Volume 18, Issue 4

Consideration of the field of events in the theory of probability gives rise to the notion of the field of events ℱ( B ) consisting of a set of subsets of some set B . On the field ℱ( B ), two algebraic structures are naturally defined. These are the Boolean algebra (ℱ( B )) with the operations of union, intersection and complement, and the lattice L (ℱ( B )), where the order is defined according to inclusion of the sets of ℱ( B ). In this paper, we consider one more algebraic structure on ℱ( B ) and the abstract variant of this structure, the so-called probabilistic structure, which is closely related to properties of the measure on ℱ( B ).

We consider a random system of discrete equations in n two-valued unknowns consisting of M = M ( n ) equations. The functions in the left-hand sides of equations are randomly selected from a finite set of functions and can depend on at most m variables. We suggest and justify a criterion of existence of a threshold function for consistency of a random system of equations defined as a function Q ( n ) for which the probability of consistency of the system tends to one or zero as n → ∞, M ( n )/ Q ( n ) → 0 or M ( n )/ Q ( n ) → ∞ respectively. It is shown that the threshold functions for consistency can be only of the form n and n 1−1/ r , 2 ≤ r ≤ m + 1, we give criteria of existence of such functions for a random system of equations. For random systems of equations with threshold functions of the form n 1−1/ r , 2 ≤ r ≤ m + 1, we estimate the probability of consistency as n → ∞ and M ∼ cn 1−1/ r (the probability decreases from one to zero, taking all intermediate values, as c increases from zero to ∞) and construct an algorithm recognising inconsistency of realisations of such system of equations. This algorithm has the same limit probability of recognising inconsistency of systems of equations as the algorithm of complete checking of possible solutions but has the lower complexity of order n 1−1/ r operations.

We obtain an upper bound for the number of spanning trees of regular graphs of degree k which is in a sense asymptotically exact as k → ∞.

We investigate the complexity of basic operations over dynamic databases including search, insertion, and deletion of records. We suggest a structure of a database and corresponding algorithms which with the use of two parallel computing processes permit to fulfil these operations over a database in a constant time on the average and with logarithmic complexity in the worst case.

On the set of partial Boolean functions, we consider the closure operator with the equality predicate branching. We prove that each closed class with respect to this operator is generated in by the set of all its functions depending on at most two variables. In the paper, all maximal and submaximal classes in and all closed classes generated by one one-place or two-place function are presented. This research was supported by the Russian Foundation for Basis Research, grant 06–01–00438.

Earlier, it was proved that even for a minimal circuit the delay T could be much less than the depth D . Namely, an infinite sequence of minimal circuits was constructed such that T < log 2 D + 6 and D → ∞. This result would be more interesting if the inequality were true for all equivalent minimal circuits. In this paper, we present an infinite sequence of Boolean functions F k , k = 1, 2,…, such that any minimal circuit for an arbitrary function F k has the depth and the delay obeying the inequality T < log 2 D + 14. This research was supported by the Program of Basic Research of Department of Applied Mathematics of Russian Academy of Sciences ‘Algebraic and Combinatorial Methods of Mathematical Cybernetics,’ project ‘Design and Complexity of Control Systems.’

The paper is devoted to describing all homomorphisms of shift registers over finite fields with feedback function which is a permutation function with respect to the input variable into linear automata. We prove that the linear automaton which is the homomorphic image of a shift register is always isomorphic to a linear shift register. Therefore, according to the theorem proved by the author earlier, the question on the homomorphisms of linear shift registers into linear automata reduces to the question on the decomposition of a function (or the polynomial representing the function) into the so-called shift-composition of two functions (polynomials), where the left function is an affine function. We also prove that any polynomial is uniquely representable in the form of a sum of shift-compositions of linear polynomials and monomials with the first variable. These linear polynomials are in correspondence with polynomials of one variable and the question on the decomposition is reduced to the search for the common divisors of these polynomials of one variable. We give some simple conditions sufficient for absence of nontrivial inner homomorphisms of shift registers into linear automata.

Proofs of security for practical signature schemes are known in idealised models only. In the present paper, we consider the tamper-proof device model that does not use ideal primitives. Instead of access to a random oracle each participant is provided with tamper-proof device implementing a private-key cryptosystem. The hash-value of a message to be signed is submitted to the tamper-proof device for encryption and this encrypted value is used in the signature generation algorithm. In this model, we prove, modulo a physical assumption, a necessary and sufficient condition for security of the GOST signature scheme.

We consider general rules of constructing of local products of the Fitting classes with the use of nonlocal factors determined by the minimal elements of the Lockett section. We give a simple method to construct local Fitting classes which are factorised by nonlocal Fitting classes of some classes of π -groups and π ′-groups. Applying the class , we simplify the procedure of construction of a nonlocal factor in the local product. It is shown that in the construction of the local product the nonlocal factor for the appropriate set of primes π is the Fitting class of the form , where is the minimal element of the Lockett section of the class . In this paper, we consider only finite solvable groups.