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...hines, the two notions of a cover coincide. To construct deterministic transfer sequences, we delete outputs from the FSM A and apply a standard technique for determinizing of the obtained automaton (=-=Hopcroft and Ullman, 1979-=-). A state s is a d-reachable state in A if and only if there exists a set {s} among the states of the deterministic automaton. An input sequence that takes the automaton to the state {s} is a determi...

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...e union of ridentifiers over all states of the FSM A is said to be an r-characterization set W of A. It is a generalization of a classical notion of a characterization set of a deterministic machine (=-=Kohavi, 1978-=-). The equivalence and reduction relations serve the conformance relations between implementations and their FSM specifications for deriving test suites. Let a specification FSM A be defined over an i...

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...l machine A with harmonized traces can often be treated as a special complete nondeterministic machine a by treating its transitions on unspecified inputs as 'don't care' transitions to a trap state (=-=Unger, 1969-=-). Such transitions are labeled with an input not specified at the current state of A and all outputs of some superset of Y. The superset Y' of Y represents all outputs in the class of implementation ...

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...tion machine. The latter is treated as a black-box, so little is usually known about the implementation machine prior to testing; yet one typically assumes an upper bound on the number of its states (=-=Gill, 1962-=-). It is required to determine by testing whether the two are equivalent. A corresponding test suite is said to be complete with respect to equivalence relation in the class of implementation machines...

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...number of states. The problem of deriving such a test suite for a given deterministic specification machine has recently attracted close attention in the literature (Vasilevski, 1973), (Chow, 1978), (=-=Sidhu and Leung, 1989-=-), (Fujiwara, Bochmann et al. 1991), (Ural, 1992), (Bochmann and Petrenko, 1994), (Yannakakis and Lee, 1995), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general p...

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...ion, fault detection, I/O nondeterministic FSMs, equivalence and reduction relations 1 INTRODUCTION Conformance testing of protocol implementations is often formalized as the FSM equivalence problem (=-=Moore, 1956-=-) and (Hennie, 1964). In particular, we are given two machines defined over the same input alphabet, one is referred to as the specification machine, the other is referred to as the implementation mac...

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...on, I/O nondeterministic FSMs, equivalence and reduction relations 1 INTRODUCTION Conformance testing of protocol implementations is often formalized as the FSM equivalence problem (Moore, 1956) and (=-=Hennie, 1964-=-). In particular, we are given two machines defined over the same input alphabet, one is referred to as the specification machine, the other is referred to as the implementation machine. The latter is...

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...eterministic specification machine has recently attracted close attention in the literature (Vasilevski, 1973), (Chow, 1978), (Sidhu and Leung, 1989), (Fujiwara, Bochmann et al. 1991), (Ural, 1992), (=-=Bochmann and Petrenko, 1994-=-), (Yannakakis and Lee, 1995), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general problem of testing a so-called reduction relation between FSMs (Petrenko, Bochma...

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... the nondeterministic FSM may correspond to the same state of its reduction. The situation is similar to the case of compatible states of a partial deterministic FSM in context of state minimization (=-=Grasselli and Luccio, 1965-=-). We now establish necessary and sufficient conditions when two states of the FSM cannot correspond to a single state in any deterministic reduction of the FSM. Given an FSM A , states s and r of A a...

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... for a given deterministic specification machine has recently attracted close attention in the literature (Vasilevski, 1973), (Chow, 1978), (Sidhu and Leung, 1989), (Fujiwara, Bochmann et al. 1991), (=-=Ural, 1992-=-), (Bochmann and Petrenko, 1994), (Yannakakis and Lee, 1995), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general problem of testing a so-called reduction relation...

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...ine has recently attracted close attention in the literature (Vasilevski, 1973), (Chow, 1978), (Sidhu and Leung, 1989), (Fujiwara, Bochmann et al. 1991), (Ural, 1992), (Bochmann and Petrenko, 1994), (=-=Yannakakis and Lee, 1995-=-), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general problem of testing a so-called reduction relation between FSMs (Petrenko, Bochmann, and Dssouli, 1993) and (...

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... attention in the literature (Vasilevski, 1973), (Chow, 1978), (Sidhu and Leung, 1989), (Fujiwara, Bochmann et al. 1991), (Ural, 1992), (Bochmann and Petrenko, 1994), (Yannakakis and Lee, 1995), and (=-=Petrenko and Bochmann, 1996-=-). Here we take a step further addressing a more general problem of testing a so-called reduction relation between FSMs (Petrenko, Bochmann, and Dssouli, 1993) and (Petrenko, Yevtushenko, and Bochmann...

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...omes essential in situations where undefined transitions are treated as 'forbidden' transitions, as explained in (Yevtushenko and Petrenko, 1990), (Petrenko, 1991), (Petrenko and Yevtushenko, 1992), (=-=Luo, Petrenko, and Bochmann, 1994-=-), and (Yannakakis and Lee, 1995). The difference from the latter work is that we consider here a wider class of partial machines that are not necessarily reduced. (Yannakakis and Lee, 1995) gives no ...

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...te nondeterministic FSM. Thus, the test derivation problem for the reduction relation is of both, theoretical and practical interests. Not much work, however, has been done to solve this problem. In (=-=Petrenko, Yevtushenko, Lebedev, and Das, 1993-=-), it is demonstrated that the problem can be solved at least for a narrow subclass of nondeterministic FSMs. In this paper, we present a refined method for test derivation based on that work and anal...

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...set of output symbols; and h - a behavior function h: S��X��P(S��Y), where P(S��Y) is the set of all nonempty subsets of S��Y, such that |{s' | (s',y) ��h(s,x )}|1 for all (s,x=-= )��S��X and all y��Y. (Starke, 1972). The m-=-achine A becomes deterministic when | h(s,x )|=1 for all (s,x)��S��X. In a deterministic FSM, instead of the behavior function which is required for expressing a nondeterministic behavior, we ...

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...col implementation. Most existing protocols allow these options. The nondeterministic machine paradigm is also useful for embedded testing. As shown in (Petrenko, Yevtushenko, and Dssouli, 1994) and (=-=Petrenko, Yevtushenko, Bochmann, and Dssouli, 1996-=-), testing a deterministic FSM embedded within a given system of communicating FSMs can be reduced to that of an appropriate nondeterministic FSM. Thus, the test derivation problem for the reduction r...

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... from the set X A * of specified input sequences becomes essential in situations where undefined transitions are treated as 'forbidden' transitions, as explained in (Yevtushenko and Petrenko, 1990), (=-=Petrenko, 1991-=-), (Petrenko and Yevtushenko, 1992), (Luo, Petrenko, and Bochmann, 1994), and (Yannakakis and Lee, 1995). The difference from the latter work is that we consider here a wider class of partial machines...

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...d Petrenko, 1994), (Yannakakis and Lee, 1995), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general problem of testing a so-called reduction relation between FSMs (=-=Petrenko, Bochmann, and Dssouli, 1993-=-) and (Petrenko, Yevtushenko, and Bochmann, 1994). Specifically, we assume that an implementation machine is deterministic, but its specification machine is not necessarily deterministic. In this case...

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...* of specified input sequences becomes essential in situations where undefined transitions are treated as 'forbidden' transitions, as explained in (Yevtushenko and Petrenko, 1990), (Petrenko, 1991), (=-=Petrenko and Yevtushenko, 1992-=-), (Luo, Petrenko, and Bochmann, 1994), and (Yannakakis and Lee, 1995). The difference from the latter work is that we consider here a wider class of partial machines that are not necessarily reduced....

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...avior which contains options left for the protocol implementation. Most existing protocols allow these options. The nondeterministic machine paradigm is also useful for embedded testing. As shown in (=-=Petrenko, Yevtushenko, and Dssouli, 1994-=-) and (Petrenko, Yevtushenko, Bochmann, and Dssouli, 1996), testing a deterministic FSM embedded within a given system of communicating FSMs can be reduced to that of an appropriate nondeterministic F...

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... a complete test suite exclusively from the set X A * of specified input sequences becomes essential in situations where undefined transitions are treated as 'forbidden' transitions, as explained in (=-=Yevtushenko and Petrenko, 1990-=-), (Petrenko, 1991), (Petrenko and Yevtushenko, 1992), (Luo, Petrenko, and Bochmann, 1994), and (Yannakakis and Lee, 1995). The difference from the latter work is that we consider here a wider class o...

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...), and (Petrenko and Bochmann, 1996). Here we take a step further addressing a more general problem of testing a so-called reduction relation between FSMs (Petrenko, Bochmann, and Dssouli, 1993) and (=-=Petrenko, Yevtushenko, and Bochmann, 1994-=-). Specifically, we assume that an implementation machine is deterministic, but its specification machine is not necessarily deterministic. In this case, the implementation to be conforming is require...

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...assumption of induction. Thus, if t is a reduction of states s and r then they are not r(k)-distinguishable for any k, i.e. they are not rdistinguishable. q Combining Theorem 2.2 with the results of (=-=Damiani, 1994-=-), we have the following fact. A state of a deterministic FSM is not a reduction of two states of the FSM A if and only if these states are r-distinguishable. The definition of r-distinguishable state...

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...s within the assumed bound on the number of states. The problem of deriving such a test suite for a given deterministic specification machine has recently attracted close attention in the literature (=-=Vasilevski, 1973-=-), (Chow, 1978), (Sidhu and Leung, 1989), (Fujiwara, Bochmann et al. 1991), (Ural, 1992), (Bochmann and Petrenko, 1994), (Yannakakis and Lee, 1995), and (Petrenko and Bochmann, 1996). Here we take a s...