Machine learning seems to offer the solution to many problems in user modelling. However, one tends to run into similar problems each time one tries to apply out-of-the-box solutions to machine learning. This article closely relates the user modelling problem to the machine learning problem. It explicates some inherent dilemmas that are likely to be overlooked when applying machine learning algorithms in user modelling. Some examples illustrate how specific approaches deliver satisfying results and discuss underlying assumptions on the domain or how learned hypotheses relate to the requirements on the user model. Finally, some new or underestimated approaches offering promising perspectives in combined systems are discussed. The article concludes with a tentative ‘‘checklist” that one might like to consider when planning to apply machine learning to user modelling techniques.

Methods for qualitative simulation allow predictions on the dynamics of a system to be made in the absence of quantitative information, by inferring the range of possible qualitative behaviors compatible with the structure of the system. This article reviews QSIM and other qualitative simulation methods. It discusses two problems that have seriously compromised the application of these methods to realistic problems in science and engineering: the occurrence of spurious behavior predictions and the combinatorial explosion of the number of behavior predictions. In response to these problems, related approaches for the qualitative analysis of dynamic systems have emerged: qualitative phase-space analysis and semi-quantitative simulation. The article argues for a synthesis of these approaches in order to obtain a computational framework for the qualitative analysis of dynamic systems. This should provide a solid basis for further upscaling and for the development of model-based reasoning applications of a wider scope.

Hadwiger's well known conjecture (see the survey of Toft [9]) states that any graph $G$ has a $K_{\chi(G)}$ minor, where $\chi(G)$ is the chromatic number of $G$. Let $\alpha(G)$ denote the independence (or stability) number of $G$, namely the maximum number of pairwise nonadjacent vertices in $G$. It was observed in [1], [4], [10] that via the inequality $\chi(G)\ge {|V(G)|\over \alpha(G)}$, Hadwiger's conjecture implies

Conjecture 1.1. Any graph G on n vertices contains a $K_{\lceil {n\over \alpha(G)}\rceil}$ as a minor.

Many applications of Szemerédi's Regularity Lemma for graphs are based on the following counting result. If ${\mathcal G}$ is an $s$-partite graph with partition $V({\mathcal G}) =\bigcup_{i=1}^{s} V_i$, $\vert V_i\vert =m$ for all $i\in [s]$, and all pairs $(V_i, V_j)$, $1\leq i < j\leq s$, are $\epsilon$-regular of density $d$, then $\mathcal{G}$ contains $(1\pm f(\epsilon))d^{({s\atop 2})}m^s$ cliques $K_{s}$, provided $\epsilon<\epsilon(d)$, where $f(\epsilon)$ tends to 0 as $\epsilon$ tends to 0.

Guided by the regularity lemma for 3-uniform hypergraphs established earlier by Frankl and Rödl, Nagle and Rödl proved a corresponding counting lemma. Their proof is rather technical, mostly due to the fact that the ‘quasi-random’ hypergraph arising after application of Frankl and Rödl's regularity lemma is ‘sparse’, and consequently difficult to handle.

When the ‘quasi-random’ hypergraph is ‘dense’ Kohayakawa, Rödl and Skokan (J. Combin. Theory Ser. A 97 307–352) found a simpler proof of the counting lemma. Their result applies even to $k$-uniform hypergraphs for arbitrary $k$. While the Frankl–Rödl regularity lemma will not render the dense case, in this paper, for $k=3$, we are nevertheless able to reduce the harder, sparse case to the dense case.

Namely, we prove that a ‘dense substructure’ randomly chosen from the ‘sparse $\delta$-regular structure’ is $\delta$-regular as well. This allows us to count the number of cliques (and other subhypergraphs) using the Kohayakawa–Rödl–Skokan result, and provides an alternative proof of the counting lemma in the sparse case. Since the counting lemma in the dense case applies to $k$-uniform hypergraphs for arbitrary $k$, there is a possibility that the approach of this paper can be adopted to the general case as well.

In a one-parameter model for evolution of random trees, which also includes the Barabási–Albert random graph [1], the law of large numbers and the central limit theorem are proved for the maximal degree. In the proofs martingale methods are applied.

We derive a generalization of a theorem of Raimi proving there is a partition of natural numbers with given densities of classes which meet structured translates of any other class of a partition of natural numbers.

The lamplighter group over $\Z$ is the wreath product $\Z_q \wr \Z$. With respect to a natural generating set, its Cayley graph is the Diestel–Leader graph $\mbox{\sl DL}(q,q)$. We study harmonic functions for the ‘simple’ Laplacian on this graph and, more generally, for a class of random walks on $\mbox{\sl DL}(q,r)$, where $q,r \geq 2$. The $\mbox{\sl DL}$-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.

A detachment of a graph $G$ is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. In this paper we consider the question of whether a graph $H$ is a detachment of some complete graph $K_n$. When $H$ is large and restricted to belong to certain classes of graphs, for example bounded degree planar triangle-free graphs, we obtain necessary and sufficient conditions which give a complete characterization.

A harmonious colouring of a simple graph $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. The results on detachments of complete graphs give exact results on harmonious chromatic number for many classes of graphs, as well as algorithmic results.

Let $T$ denote a real function defined on random subsets of a given family of finite sets. The random variable $T$ is decomposed into the sum of the linear, the quadratic, the cubic etc. parts which are mutually uncorrelated. Applications of this decomposition to the asymptotics of the probability distribution of $T$ (as the sizes of random subsets and of finite sets increase) are discussed.

A stable set $I$ of a graph $G$ is called $k$-extendable, $k \,{\ge}\, 1$, if there exists a stable set $X \,{\subseteq}\,V(G) {\setminus} I$ such that $|X| \,{\le}\, k$ and $|N(X) \,{\cap}\, I| \,{<}\, |X|$. A graph $G$ is called $k$-extendable if every stable set in $G$, which is not maximum, is $k$-extendable. Let us denote by ${\rm E}(k)$ the class of all $k$-extendable graphs.

We present a finite forbidden induced subgraph characterization of the maximal hereditary subclass ${\rm PE}(k)$ in ${\rm E}(k)$ for every $k \,{\ge}\,1$.

Thus, we define a hierarchy ${\rm PE}(1) \,{\subset}\, {\rm PE}(2) \,{\subset}\,{\cdots}\,{ \subset}\, {\rm PE}(k) \,{\subset}\,{ \cdots}\,$ of hereditary classes of graphs, in each of which a maximum stable set can be found in polynomial time. The hierarchy covers all graphs, and all its classes can be recognized in polynomial time.

In 1978 Erdős, Faudree, Rousseau and Schelp conjectured that \[ r ( C_{p},K_{r} ) = ( p-1 ) (r-1) +1 \] for every $p\,{\geq}\,r\,{\geq}\,3$, except for $p\,{=}\,q\,{=}\,3$. This has been proved for $r\,{\leq}\,6$, and for \[ p \geq r^{2}-2r\].

In this note we prove the conjecture for $p\,{\geq}\,4r+2$.

We prove that for all $\alpha,c>0$ and for all bipartite graphs $H$, all but at most $\alpha n$ vertices of every $cn$-regular graph $G$ whose order $n$ is sufficiently large can be covered by vertex-disjoint copies of $H$. If the vertex classes of $H$ have different size, then even all but a constant number of vertices of $G$ can be covered. This implies that for all $c>0$ and all $r\geq 4$ there exists a constant $C$ such that, in every $cn$-regular graph $G$, all but at most $C$ vertices can be covered by vertex-disjoint subdivisions of $K_r$. We also show that for $r=4,5$ one can take $C=0$.

In a 1947 paper [6], M. Kac derived the eigenvalues and eigenvectors of the probability transition matrix associated with the Ehrenfest urn model with $n$ balls, in which a ball is selected at random and moved to the other urn. The connection (see, for instance, [1, p. 19]) between the Markov chain defined by the Ehrenfest model and the nearest-neighbour uniform random walk on the abelian group $\Z^n_2$ prompted M. Kac to ask when a Markov chain may be lifted to a random walk on a group. More specifically, given a Markov chain $\{X_m\}_{m \geq 0}$ with state space $S = \{1, 2, \ldots, n \}$ and probability transition matrix ${\mbox{\bf $P$}} = [p_{ij}] (\mbox{here } p_{ij}:= P(X_{r + 1} = j | X_r = i) \mbox{ for all } r \geq 0)$, we say that the Markov chain $\{X_m\}_{m \geq 0}$ or equivalently ${\mbox{\bf $P$}}$ lifts to a random walk on a finite group $G$ if there exists a probability measure $\mu$ on $G$ and a surjective map $L:G \rightarrow S$ such that, for all $i, j \in S$, and for each $g \in L^{-1}(i)$, $$ p_{ij} = \sum_{h \in L^{-1}(j)} \!\! \mu (g^{-1}h).$$

Design is situated. “Situatedness” in designing entails the explicit consideration of the state of the environment, the knowledge and experiences of the designer, and the interactions between the designer and the environment during designing. Central to the notion of situatedness is the notion of design situation and constructive memory. A design situation models a particular state of interaction between a design agent and the environment at a particular point in time. Memory construction occurs whenever a design agent uses past experiences and knowledge within the current design environment in a situated manner. This paper is concerned with the development of an agent-based computational design tool that takes into consideration the notion of situatedness in designing. A key element of this tool is a constructive memory system that supports the dynamic nature of designing. Memories of past experiences are constructed as required by the current situation, and past experiences are refined for future utility according to the current interactions between the agent and the environment. This latter case of knowledge improvement is illustrated through a series of experiments that demonstrates the effect of grounding on the operating modes and responses of a constructive memory system.

Automated support of design teams, consisting of both human and automated systems, requires an understanding of the role of trust in distributed design processes. By explicating trust, an individual designer's decisions become better understood and may be better supported. Each individual designer has his or her private goals in a cooperative design setting, in which requirement conflicts and resource competitions abound. However, there are group goals that also need to be reached. This paper presents an overview of research related to trust in the context of agents and design, a computational knowledge-level model of trust based on the seven beliefs distinguished by Castelfranchi and Falcone, and an example of the use of the trust model in a specific design process, namely, Website design from the perspective of a single designer. The results are discussed in the context of distributed design in open systems.

Organizational research has shown that effectively structuring the resources (human, informational, computational) available to an organization can significantly improve its collective computational capacity. Central to this improved capacity is the manner in which the organization's member agents are related. This study is an initial investigation into the role and potential of interagent ties in computational teaming. A computational team-based model, designed to more fully integrate agent ties, is created and presented. It is applied to a bulk manufacturing process-planning problem and its performance compared against a previously tested agent-based algorithm without these agent relationships. The performance of the new agent method showed significant improvement over the previous method: improving solution quality 280% and increasing solution identification per unit time an entire order of magnitude. A statistical examination of the new algorithm confirms that agent interdependencies are the strongest and most consistent performance effects leading to the observed improvements. This study illustrates that the interagent ties associated with team collaboration can be a highly effective method of improving computational design performance, and the results are promising indications that the application of organization constructs within a computational context may significantly improve computational problem solving.

This Special Issue had its genesis in an international Workshop on Agents in Design held in June 2002, at MIT by the Guest Editors. Computational agents have been developed within the artificial intelligence community over an extended period. The concept of an agent can be traced to Carl Hewitt's 1977 work on “actors.” Hewitt defined actors as self-contained, interactive, and concurrently executing objects. Since then, considerable research has gone into developing the concept of an agent and into formalizing agents, developing multiagent systems, and exploring their use. The use of agents in design is more recent, and the first PhDs in the area appeared in the early 1990s. Although a precise and unique definition of an agent has yet to be agreed upon, one distinguishing characteristic of an agent is that it exhibits autonomous behavior. Research on agents in design focuses on two primary areas: how to make agents useful in design, and how to apply them to design tasks. This Special Issue has papers from both areas.

This paper presents a possible future direction for agent-based simulation using complex agents that can learn from experience and report their individual evaluations. Adding learning to the agent model permits the simulation of potentially important agent behavior such as curiosity. The agents can then report evaluations of a design that are situated in their individual experience. The paper describes the architecture of curious agents used in the situated evaluation of designs. It then describes an example of the application of such curious agents in the evaluation of the curating of an exhibition in an art gallery.