Lecture Notes in Computer Science 3472

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14 Sven Sandberg Proof. We prove the equivalent, contrapositive form: ρi+1 �= ρi ⇒ ρi �= ρi−1 for all i ≥ 1. If ρi+1 �= ρi then there are two states s, t ∈ S with a shortest separating sequence of length i +1, say ax ∈ I i+1 (i.e., a is the first letter and x the tail of the sequence). Since ax is separating for s and t but a is not, x must be separating for δ(s, a) andδ(t, a). It is also a shortest separating sequence, because if y ∈ I ∗ was shorter than x ,thenay would be a separating sequence for s and t, and shorter than ax . This proves that there are two states δ(s, a),δ(t, a) with a shortest separating sequence of length i, soρi �= ρi−1. ⊓⊔ Since a partition of n elements can only be refined n times, the sequence ρ0,ρ1,... of partitions becomes constant after at most n steps. And since the machine is minimized, after this point the partitions contain only singletons. So any two states have a separating sequence of length at most n − 1, and the homing sequence has length at most (n − 1) 2 . Hibbard [Hib61] improved this bound, showing that the homing sequence computed by Algorithm 1 has length at most n(n −1)/2, provided we choose two states with the shortest possible separating sequence in each iteration. Moreover, for every n there is an n-state machine whose shortest homing sequence has length n(n − 1)/2, so the algorithm has the optimal worst case behavior in terms of output length. Computing Separating Sequences. We are now ready to fill in the last detail of the algorithm. To compute separating sequences, we first construct the partitions ρ1,ρ2,...,ρr described above, where r is the smallest index such that ρr contains only singletons. Two states s, t ∈ S belong to different blocks of ρ1 if and only if there is an input a ∈ I so that λ(s, a) �= λ(t, a), and thus ρ1 can be computed. Two states s, t ∈ S belong to different blocks of ρi for i > 1if and only if there is an input a such that δ(s, a) andδ(t, a) belong to different blocks of ρi−1, and thus all ρi with i > 1 can be computed. To find a separating sequence for two states s, t ∈ S, find the smallest index i such that s and t belong to different blocks of ρi. As argued in the proof of Lemma 1.12, the separating sequence has the form ax ,wherexis a shortest separating sequence for δ(s, a) andδ(t, a). Thus, we find the a that takes s and t to different blocks of ρi−1 and repeat the process until we reach ρ0. The concatenation of all such a is our separating sequence. This algorithm can be modified to use only O(n) memory,notcountingthe space required by the output. Typically, the size of the output does not contribute to the memory requirements, since the sequence is applied to some machine on the fly rather than stored explicitly. Exercise 1.4. Give an example of a Mealy machine that is not minimized but has a homing sequence. Is there a Mealy machine that is not minimized and has a homing but no synchronizing sequence?
1.3.2 Computing Synchronizing Sequences 1 Homing and Synchronizing Sequences 15 Synchronizing sequences are computed in a way similar to homing sequences. The algorithm also concatenates many short sequences into one long, but this time the sequences take two states to the same final state. Analogously to separating sequences, we define merging sequences: Definition 1.13. A merging sequence for two states s, t ∈ S is a sequence x ∈ I ∗ such that δ(s, x )=δ(t, x ). ⊓⊔ The Algorithm. Algorithm 2 first finds a merging sequence y for two states. This ensures that |δ(S, y)| < |S|, because each state in S gives rise to at most one state in δ(S, y), but the two states for which the sequence is merging give rise to the same state. This process is repeated, in each step appending a new merging sequence for two states in δ(S, x )totheresultx , thus decreasing |δ(S, x )|. If at some point the algorithm finds two states that do not have a merging sequence, then there is no synchronizing sequence: if there was, it would merge them. And if the algorithm terminates by finishing the loop, the sequence x is clearly synchronizing. This shows correctness. Since |δ(S, x )| is n initially and 1 on successful termination, the algorithm needs at most n − 1 iterations. Algorithm 2 Computing a synchronizing sequence. 1 function Synchronizing(Mealy machine M) 2 x ← ε 3 while |δ(S, x )| > 1 4 find two different states s0, t0 ∈ δ(S, x ) 5 let y be a merging sequence for s0 and t0 (if none exists, return Failure) 6 x ← xy 7 return x As a consequence of this algorithm, we have (see, e.g., Starke [Sta72]): Theorem 1.14. A machine has a synchronizing sequence if and only if every pair of states has a merging sequence. Proof. If the machine has a synchronizing sequence, then it is merging for every pair of states. If every pair of states has a merging sequence, then Algorithm 2 computes a synchronizing sequence. ⊓⊔ To convince yourself, it may be instructive to go back and see how this algorithm works on Exercise 1.1.
Page 1 and 2: Lecture Notes in Computer Science 3
Page 3 and 4: Volume Editors Manfred Broy Martin
Page 5 and 6: VI Preface The last part of the boo
Page 7 and 8: VIII Contents 13 Real-Time and Hybr
Page 9 and 10: 2 Part I. Testing of Finite State M
Page 11 and 12: 1 Homing and Synchronizing Sequence
Page 17 and 18: s2 a/1 1 Homing and Synchronizing S
Page 19: 1 Homing and Synchronizing Sequence
Page 35 and 36: A1 A2 Bad 1 Homing and Synchronizin
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66 Moez Krichen The two authors als
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3 State Verification Henrik Björkl
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a/1 s1 s3 b/1 a/1 b/1 a/0 s2 s4 3 S
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3 State Verification 73 Thus (s, Q)
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3 State Verification 75 0, and x ca
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s1 s3 a/0 b/0 a/1 s2 s4 3 State Ver
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3.4.2 Naik’s Algorithm 3 State Ve
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s1 s2 s3 s1 s3 s3 a/0 a/0 s1 s2 s3
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3 State Verification 83 Since the m
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3 State Verification 85 x = a1a2 ..
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4 Conformance Testing Angelo Gargan
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4 Conformance Testing 89 the test u
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4 Conformance Testing 91 state. For
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4 Conformance Testing 93 This check
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s1 s1 a b s2 s2 a b s3 4 Conformanc
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4 Conformance Testing 97 Using a Q
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4 Conformance Testing 99 this case
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4.4.5 Cost and Length 4 Conformance
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t τ (t,si−1) x τti−1 ,si si
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si 1 2 w1 t i w 2 3 a: in MS si 1 w
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4 Conformance Testing 107 ti = δ(s
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4 Conformance Testing 109 Example.
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4 Conformance Testing 111 • If on
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114 Part II. Testing of Labeled Tra
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5 Preorder Relations ∗ Stefan D.
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5 Preorder Relations 119 power of r
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5 Preorder Relations 121 To summari
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5 Preorder Relations 123 Two import
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∧ ⊥⊤ ⊥ ⊥⊥ ⊤ ⊥⊤
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5 Preorder Relations 127 Definition
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5 Preorder Relations 129 For one th
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5 Preorder Relations 131 We close t
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s a b t 5 Preorder Relations 133 y
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5 Preorder Relations 135 (1) p ⊑m
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a p ′ τ b p ′′ τ τ a a b 5
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5 Preorder Relations 139 It should
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coin coin e c coin coin τ 5 Preord
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5 Preorder Relations 143 for free.
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5 Preorder Relations 145 condition
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5 Preorder Relations 147 ⊑←→
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5 Preorder Relations 149 The utilit
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152 Valéry Tschaen The labels repr
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154 Valéry Tschaen Intuitively, th
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156 Valéry Tschaen Definition 6.2.
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158 Valéry Tschaen By this theorem
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160 Valéry Tschaen The test suite
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162 Valéry Tschaen each sequence (
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164 Valéry Tschaen and B2 are LOTO
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166 Valéry Tschaen 6.4.2 The CO-OP
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168 Valéry Tschaen a τ q0 τ τ q
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170 Valéry Tschaen Concerning the
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7 I/O-automata Based Testing Machie
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7 I/O-automata Based Testing 175 th
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7 I/O-automata Based Testing 177 in
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7 I/O-automata Based Testing 179 sp
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7 I/O-automata Based Testing 181 na
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7 I/O-automata Based Testing 183 Un
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7 I/O-automata Based Testing 185 We
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7 I/O-automata Based Testing 187 fa
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7 I/O-automata Based Testing 189 Th
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7 I/O-automata Based Testing 191 Ac
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7 I/O-automata Based Testing 193 vi
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7 I/O-automata Based Testing 195 De
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7 I/O-automata Based Testing 197 To
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7 I/O-automata Based Testing 199 In
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8 Test Derivation from Timed Automa
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s0 on, true, {c} off , c =5, ∅ 8
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8.3 Testing Event Recording Automat
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[s1, z], z = con < 5, 8 Test Deriva
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Ψ(S ′ )={PE ′ | E ′ ∈ 2E(S
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{s0}, p0 p0 : tt 8 Test Derivation
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s0 c = ∞ 8 Test Derivation from T
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Example. The Light Switch can be de
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��s0� ,c=0.0 s0 en, on, {c :=
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��s0� ,c=0 s0 8 Test Derivati
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234 Verena Wolf to probabilistic tr
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236 Verena Wolf • A finite trace
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238 Verena Wolf (a, 0.2) v1 sP (a,
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240 Verena Wolf 1 2 sP a b µ λ 1
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242 Verena Wolf (a, 1 4 ) (a, 1 4 )
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244 Verena Wolf 9.6 Test Processes
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246 Verena Wolf We present two appr
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248 Verena Wolf specification proce
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250 Verena Wolf sP (τ, 1 1 ) (c, 3
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252 Verena Wolf sP (a, 1 1 ) (a, 2
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254 Verena Wolf v1 u1 w1 1 2 sP λ
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256 Verena Wolf • P ⊑ must JY Q
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258 Verena Wolf u1 s M ′ 1 (a, r1
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260 Verena Wolf Proposition 9.22. L
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262 Verena Wolf Now let Q min be th
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264 Verena Wolf (8) (⊑BC , ⊑CH
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266 Verena Wolf is that an action a
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268 Verena Wolf can perform τ-tran
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270 Verena Wolf T np,re seq ⊑ tr
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272 Verena Wolf t2 t4 a sT t1 b b t
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274 Verena Wolf model test processe
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Part III Model-Based Test Case Gene
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Part III. Model-Based Test Case Gen
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282 Alexander Pretschner and Jan Ph
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11 Evaluating Coverage Based Testin
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12 Technology of Test-Case Generati
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Increment ∆Counter 12 Technology
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(12.4) ✟ nat(X ′ ) {A/0, B/X
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a: b: (6) α1+1−α2 α2 a: b: PC:
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12.5 Summary 12 Technology of Test-
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13 Real-Time and Hybrid Systems Tes
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Test Driver SUT 13 Real-Time and Hy
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fitness 13 Real-Time and Hybrid Sys
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13.5 Summary 13 Real-Time and Hybri
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390 Part IV. Tools and Case Studies
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392 Axel Belinfante, Lars Frantzen,
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440 Wolfgang Prenninger, Mohammad E
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Part V Standardized Test Notation a
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466 George Din inter-operability te
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468 George Din • Tabular Presenta
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470 George Din 16.3.1 Test Building
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472 George Din Abstract Test System
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474 George Din field can be marked
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476 George Din • omit: the value
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478 George Din altstep Default() ru
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480 George Din alt { [] httpPort.re
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482 George Din • Test Management
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484 George Din Value Type getType()
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486 George Din This task is rather
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488 George Din hosts, of preparing
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490 George Din A B C D MEM = 256 MB
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492 George Din A hardware load moni
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494 George Din ptcType2 ptcType3
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496 George Din communicate with the
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498 Zhen Ru Dai U2TP document is no
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500 Zhen Ru Dai state machines, sta
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502 Zhen Ru Dai TestResult TestCase
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504 Zhen Ru Dai can be mapped to JU
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506 Zhen Ru Dai 17.4 Test Developme
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508 Zhen Ru Dai Slave: Master: SRI
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510 Zhen Ru Dai sa: Slave− Appli
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512 Zhen Ru Dai sdConnect_To_Master
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514 Zhen Ru Dai BluetoothUnitTest
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516 Zhen Ru Dai is returned to the
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518 Zhen Ru Dai (1) /* test configu
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520 Zhen Ru Dai (1) /* test case */
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Part VI Beyond Testing This last pa
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526 Séverine Colin and Leonardo Ma
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19 Model Checking Therese Berg 1 an
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19 Model Checking 559 The first sta
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19 Model Checking 561 function I :
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coffee coin tea 19 Model Checking 5
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AX (φ) :=¬EX (¬φ) AF (φ) :=Atr
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19 Model Checking 567 Another appro
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19 Model Checking 569 has a transit
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19 Model Checking 571 The alternati
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19 Model Checking 573 Obviously the
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19 Model Checking 575 purposes ther
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19 Model Checking 577 otherwise it
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19 Model Checking 579 Expanding a P
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19 Model Checking 581 We conduct an
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• SA ⊆ Σ ∗ is a nonempty fin
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19 Model Checking 585 When OT is cl
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19 Model Checking 587 • se = s
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19 Model Checking 589 the new colum
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19 Model Checking 591 Henceforth th
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19 Model Checking 593 DT , the tran
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19 Model Checking 595 queries. At t
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19 Model Checking 597 Assistant 4.
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19 Model Checking 599 have created
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19 Model Checking 601 Logic (see Se
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19 Model Checking 603 linear time l
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20 Model-Based Testing - A Glossary
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20 Model-Based Testing - A Glossary
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612 Bengt Jonsson a/0 b/0 b/1 s2 s4
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614 Bengt Jonsson of input symbols
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616 Joost-Pieter Katoen The followi
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618 Literature [AFH94] Rajeev Alur,
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620 Literature [BCMD90] J. R. Burch
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622 Literature [BRV95] Ed Brinksma,
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624 Literature [CL97a] Duncan Clark
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626 Literature [DGNP04] Z. R. Dai,
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628 Literature [FHP02] Eitan Farchi
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630 Literature [GL02] Hubert Garave
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632 Literature [Hib61] Thomas N. Hi
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634 Literature [HR01b] Klaus Havelu
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636 Literature [KKL + 01] M. Kim, S
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638 Literature [LPY97] Kim Guldstra
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640 Literature [Müh97] H. Mühlenb
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642 Literature [Pha93] M. Phalippou
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644 Literature [RdBJ00] V. Rusu, L.
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646 Literature [Sch00] Johann M. Sc
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648 Literature [SVG02] S. Schulz an
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650 Literature [Vas73] M. P. Vasile
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Index =⇒ , 175 ioco, 187 ioconf,
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homing sequence, 8 - adaptive, 8, 2
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eset sequence, see synchronizing se
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timed transition system, 223, 360,
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