In 1972, Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere Sd in such a way that two vertices joined by an edge have distance more than (ie, distance more than 2π/3 on the sphere). In 1978, Larman [LAR] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed . In addition, he conjectured that the right answer would be , which is not better than the class of all graphs. Larman'sconjecture was independently proved by Rosenfeld [MR] and Rödl [VR[. In this last paper it was shown that no bound better than can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is stilltrue for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree.

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].

We define the γ invariant as the coefficient of x1 in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1 in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1 when n true twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

This paper describes the development of the PALS system, an implementation of Prolog capable of efficiently exploiting or-parallelism on distributed-memory platforms—specifically Beowulf clusters. PALS makes use of a novel technique, called incremental stack-splitting. The technique proposed builds on the stack-splitting approach, previously described by the authors and experimentally validated on shared-memory systems, which in turn is an evolution of the stack-copying method used in a variety of parallel logic and constraint systems—e.g., MUSE, YAP, and Penny. The PALS system is the first distributed or-parallel implementation of Prolog based on the stack-splitting method ever realized. The results presented confirm the superiority of this method as a simple yet effective technique to transition from shared-memory to distributed-memory systems. PALS extends stack-splitting by combining it with incremental copying; the paper provides a description of the implementation of PALS, including details of how distributed scheduling is handled. We also investigate methodologies to effectively support order-sensitive predicates (e.g., side-effects) in the context of the stack-splitting scheme. Experimental results obtained from running PALS on both Shared Memory and Beowulf systems are presented and analyzed.

In this article we determine a formula for fractal and resistance dimensions of two models of uniformly bounded random trees. The type (transient or recurrent) of the random walk on such trees is ascribed, to some extent, to these dimensions. The results presented in this article generalize some of the results of [6] and [7].

Consider a system of queuing stations in tandem having both flexible servers (who are capable of working at multiple stations) and dedicated servers (who can only work at the station to which they are dedicated). We study the dynamic assignment of servers to stations in such systems with the goal of maximizing the long-run average throughput. We also investigate how the number of flexible servers influences the throughput and compare the improvement that is obtained by cross-training another server (i.e., increasing flexibility) with the improvement obtained by adding a resource (i.e., a new server or a buffer space). Finally, we show that having only one flexible server is sufficient for achieving near-optimal throughput in certain systems with moderate to large buffer sizes (the optimal throughput is attained by having all servers flexible). Our focus is on systems with generalist servers who are equally skilled at all tasks, but we also consider systems with arbitrary service rates.

We study a cumulative storage system that is totally cleared sporadically at stationary renewal times and whenever a finite-capacity threshold is exceeded. The independent and identically distributed inputs occur at time epochs that also form a stationary renewal process. We determine the distribution of the interoverflow times. Although this distribution is quite intricate when both underlying renewal processes are general, in the special case of Poisson sporadic clearings we obtain a neat formula for its Laplace transform.

Considered are semi-Markov decision processes (SMDPs) with finite state and action spaces. We study two criteria: the expected average reward per unit time subject to a sample path constraint on the average cost per unit time and the expected time-average variability. Under a certain condition, for communicating SMDPs, we construct (randomized) stationary policies that are ε-optimal for each criterion; the policy is optimal for the first criterion under the unichain assumption and the policy is optimal and pure for a specific variability function in the second criterion. For general multichain SMDPs, by using a state space decomposition approach, similar results are obtained.

Let X1, … , Xn be independent random variables with Xi having survival function λi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution , where = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n − X1:n is larger than Yn:n − Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

Consider a single machine that can process multiple jobs in batch mode. We have n jobs and the processing time of job j is a random variable Xj with distribution Fj. Up to b jobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions, the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time but different variances. We show that for certain special classes of processing time distributions the Smallest Variance First rule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.

In this article we investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative. The main structural results are applied to several kinds of ordered random variable, such as delayed record values, continuous and discrete ℓ∞⩽-spherical order statistics, epoch times of mixed Poisson processes, generalized order statistics, discrete weak record values, and epoch times of modified geometric processes. These applications extend the main results for ordinary order statistics in Qi [28] and for usual record values in Nagaraja and Nevzorov [25].

The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

Most programming languages adopt static binding, but for distributed programming an exclusive reliance on static binding is too restrictive: dynamic binding is required in various guises, for example, when a marshalled value is received from the network, containing identifiers that must be rebound to local resources. Typically, it is provided only by ad hoc mechanisms that lack clean semantics. In this paper, we adopt a foundational approach, developing core dynamic rebinding mechanisms as extensions to the simply typed call-by-value λ calculus. To do so, we must first explore refinements of the call-by-value reduction strategy that delay instantiation, to ensure computations make use of the most recent versions of rebound definitions. We introduce redex-time and destruct-time strategies. The latter forms the basis for a λmarsh calculus that supports dynamic rebinding of marshalled values, while remaining as far as possible statically typed. We sketch an extension of λmarsh with concurrency and communication, giving examples showing how wrappers for encapsulating untrusted code can be expressed. Finally, we show that a high-level semantics for dynamic updating can also be based on the destruct-time strategy, defining a λupdate calculus with simple primitives to provide type-safe updating of running code. We show how the ideas of this simple calculus extend to more real-world, module-level dynamic updating in the style of Erlang. We thereby establish primitives and a common semantic foundation for a variety of real-world dynamic rebinding requirements.

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models called classical categories that is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models called Dummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.

Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

We propose a mathematical semantics for event-based architectures that serves two main purposes: to characterise the modularisation properties that result from the algebraic structures induced on systems by this discipline of coordination; and to further validate and extend the categorical approach to architectural modelling that we have been building around the language CommUnity with the ‘implicit invocation’, also known as ‘publish/subscribe’ architectural style. We then use this formalisation to bring together synchronous and asynchronous interactions within the same modelling approach. We see this effort as a first step towards a form of engineering of architectural styles. Our approach adopts transition systems extended with events as a mathematical model of implicit invocation, and a family of logics that support abstract levels of modelling.

It is an empirical observation arising from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene–Kreisel continuous functionals, its effective substructure Ceff and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously.

In this paper we present some new results that go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, Ceff or HEO (as appropriate). We obtain versions of our results for both the ‘standard’ and ‘modified’ extensional collapse constructions. The proofs make essential use of a technique due to Normann.

Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the three type structures under consideration are highly canonical mathematical objects.

Object-oriented (OO) programming techniques can be applied to equational specification logics by distinguishing visible data from hidden data (that is, by distinguishing the output of methods from the objects to which the methods apply), and then focusing on the behavioural equivalence of hidden data in the sense introduced by H. Reichel in 1984. Equational specification logics structured in this way are called hidden equational logics, HELs. The central problem is how to extend the specification of a given HEL to a specification of behavioural equivalence in a computationally effective way. S. Buss and G. Roşu showed in 2000 that this is not possible in general, but much work has been done on the partial specification of behavioural equivalence for a wide class of HELs. The OO connection suggests the use of coalgebraic methods, and J. Goguen and his collaborators have developed coinductive processes that depend on an appropriate choice of a cobasis, which is a special set of contexts that generates a subset of the behavioural equivalence relation. In this paper the theoretical aspects of coinduction are investigated, specifically its role as a supplement to standard equational logic for determining behavioural equivalence. Various forms of coinduction are explored. A simple characterisation is given of those HELs that are behaviourally specifiable. Those sets of conditional equations that constitute a complete, finite cobasis for a HEL are characterised in terms of the HEL's specification. Behavioural equivalence, in the form of logical equivalence, is also an important concept for single-sorted logics, for example, sentential logics such as the classical propositional logic. The paper is an application of the methods developed through the extensive work that has been done in this area on HELs, and to a broader class of logics that encompasses both sentential logics and HELs.

This is the second part of a special issue in honour of Klaus Keimel – the first part appeared in Mathematical Structures in Computer Science16 (2). This second part consists of a single paper by John Longley, which could not be included in the earlier issue for reasons of size.

We consider the parallel approximability of two problems arisingfrom high multiplicity scheduling, namely the unweightedmodel with variable processing requirements and the weighted model with identical processing requirements. These twoproblems are known to be modelled by a class of quadratic programsthat are efficiently solvable in polynomial time. On the parallelsetting, both problems are P-complete and hence cannot beefficiently solved in parallel unless P = NC. To deal with theparallel approximablity of these problems, we show first aparallel additive approximation procedure to a subclass ofmulti-valued quadratic programming, called smooth multi-valuedQP, which is defined by imposing certain restrictions onthe coefficients of the instance. We use this procedure to obtainparallel approximation to dense instances of the two problems by observing that denseinstances of these problems are instances of smooth multi-valuedQP. The dense instances of the problemsconsidered here are defined similarly as for other combinatorialproblems in the literature. For such instances we can find inparallel a near optimal schedule. The definition of smoothmulti-valued QP as well as the procedure forapproximating it in parallel are of interest independently of theapplication to the scheduling problems considered in this paper.

We study the problem of learning regular tree languages from text. We show that the framework of function distinguishability, as introduced by the author in [Theoret. Comput. Sci.290 (2003) 1679–1711], can be generalized from the case of string languages towards tree languages. This provides a large source of identifiable classes of regular tree languages. Each of these classes can be characterized in various ways. Moreover, we present a generic inference algorithm with polynomial update time and prove its correctness. In this way, we generalize previous works of Angluin, Sakakibara and ourselves. Moreover, we show that this way all regular tree languages can be approximately identified.