Homing and synchronizing sequences (HSs and SSs) are used in Finite State Machine (FSM)-based testing for state identification when deriving a test suite with guaranteed fault coverage or when performing a non-intrusive (passive) testing or monitoring. However, such preset sequences do not always exist for nondeterministic FSMs or can be rather long if existing. Adaptive HSs and SSs are known to exist more often and can be much shorter which makes them attractive for deriving test suites and adaptive checking sequences. As nowadays, a number of specifications are represented by nondeterministic FSMs, there is a need for the deeper study of such sequences, their derivation strategies, and related complexity estimation / reduction. In this paper, given an FSM and a subset \(S'\) of its states, we are concerned with the existence check and derivation of \(S'\)-homing and \(S'\)-synchronizing adaptive sequences (test cases) which allow to reduce the uncertainty about the current state of the FSM up to a state of the subset \(S'\). There are a number of research papers on evaluating the complexity of the existence check of adaptive HS / SS as well as on estimating the length of an HS / SS, and in the ICTSS’19 conference, we presented the complexity of their derivation for non-initialized complete nondeterministic FSMs. In this paper, we utilize the criteria of the ICTSS’19 paper to estimate the complexity of the existence check and derivation of \(S'\)-homing and \(S'\)-synchronizing test cases for non-initialized observable nondeterministic FSMs. Some sufficient conditions for weakly initialized FSMs to have adaptive \(S'\)-homing and \(S'\)-synchronizing test cases are also established.

在基于有限状态机（FSM）的测试中，使用归巢和同步序列（HS和SS）进行状态识别时，将派生具有保证的故障范围的测试套件，或者执行非侵入式（被动）测试或监视。但是，此类预设序列对于不确定性FSM并不总是存在，或者如果存在则可能会很长。已知自适应HS和SS的存在频率更高，并且可以更短，这使得它们对于派生测试套件和自适应检查序列具有吸引力。当今，不确定性FSM代表了许多规范，因此需要更深入地研究此类序列，其推导策略以及相关的复杂性估算/归约。在本文中，给定一个FSM和一个子集\（S'\）关于它的状态，我们关注的是\（S'\）- homing和\（S'\）-同步自适应序列（测试用例）的存在检查和推导，它们可以减少FSM当前状态的不确定性直到子集\（S'\）的状态。关于评估自适应HS / SS的存在性检查的复杂性以及估计HS / SS的长度，有许多研究论文，在ICTSS'19会议上，我们介绍了非HS / SS的推导的复杂性。 -初始化的完全不确定的FSM。在本文中，我们利用ICTSS'19论文的标准来估计\（S'\）- homing和\（S'\）的存在性检查和推导的复杂性-同步未初始化的可观察的不确定性FSM的测试用例。还建立了一些弱初始化FSM具有自适应\（S'\）-归位和\（S'\）-同步测试用例的充分条件。