@ 446