Computing the image of a regular language by the transitive closure of a relation is a central question in regular model checking. In a recent paper Bouajjani et al. [IEEE Comput. Soc. (2001) 399–408] proved that the class of regular languages L – called APC – of the form UjL0,jL1,jL2,j...Lkj,j, where the union is finite and each Li,j is either a single symbol or a language of the form B* with B a subset of the alphabet, is closed under all semi-commutation relations R. Moreover a recursive algorithm on the regular expressions was given to compute R*(L). This paper provides a new approach, based on automata, for the same problem. Our approach produces a simpler and more efficient algorithm which furthermore works for a larger class of regular languages closed under union, intersection, semi-commutation relations and conjugacy. The existence of this new class, PolC, answers the open question proposed in the paper of Bouajjani et al.

Real functions on the domain [0,1)n – often used to describe digital images – allow for different well-known types of binary operations. In this note, we recapitulate how weighted finite automata can be used in order to represent those functions and how certain binary operations are reflected in the theory of these automata. Different types of products of automata are employed, including the seldomly-used full Cartesian product. We show, however, the infeasibility of functional composition; simple examples yield that the class of automatic functions (i.e., functions computable by automata) is not closed under this operation.

We consider the parallel approximability of two problems arising from high multiplicity scheduling, namely the unweighted model with variable processing requirements and the weighted model with identical processing requirements. These two problems are known to be modelled by a class of quadratic programs that are efficiently solvable in polynomial time. On the parallel setting, both problems are P-complete and hence cannot be efficiently solved in parallel unless P = NC. To deal with the parallel approximablity of these problems, we show first a parallel additive approximation procedure to a subclass of multi-valued quadratic programming, called smooth multi-valued QP, which is defined by imposing certain restrictions on the coefficients of the instance. We use this procedure to obtain parallel approximation to dense instances of the two problems by observing that dense instances of these problems are instances of smooth multi-valued QP. The dense instances of the problems considered here are defined similarly as for other combinatorial problems in the literature. For such instances we can find in parallel a near optimal schedule. The definition of smooth multi-valued QP as well as the procedure for approximating it in parallel are of interest independently of the application to the scheduling problems considered in this paper.

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within $\Delta^{0}_2$. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class $\Delta^{B}_2$. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and – if infinite pre/post sets of transitions are allowed – even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.

J. Hromkovic et al. have given an elegant method to convert a regular expression of size n into an ε-free nondeterministic finite automaton having O(n) states and O(nlog2(n)) transitions. This method has been implemented efficiently in O(nlog2(n)) time by C. Hagenah and A. Muscholl. In this paper we extend this method to weighted regular expressions and we show that it can be achieved in O(nlog2(n)) time.

We investigate the density and distribution behaviors of the chinese remainder representation pseudorank. We give a very strong approximation to density, and derive two efficient algorithms to carry out an exact count (census) of the bad pseudorank integers. One of these algorithms has been implemented, giving results in excellent agreement with our density analysis out to 5189-bit integers.

We investigate the computational structure of the biological kinship assignment problem by abstracting away all biological details that are irrelevant to computation. The computational structure depends on phenotype space, which we formally define. We illustrate this approach by exhibiting an approximation algorithm for kinship assignment in the case of the Simpson index with a priori error bound and running time that is polynomial in the bit size of the population, but exponential in phenotype space size. This algorithm is based on a relaxed version of the assignment problem, where fractional assignments (over the reals) are permitted.

The cyclic shift of a language L, defined as SHIFT(L) = {vu | uv ∈ L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov's pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl.11 (1970) 1373–1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)! . 2(n-1)(n-2), which shows that the state complexity of cyclic shift is 2n2+ nlogn - O(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove $2^{\Theta(n^2)}$ state complexity. We also establish a tight 2n2+1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.

A new algorithm is presented for the D0L sequence equivalence problem which, when the alphabets are fixed, works in time polynomial in the rest of the input data. The algorithm uses a polynomial encoding of words and certain well-known properties of $\mathbb{Z}$-rational sequences.

We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time.

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.