Volume 29, Issue 1

We prove a criterion of perfect balance for sliding superposition of functions over an arbitrary finite alphabet. We also give examples of applying this result to the construction of perfectly balanced functions that are not permutations with respect to the first and to the last variable.

In 1995 T. Matsui considered a special family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation. In 2012 the author demonstrated that all polytopes of this family appear as faces of polytopes associated with the following NP-complete problems: the travelling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. Here it is shown that none of the polytopes of the aforementioned special family (with the exception of the one-dimensional segment) can appear as a face in a polytope associated with the problem of the maximum independent set, the set packing problem, the set partitioning problem, and the problem of 3-assignments.

The formula for the frequencies of symbols in the output sequence of multi-cyclic generator over the direct sum of residue groups modulo 2 is obtained. In the case when cells of registers contain independent random values with the uniform distribution the limit joint distribution of the frequencies (as lengths of the registers tend to infinity) is obtained.

We obtain nontrivial lower bounds for lengths of minimal single fault detection and diagnostic tests for Boolean circuits in wide classes of bases in presence of stuck-at faults at outputs of circuit gates.

A constructive proof is given that in each of the bases B ′ = { x & y , x ⊕ y , x ∼ y }, B 1 = { x & y , x ⊕ y , 1} any n -place Boolean function may be implemented: by an irredundant combinational circuit with n inputs and one output admitting (under single stuck-at faults at inputs and outputs of gates) a single fault detection test of length at most 16, by an irredundant combinational circuit with n inputs and one output admitting (under single stuck-at faults at inputs and outputs of gates and at primary inputs) a single fault detection test of length at most 2 n −2log 2 n + O (1); besides, there exists an n -place function that cannot be implemented by an irredundant circuit admitting a detecting test whose length is smaller than 2 n −2log 2 n − Ω (1), by an irredundant combinational circuit with n inputs and three outputs admitting (under single stuck-at faults at inputs and outputs of gates and at primary inputs) a single fault detection test of length at most 17.

The paper is concerned with representations of predicates over a finite set in the form of generalized conjunctive normal forms (GCNF). Properties of predicates GCNF are found which are preserved by some majority function. Such predicates are called generalized bijunctive predicates. These properties are used to construct new faster polynomial algorithms for the generalized satisfiability problem in the case when some majority function preserves all the original predicates.

We consider the classical model of embeddings in a simple binary Markov chain with unknown transition probability matrix. We obtain conditions on the asymptotic growth of lengths of the original and embedded sequences sufficient for the consistency of the proposed statistical embedding detection test.