Setting a Complex System to a Known Initial State

Abstract

Nowadays nondeterministic models are widely used when describing the behavior of complex systems. Nondeterminism occurs due to various reasons such as physical properties, limited controllability, and/or observability of a system under analysis, abstraction level, etc. [1]. Given a complex system, if its behavior can be described using finite sets of states and transitions between them, then homing experiments with nondeterministic Finite State Machines (FSMs) can be applied in order to set the system into a known initial state [2]. If the technical system behavior is described by a deterministic FSM then such experiments are well established. In particular, a complete deterministic FSM with n pairwise distinguishable states always has a homing sequence of length at most n(n – 1)/2. Moreover, the order of length of the homing sequence cannot be reduced when the next input depends on output responses of a system under experiment to the previous inputs, i.e., the complexity of homing experiments cannot be reduced using adaptive experiments instead of preset. For systems with the nondeterministic behavior, investigations of homing experiments are just beginning to develop [3, 4, 5]. In this paper, we briefly present the novel results obtained in this area under the assumption that the behavior of the system under experiment is described by a complete observable nondeterministic FSM. In other words, the behavior of the system in each state is defined for any input, but there can be several output responses to the same input in some states. For the observable FSM, given an input/output pair and a current state, the next state of the system can be uniquely identified. A finite state machine, or simply a machine is a 5-tuple S = (S, I, O, hS, S′), where S is a finite nonempty set of states with a nonempty subset S' of initial states, I and O are finite input and output alphabets, and hS ⊆ S × I × O × S is a transition relation that describes the behavior of the system which moves from state to state when an input of the set I is applied and produces an output response of the set O. Thus, an FSM is a finite representation of a system behavior with respect to the infinite number of input sequences. In this paper, we are interested in nondeterministic complete FSMs, i.e., at each state there is defined transition under each input, while different outputs can be produced at some state under some inputs. We are also interested in observable nondeterministic machines where the next state at each transition can be defined by observing corresponding input/output pair. Given a nondeterministic complete observable FSM S = (S, I, O, hS, S'), an input sequence α ∈ I is a homing sequence for S if for each pair s1, s2 ∈ S ′ of initial states at which the system responses to α with the output sequence β, the input/output pair α/β takes the FSM from states s1, s2 to one and the same single state, i.e., after applying α and observing β, we certainly know which state is reached by the system under experiment. Differently from deterministic FSMs, a homing sequence may not exist for a nondeterministic FSM without equivalent states [3]. Moreover, in [4] it has been shown that the length of a homing sequence for a nondeterministic machine (if such sequence exists) exponentially depends on the FSM size, i.e., on the number of FSM transitions. Different output sequences can be observed when applying the same input sequence to a nondeterministic FSM, and thus adaptive experiments can result in a shorter input sequence that sets the system under experiment into a given initial state, i.e., adaptive experiments can be more efficient than preset. During adaptive experiment, the next input significantly depends on the system outputs produced to the previous inputs. In order to formally define