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Abstract.A predicate linear temporal logic LT L λ= without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LT L λ= can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LT L λ= for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LT L λ= is not recursively axiomatizable and, therefore, fully automated verification of LT L λ= specifications is not, in general, possible.

A gossiping is a communication primitive in which each node of the network possesses a unique message that is to be communicated to all other nodes in the network. We study the gossiping problem in known ad hoc radio networks, where during each transmission only unit messages originated at any node of the network can be transmitted successfully. We survey a number of radio network topologies. Assuming that the size (a number of nodes) of the network is n we show that the exact complexity of radio gossiping in stars is 2n-1, in rings is 2n±O(l), and on a line of processors is 3n ± 0(1). We later prove that radio gossiping in free trees is harder and it requires at least 3~n -16 time steps to be completed. For free trees we also show a gossiping algorithm with time complexity 5n + 8. In conclusion we prove that in general graphs radio gossiping requires fl(n log n) time, and we propose radio gossiping algorithm that works in time O(n log 2 n).

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Abstract. We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability of the mortality problem in two dimensions remains a long standing open problem although in dimension three is known to be undecidable as was shown by Paterson in 1970. We also show a lower bound on the minimum length solution to the Mortality Problem, which is exponential in the number of matrices of the generator set and the maximal element of the matrices.

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equationwhere Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general.The restricted problem where k = 2 for commutative matrices is known as the "A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field.

The main result of this paper is the decidability of the membership problem for 2 × 2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2 × 2 integer matrices M1, . . . , Mn and M decides whether M belongs to the semigroup generated by {M1, . . . , Mn}. Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words. It also makes use of some algebraic properties of well-known subgroups of GL(2, Z) and various new techniques and constructions that help to convert matrix equations into the emptiness problem for intersection of regular languages.

Abstract. We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.