Using synchronizing heuristics to construct homing sequences

Computing a shortest synchronizing sequence of an automaton is an NP-Hard problem. There are well-known heuristics to find short synchronizing sequences. Finding a shortest homing sequence is also an NP-Hard problem. Unlike existing heuristics to find synchronizing sequences, homing heuristics are not widely studied. In this paper, we discover a relation between synchronizing and homing sequences by creating an automaton called homing automaton. By applying synchronizing heuristics on this automaton we get short homing sequences. Furthermore, we adapt some of the synchronizing heuristics to construct homing sequences

Adaptive Homing is in P

Homing preset and adaptive experiments with Finite State Machines (FSMs) are widely used when a non-initialized discrete event system is given for testing and thus, has to be set to the known state at the first step. The length of a shortest homing sequence is known to be exponential with respect to the number of states for a complete observable nondeterministic FSM while the problem of checking the existence of such sequence (Homing problem) is PSPACE-complete. In order to decrease the complexity of related problems, one can consider adaptive experiments when a next input to be applied to a system under experiment depends on the output responses to the previous inputs. In this paper, we study the problem of the existence of an adaptive homing experiment for complete observable nondeterministic machines. We show that if such experiment exists then it can be constructed with the use of a polynomial-time algorithm with respect to the number of FSM states.Comment: In Proceedings MBT 2015, arXiv:1504.0192

The upper bound for the length of the shortest homing sequences

Homing sequences are special input sequences that are used by various techniques of finite state machine based testing. Using a shorter homing sequence is typically preferred since it would yield a shorter test sequence. Finding a shortest homing sequence is known to be an NP–hard problem. The upper bound of shortest homing sequences is also a problem studied in the literature. A tight upper bound for the length of shortest homing sequence for a finite state machine with n states is known to be n(n−1)/2 . However, the known examples of finite state machines hitting to this upper bound also use n−1 input symbols, i.e. the size of the input alphabet also grows with the number of states. Is this upper bound reachable for a finite state machine with a constant number of inputs? In this work, we use an experimental analysis and we answer this question negatively. By exhaustively enumerating all finite state machines with two input symbols and two output symbols, we experimentally compute the upper bound for the length of the shortest homing sequence for finite state machines with 10 or less states. In order to make this computation feasible in practice, we apply several techniques to eliminate from our search those finite state machines which would not affect the result of the computatio

Experiments with finite state machines

The purpose of this study is to introduce and illustrate the various types of experiments with finite state machines. A finite state machine is an abstract object composed of a finite number of input, output and state symbols. The behavior of the machine is described by a functional relationship between input, output and state. In designing a finite state machine it often happens that two states represent the same internal condition. Therefore it is desirable to develop a technique for transforming one machine into another which has no redundant states, so that both have the same behavior. The definition of k-equivalent and k-distinguishable are useful in an algorithm which is developed to determine which states of the machine are equivalent. The machine with no two equivalent states is a reduced machine. An experiment on a reduced state machine consists of applying an input sequence and observing the output. The classification of experiments is (1) simple and multiple experiments; (2) preset and adaptive experiments; (3) distinguishing and homing experiments. The successor tree is a useful device in designing the experiment. The successor tree is terminated by specific rules in each experiment. Homing and distinguishing experiments can be either preset and adaptive. All reduced machines have a homing sequence. A distinguishing sequence is sometimes possible in both preset and adaptive form. However, some reduced machines have only an adaptive experiment and some do not have any simple distinguishing sequence at all

Slowly synchronizing automata and digraphs

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.Comment: 13 pages, 5 figure

Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

Complexity of checking whether two automata are synchronized by the same language

A deterministic finite automaton is said to be synchronizing if it has a reset word, i.e. a word that brings all states of the automaton to a particular one. We prove that it is a PSPACE-complete problem to check whether the language of reset words for a given automaton coincides with the language of reset words for some particular automaton.Comment: 12 pages, 4 figure