We prove a general strong normalisation theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of Gödel's system T, but also to various forms of barrecursion for which strong normalisation was hitherto unknown.

Lipschitz continuity is used as a tool for analysing the relationship between incomputability and randomness. We present a simpler proof of one of the major results in this area – the theorem of Yu and Ding, which states that there exists no cl-complete c.e. real – and go on to consider the global theory. The existential theory of the cl degrees is decidable, but this does not follow immediately by the standard proof for classical structures, such as the Turing degrees, since the cl degrees are a structure without join. We go on to show that strictly below every random cl degree there is another random cl degree. Results regarding the phenomenon of quasi-maximality in the cl degrees are also presented.

The MATRIX PACKING DOWN problem asks to find a row permutation of a given (0,1)-matrix in such a way that the total sum of the first non-zero column indexes is maximized. We study the computational complexity of this problem. We prove that the MATRIX PACKING DOWN problem is NP-complete even when restricted to zero trace symmetric (0,1)-matrices or to (0,1)-matrices with at most two 1's per column. Also, as intermediate results, we introduce several new simple graph layout problems which are proved to be NP-complete.

In secret sharing, different access structures have different difficulty degrees for acceding to the secret. We give a numerical measure of how easy or how difficult is to recover the secret, depending only on the structure itself and not on the particular scheme used for realizing it. We derive some consequences.

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations.Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones.For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n).Some results can be extended to labeled trees considering coloredDyck paths or, equivalently, colored one-stack sortable permutations.

We consider conversions of regular expressions into k-realtimefinite state automata, i.e., automata in which the number ofconsecutive uses of ε-transitions, along any computation path,is bounded by a fixed constant k. For 2-realtime automata,i.e., for automata that cannot change the state, without readingan input symbol, more than two times in a row, we show that theconversion of a regular expression into such an automaton producesonly O(n) states, O(nlogn) ε-transitions, and O(n)alphabet-transitions. We also show how to easily transform these2-realtime machines into 1-realtime automata, still with onlyO(nlogn) edges. These results contrast with the known lowerbound Ω(n(logn)2 / loglogn), holding for 0-realtimeautomata, i.e., for automata with no ε-transitions.

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.

In the infinite Post Correspondence Problem an instance (h,g)consists of two morphisms h and g, and the problem is todetermine whether or not there exists an infinite word ωsuch that h(ω) = g(ω). This problem was shown to beundecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36(2003) 231–245) showed that this problem is undecidable for domainalphabets of size 105. Here we give a proof that the infinite PostCorrespondence Problem is undecidable for instances where themorphisms have domains of 9 letters. The proof uses a recentresult of Matiyasevich and Sénizergues and a modification of aresult of Claus.

We introduce doubly-ranked (DR) monoids in order to study picturecodes. We show that a DR-monoid is free iff it is pictoriallystable. This allows us to associate with a set C of pictures apicture code B(C) which is the basis of the least DR-monoidincluding C.A weak version of the defect theorem for pictures is established.A characterization of picture codes through picture series isalso given.

In this note we consider the longest word, which has periods p1,...,pn , and does not have the period gcd(p1,...,pn ).The length of such a word can be established by a simple algorithm. We give a short and natural way to prove that the algorithm is correct. We also give a new proof that the maximal word is a palindrome.

We say that $n$-vertex graphs $G_1,G_2,\ldots,G_k$ pack if there exist injective mappings of their vertex sets onto $[n] = \{1, \ldots,n \}$ such that the images of the edge sets do not intersect. The notion of packing allows one to make some problems on graphs more natural or more general. Clearly, two $n$-vertex graphs $G_1$ and $G_2$ pack if and only if $G_1$ is a subgraph of the complement $\overline{G}_2$ of $G_2$.

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

In this article, we study the problem of finding tight bounds on the expected value of the kth-order statistic E [Xk:n] under first and second moment information on n real-valued random variables. Given means E [Xi] = μi and variances Var[Xi] = σi2, we show that the tight upper bound on the expected value of the highest-order statistic E [Xn:n] can be computed with a bisection search algorithm. An extremal discrete distribution is identified that attains the bound, and two closed-form bounds are proposed. Under additional covariance information Cov[Xi,Xj] = Qij, we show that the tight upper bound on the expected value of the highest-order statistic can be computed with semidefinite optimization. We generalize these results to find bounds on the expected value of the kth-order statistic under mean and variance information. For k < n, this bound is shown to be tight under identical means and variances. All of our results are distribution-free with no explicit assumption of independence made. Particularly, using optimization methods, we develop tractable approaches to compute bounds on the expected value of order statistics.

We study a stochastic scheduling problem of processing a set of jobs on a single machine. Each job has a random processing time Pi and a random due date Di, which are independently and exponentially distributed. The machine is subject to stochastic breakdowns in either preempt-resume or preempt-repeat patterns, with the uptimes following an exponential distribution and the downtimes (repair times) following a general distribution. The problem is to determine an optimal sequence for the machine to process all jobs so as to minimize the expected total cost comprising asymmetric earliness and tardiness penalties, in the form of E[[sum ]αi max{0,Di − Ci} + βi max{0,Ci − Di}]. We find sufficient conditions for the optimal sequences to be V-shaped with respect to {E(Pi)/αi} and {E(Pi)/βi}, respectively, which cover previous results in the literature as special cases. We also find conditions under which optimal sequences can be derived analytically. An algorithm is provided that can compute the best V-shaped sequence.

We consider a multiclass make-to-stock system served by a single server with adjustable capacity (service rate). At any point in time, the decision-maker must determine the capacity level, make a production decision (i.e., whether to produce an item to stock or to satisfy a backorder), and make a rationing decision (i.e., whether to satisfy a new order from stock or place it on backorder). In this article we characterize the structure of optimal capacity adjustment, production, and stock rationing policy for both finite- and infinite-horizon problems. We show that an optimal policy is monotone in current inventory and backorder levels, and we characterize its properties. In a numerical study we compare the optimal policy with heuristic policies and show that the savings from using an optimal policy can be significant.

Recently, Bartoszewicz [5,6] considered mixtures of exponential distributions treated as the Laplace transforms of mixing distributions and established some stochastic order relations between them: star order, dispersive order, dilation. In this article the preservation of the likelihood ratio, hazard rate, reversed hazard rate, mean residual life, and excess wealth orders under exponential mixtures is studied. Some new preservation results for the dispersive order are given, as well as the preservation of the convex transform order, and the star one is discussed.

We start with a simple derivation of an identity connecting the conditional expected residual service time as seen by an arrival and the steady-state tail distribution function of the number of customers in the system, which was previously proven by Mandelbaum and Yechiali. We then show how to use it to obtain bounds on the the stationary distribution of the number of customers in the M/G/1 queue.

In this article we introduce isotone differences stochastic ordering of Markov processes on lattice ordered state spaces as a device to compare the internal dependencies of two such processes. We derive a characterization in terms of intensity matrices. This enables us to compare the internal dependency structure of different degradable Jackson networks in which the nodes are subject to random breakdowns and repairs. We show that the performance behavior and the availability of such networks can be compared.

For the GI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

This article discusses a real-world application of a terminating two-person stochastic game. The problem comes from a Dutch television game show in which two finalists play a dice game. Each player chooses a number of dice to be rolled. The score of the roll is added to the player's total provided that none of the dice showed the outcome one. The first player reaching a prespecified number of points is the winner. This article discusses the computation and the structure of an optimal strategy.