Set Of Output Sequences Research Articles

Deciding Irreducibility/Indecomposability of Feedback Shift Registers Is NP-Hard

Feedback shift registers (FSRs) are used as a fundamental component in electronics and confidential communication. A FSR f is said to be reducible if all the output sequences of another FSR g can also be generated by f and the FSR g costs less memory than f. A FSR is said to be decomposable if it has the same set of output sequences as a cascade connection of two FSRs. Two polynomial-time computable transformations from Boolean circuits to FSRs are proposed such that the output FSR of the first (resp. second) transformation is irreducible (resp. indecomposable) if and only if the input Boolean circuit is satisfiable. Through the two transformations, it is proved that deciding irreducibility (indecomposability) of FSRs is NP-hard. Additionally, FSRs are constructed to show that there exist infinitely many irreducible (resp. indecomposable) FSRs which are decomposable (resp. reducible).

A New Upper Bound on the Order of Affine Sub-families of NFSRs

Nonlinear feedback shift registers (NFSRs) are widely used as building blocks in the design of stream ciphers. Let NFSR(f) be an NFSR with the characteristic function f and let G(f) be the set of output sequences of NFSR(f). For a given NFSR(f), if there exists an affine Boolean function l such that G(l) ⊆ G(f), then G(l) is called an affine sub-family of NFSR(f). In this paper, by skillfully combining previous ideas, the authors give a new upper bound on the order of affine sub-families of NFSR(f). Compared with the four known bounds, the bound is better than three of them, and in some cases is also better than the rest one.

Fault-detection experiment with deterministic realizations and a nondeterministic model

The algorithmic approach to the design of fault-detection experiments for testing of control devices requires a mathematical model that describes the behavior of the tested device and its reference model. A finite automaton usually provides a suitable mathematical model. The classical problem assumes that the reference model and the tested device behave as a deterministic f'mite automaton (in what follows, we use the term "automaton" for a deterministic f'mite automaton). The class of faults is limited to faults that do not increase the number of automaton states. It is also assumed that the experimenter has direct access to the inputs and outputs of the device being tested. The construction of multiple fault-detection experiments assumes the existence of an input symbol that allows the model automaton to return from any state to some fixed (initial) state. This special symbol remains a reset symbol under all faults [1]. Recent studies use a nondeterministic automaton (nd-automaton) as a reference model for testing. In particular, an ndautomaton is used to construct test cases that check computer network protocols for conformity [2-4]. It has been shown [5] that if the output of the device being tested is observable only on the output of another device, then a fault,detection experiment also can be constructed using a nondeterministic model. The construction of a fault-detection experiment usually focuses mainly on the design of input sequences. The set of admissible output responses is identified with the entire set of model output sequences. In this article we analyze the possibility of reducing the length of the fault-detection experiment by reducing the set of admissible output responses. The article is organized as follows. Section 1 introduces the main def'mitions and notation. Section 2 describes the possibility of reducing the admissible set of output sequences in a fault-detection experiment. Section 3 provides necessary and sufficient conditions for the realization of a nondeterministic automaton by a deterministic automaton. We examine the conditions that permit passing from the nd-automaton P to an nd-automaton P,~ with fewer states and fewer output responses. The set of automata realizing P is identical with the set of automata realizing P~.. We can thus construct a fault~etection experiment for the nd-automaton P based on the nd-automaton P~.. This possibility is illustrated in Section 4.

Testing in context: framework and test derivation

The paper addresses the problem of testing a component embedded within a given modular system. A context of the component represents the rest of the system and serves as its operational or testing environment. A framework for testing in context is presented based on the model of a system of communicating finite state machines. In particular, the problems of test executability and fault propagation in the presence of the context are identified and discussed. The proposed solution to these problems consists in computing so-called approximation of the specification in context, i.e. the FSM model of the component's properties that can be controlled and observed through the context. The approximation assures executability of tests and fault propagation through the context and serves as a base for test derivation. A conformance relation used for test derivation is shown to be the reduction relation between an implementation and the approximation of the given specification. This relation requires that the implementation produces a (sub)set of output sequences that can be produced by its specification in response to every input sequence. An approach to test generation for the reduction relation and deterministic implementations is also presented.

Comma free error correcting codes of variable length, generated by finite-state encoders

A finite-state code (FSC) is the set of output sequences of a finite-state encoder. The case is considered where the period of the eneoder is one and the code is used as an error-correcting code. In this context the Levenshtein distance will measure the discrepancy between an information sequence and its estimate, and the minimum Hamming distance will be taken as the criterion of efficiency. The class of binary autocomplementary FSC's (BAC's) satisfies a lower bound on the minimum Hamming distance that is essentially equivalent to the well-known Gilbert bound for block codes. Examples of good BAC's are provided in the last section.