Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main results of the paper, concerning when a category obtained by Artin glueing is a topos, we feel it is appropriate to publish this correction in the hope that it may prevent future readers of Carboni and Johnstone (1995) from being misled. We are grateful to Tom Leinster and to Marek Zawadowski for drawing our attention to the two errors.

In this paper we interpret (fragments of) intuitionistic logic in categories with weak closure properties, such as quasi left exact categories and locally cartesian closed categories (LCCC) with sums. We also interpret the full choice scheme in an LCCC. The interpretation can be seen as a categorical form of the usual Brouwer–Heyting– Kolmogorov (BHK) interpretation. The standard interpretation of geometric logic in a pretopos is obtained by applying the image functor to the BHK-interpretation.

This paper initiates the study of a process algebra based on atomic actions that are assigned resources, and that supports true concurrency. By true concurrency we mean that the parallel composition of concurrent processes does not rely on an interleaving of concurrent actions for its definition. Our process algebra includes a number of interesting operators that can be defined using resources of atomic actions to control their behaviour: of particular note is a (weak) sequential composition operator that exploits the truly concurrent nature of the semantics; this operator extends significantly the operation of prefixing by atomic actions that is supported in most truly concurrent semantics. Our language also includes a parallel composition operator that allows local events to execute asynchronously, while requiring synchronising events to execute simultaneously. In addition, the language supports a restriction operator and includes (unguarded) recursion.

We present both a denotational semantics and a companion operational semantics for our language. The denotational semantics supports true concurrency, so that parallel composition is defined without non-determinism or interleaving. This semantics also is novel for its treatment of recursion. The meaning of a recursive process is defined using a least fixed point on a subdomain that is determined by the body of the recursion, and that varies from one process to another. Nonetheless, the recursion operators in the language have continuous interpretations in the denotational model. In fact, our denotational model is based on a domain-theoretic generalisation of Mazurkiewicz traces in which the concatenation operator, as well as the other operators from our language, can be given continuous interpretations.

The operational model is presented in a natural SOS style. We prove a congruence theorem relating the two semantics, which implies the operational model itself is compositional. The congruence theorem also implies the denotational model is adequate with respect to the operational semantics, and we characterise the relatively mild conditions under which the denotational semantics is fully abstract with respect to the operational semantics.

This paper introduces $\lambda^\widehat$, a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system $\lambda_{\mathcal{G}}$ in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to support coinductive types and corecursive function definitions also.

In the paper Métayer (2001), Métayer transforms multiplicative proof-structures into orientable surfaces with boundaries. He investigates the link between the topological complexity and the number of exchanges in a sequentialisation. The theorem he achieves is about a particular rule of exchange (transpositions by blocks). We complete his approach by showing that the topological complexity does not provide any information in other cases (arbitrary exchange, upper bound of the number of exchanges). Then, we show that, on the other hand, the surface associated to a proof-structure is the surface of minimal complexity on which the proof can be drawn without crossing and respecting the local orientation.

We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is, in a sense, possible. The resulting structure is called a wheel. Wheels are similar to rings, but $0x=0$ does not hold in general; the subset $\{x|0x=0\}$ of any wheel is a commutative ring (or semiring), and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural, and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings.

Supply interruptions resulting from unpredictable events, such as machine breakdowns, order cancellations, unscheduled maintenance, and labor strikes can produce adverse effects on the production/inventory system. In this article, we consider a periodic-review inventory system subject to random demand and unreliable supply. The availability of supply is modeled as an alternating renewal process with general distributions for the durations of the UP and DOWN cycles. We consider the lost-sales case and also discuss the backorder case, for both the discounted and long-run average cost criteria. For the linear cost model, we derive the structural properties and bounds of the optimal policy. We also propose the “end-of-cycle” inventory return contract and show that it may be mutually beneficial to both the firm and the supplier.

A functional programming language can be taught successfully as a first language, but if there is no follow up the students do not appreciate the functional approach. Following discussions concerning this issue at the 1995 FPLE conference (Hartel & Plasmeijer, 1995), we decided to develop such a follow up by writing a book that teaches C to students who can write simple functional programs. This paper summarises the essence of our approach, which is based on program transformation, and presents our experience teaching functional C at the Universities of Southampton and Bristol.

Let F be the common distribution function of the increments of a random walk {Sn, n ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

Carousels are rotatable closed-loop storage systems for small items, where items are stored in bins along the loop. An order at a carousel consists of (say) n different items stored there. We analyze two problems: (1) minimizing the total time to fill an order (travel time) and (2) order delays as they arrive, are filled, and depart. We define clumpy orders and the nearest-end-point heuristic (NEPH) for picking them. We determine conditions for NEPH to be optimal for problem (1), and under a weak stochastic assumption, we derive the distribution of travel time. We compare NEPH with the nearest-item heuristic. Under Poisson arrivals and assumptions much weaker than in the literature, we show that problem (2) may be modeled as an M/G/1 queue.

We derive convexity results and related properties for the value functions of tandem queuing systems. The results for standard multiserver queues are new. For completeness, we also prove and generalize existing results on tandems of controllable queues. The results can be used to compare queuing systems. This is done for systems with and without batch arrivals and for systems with different numbers of on–off sources.