We have developed an ensemble-based approach for online machine learning: adaptive rotation forest and AD-WIN adaptive rotation forest. We focused on rotation forest, an offline supervised ensemble algorithm with a particularly high prediction accuracy while all the features are continuous. Our objective was to develop a high-performance online ensemble method that uses a process similar to that of rotation forest in an online environment. Our experiments demonstrated that the proposed approach simplifies the tree structure used for the base learners, reduces memory consumption, and improves prediction accuracy for some data streams.

Factorization machines (FMs) are machine learning predictive models based on second-order feature interactions and FMs with sparse regularization are called sparse FMs. Such regularizations enable feature selection, which selects the most relevant features for accurate prediction, and therefore they can contribute to the improvement of the model accuracy and interpretability. However, because FMs use second-order feature interactions, the selection of features often causes the loss of many relevant feature interactions in the resultant models. In such cases, FMs with regularization specially designed for feature interaction selection trying to achieve interaction-level sparsity may be preferred instead of those just for feature selection trying to achieve feature-level sparsity. In this paper, we present a new regularization scheme for feature interaction selection in FMs. For feature interaction selection, our proposed regularizer makes the feature interaction matrix sparse without a restriction on sparsity patterns imposed by the existing methods. We also describe efficient proximal algorithms for the proposed FMs and how our ideas can be applied or extended to feature selection and other related models such as higher-order FMs and the all-subsets model. The analysis and experimental results on synthetic and real-world datasets show the effectiveness of the proposed methods.

A number of knowledge graph (KG)-based recommendation algorithms have been introduced; KGs enable users and items and their attributes to be treated in an integrated way and structural information to be captured through graphs. There are many variations of the KG based recommendation algorithms. Among them, KG embedding is often used, but doing this does not take advantage of the meta-path-level proximity between users and items. This paper presents a flexible framework combining random walk and KG embedding methods. The random walk model is formulated on the basis of the similarity between nodes revealed by the KG embedding. This enables the metapath level proximity of users and items to be efficiently utilized. Comparison testing demonstrated that the proposed framework performs better than random- walk-only methods and KG-embedding-only methods, and slightly better than the existing method we have extended.

Counterfactual explanations help users to understand the behaviors of machine learning models by changing the inputs for the existing outputs. For an image classification task, an example counterfactual visual explanation explains: "for an example that belongs to class A, what changes do we need to make to the input so that the output is more inclined to class B." Our research considers changing the attribute description text of class A on the basis of the attributes of class B and generating counterfactual images on the basis of the modified text. We can use the prediction results of the model on counterfactual images to find the attributes that have the greatest effect when the model is predicting classes A and B. We applied our method to a fine-grained image classification dataset and used the generative adversarial network to generate natural counterfactual visual explanations. To evaluate these explanations, we used them to assist crowdsourcing workers in an image classification task. We found that, within a specific range, they improved classification accuracy.

Molecular graphs are one of the established representations for small molecules, and even steric or electronic information can be encoded as node and edge features. Naturally, graph neural networks have been intensively investigated to solve various chemical problems at molecular levels. However, it remains unclear how to encode relevant chemical information into graphs. We investigate this problem by proposing three models of graph neural networks with self-attention mechanisms at different levels to adaptively select relevant chemical information for each input. Using neural graph fingerprint (NFP) as a baseline, we introduce three types of attention mechanisms on the top of NFPs. Our experimental evaluations suggest that introducing these self-attention mechanisms contributes to not only improving the prediction accuracy but also providing quantitative interpretation using obtained attention coefficients.

Recommender systems using a knowledge graph can comprehensively organize users and items and their attributes and thereby improve recommendation performance. In addition, the relationship between users and items can be easily interpreted on the basis of entities and relations, thus giving explanations to recommendations. The algorithms and knowledge graphs used for generating explanations have not utilized review text. We have developed a recommendation method for predicting interactions between users and items using a knowledge graph and review text. The underlying user-item relationships are reflected and explanations are generated by predicting user-item interactions from the paths between a user and an item. The modeling is done using a recurrent neural network or a factorization machine. Items' aspects that interest users are extracted from review text and leveraged using an attention-like mechanism. Since the path between a user and an item can be easily interpreted, and the important aspects between a user and an item can be interpreted by observing the attention weight, the proposed model can generate a reasonable recommendation explanation. Testing using a real-world dataset demonstrated that the proposed model can explain the recommendations.

Although supervised learning requires a labeled dataset, obtaining labels from experts is generally expensive. For this reason, crowdsourcing services are attracting attention in the field of machine learning as a way to collect labels at relatively low cost. However, the labels obtained by crowdsourcing, i.e., from non-expert workers, are often noisy. A number of methods have thus been devised for inferring true labels, and several methods have been proposed for learning classifiers directly from crowdsourced labels, referred to as "learning from crowds." A more practical problem is learning from crowdsourced labeled data and unlabeled data, i.e., "semi-supervised learning from crowds." This paper presents a novel generative model of the labeling process in crowdsourcing. It leverages unlabeled data effectively by introducing latent features and a data distribution. Because the data distribution can be complicated, we use a deep neural network for the data distribution. Therefore, our model can be regarded as a kind of deep generative model. The problems caused by the intractability of latent variable posteriors is solved by introducing an inference model. The experiments show that it outperforms four existing models, including a baseline model, on the MNIST dataset with simulated workers and the Rotten Tomatoes movie review dataset with Amazon Mechanical Turk workers.

Explaining recommendations helps users to make more accurate and effective decisions and improves system credibility and transparency. Current explainable recommender systems tend to provide fixed statements such as "customers who purchased this item also purchased....". This explanation is generated only on the basis of the purchase history of similar customers, so it does not include the preferences of customers who have purchased the item or a description of the item. Since user-generated reviews generally contain information about the reviewer's preferences and a description of the item, such reviews typically have more effect on purchase decisions. Therefore, using reviews to explain recommendations should be more useful than providing only a fixed statement explanation. Aiming to create a system that provides personalized explanations for recommendations, we have developed a recurrent neural network model that uses multicriteria evaluation data to generate reviews.

Artificial intelligence has shown remarkable performance in perfect information games. However, it is still no match for human players when it comes to most imperfect information games. Information set Monte Carlo tree search (ISMCTS) has been developed to reduce the effects of strategy fusion caused by determinization of the imperfect information and demonstrated advantages over the conventional Monte Carlo tree search (MCTS) that uses determinization. Because ISMCTS has only been used for games with relatively simple structure, it is still unknown whether it works effectively for more complex games. In this study, we take Pokemon as an example of a complex imperfect information game and implement a simulator to evaluate the effectiveness of ISMCTS. Experimental results show that ISMCTS outperforms the conventional MCTS that uses determinization.

A method is presented for predicting the weights used in neural networks that shortens the training time by omitting some weight updates. In general neural network training, convergence slows as the dimensions of the input data, the amount of data, and/or the complexity of the model increases. The proposed method advances convergence and facilitates the implementations of neural networks on low-performance hardware such as mobile devices. Experiments using the MNIST, CIFAR-10, and SVHN datasets demonstrated that the proposed weight prediction method shortens the training time.

Many organizations provide open data, and important insights can be gained by analyzing such data. Analysis of the potential causal relationships is a complex task. We have developed a framework for analyzing causal relationships that combines the intelligence of the crowd with state-of-the-art machine learning methods. The proposed framework takes into account the effect of possible confounding in causal analysis by collecting explanations of the correlation between variables. The validity of the collected explanations is tested using a causal discovery workflow including a conditional independence test step and a causal direction inference step. Application of this framework to data obtained from the World Bank Data website and open government data sources revealed several interesting causal relationships. The results demonstrate that the proposed framework can efficiently perform causal analysis of open data.

We present a comprehensive framework of search methods, such as simulated annealing and batch training, for solving nonconvex optimization problems. These methods search a wider range by gradually decreasing the randomness added to the standard gradient descent method. The formulation that we define on the basis of this framework can be directly applied to neural network training. This produces an effective approach that gradually increases batch size during training. We also explain why large batch training degrades generalization performance, which previous studies have not clarified.

Ratings by users on various items such as hotels and movies have become easily available on the Web. In many cases, other than overall rating for each item by each user, more detailed information such as ratings from different viewpoints and free text comments, as well as aggregated information such as the average of ratings by different users, are also available. We investigated the effectiveness of six existing collaborative filtering methods for large-scale sparse multicriteria rating data. We formulated rating aggregation as a collaborative filtering problem and applied six collaborative filtering methods to it. Furthermore, we extended three of the methods to calculate user similarity using indirect users and review comments and applied them to collaborative filtering and rating aggregation. The results show that multicriteria rating approaches perform better than single criterion rating approaches. The extended methods had better performance both in collaborative filtering and in rating aggregation.

Theory exploration has been investigated as the lemma generation methods which play important role in automation of theorem provers. In order to enlarge the scope of provable theorems in the exploration, in this paper we propose an approach of applying the rewriting induction technique in exploration of inductive theorems. Especially, we propose some heuristics for proof search in the rewriting induction. In the experimentation using two examples, the proposed heuristics improve the efficiency without sacrificing the number of successes in the proof search.

Machine learning is the basis of important advances in artificial intelligence. Unlike the general methods of machine learning, which use the same tasks for training and testing, the method of transfer learning uses different tasks to learn a new task. Among the various transfer learning algorithms in the literature, we focus on the attribute-based transfer learning. This algorithm realizes transfer learning by introducing attributes and transferring the results of training to another task with the common attributes. However, the existing method does not consider the frequency in which each attribute appears in feature vectors (called the observation probability). In this paper, we present a generative model with the observation probability. By the experiments, we show that the proposed method has achieved a higher accuracy rate than the existing method. Moreover, we see that it makes possible the incremental learning that was impossible in the existing method.

Software testing in object-oriented programming requires not only test input data, but also method sequences. Method sequences create new instances and change object states as desired. Method sequence generation to get desired object states is challenging. Seeker is one of the state-of-the-art implementation to generate test cases with method sequences. However, the technique generates method sequences to change a value of only a single variable. Therefore, it cannot cover the branches that need to modify values of multiple variables. In this paper, we extend the technique in order to cover branches which require multiple desired object states. The proposed approach detects all related variables in uncovered branches and uses a fitness function to give method sequences an evaluation value to reduce candidate methods. We applied Seeker and the proposed approach to four open source projects. The result shows a 1-5% improvement of branch coverage over Seeker and also indicates that the effectiveness of the proposed approach varies depending on some specific features of projects.

Monophonic sound source separation is an essential subject on the fields where sound, such as voice, music and noise, is dealt with. In particular, unsupervised approaches to this problem have high versatility in comparison with supervised approaches. Non-negative matrix factorization is the most frequently used algorithm for the monophonic sound source separation without prior knowledge. This is also applied to various applications, including data clustering, face recognition, gene expression classification. However, non-negative matrix factorization cannot be efficiently used in online learning. In order to solve this difficulty, the non-negative sparse autoencoder was proposed in the literature. Although several successful applications have been reported, this is not yet applied to the monophonic sound source separation. This paper shows that the non-negative sparse autoencoder can perform the monophonic sound source separation without prior knowledge in online learning.

Self-organizing neural networks are important tools for realizing unsupervised learning. Recently, a difficult task has involved the incremental, efficient and robust learning in noisy environments. Most of the existing techniques are poor in this regard. In this paper, we first propose a new topology generating method called enhanced competitive Hebbian learning (enhanced CHL), and then propose a novel incremental self-organizing neural network based on the enhanced CHL method, called enhanced incremental growing neural gas (Hi-GNG). The experiments presented in this paper show that the Hi-GNG algorithm can automatically and efficiently generate a topological structure with a suitable number of neurons and that the proposed algorithm is robust to noisy data.

Knuth–Bendix completion is a classical calculus in automated deduction for transforming a set of equations into a confluent and terminating set of directed equations which can be used to decide the induced equational theory. Multi-completion with termination tools constitutes an approach that differs from the classical method in two respects: (1) external termination tools replace the reduction order—a typically critical parameter—as proposed by Wehrman et al. (2006), and (2) multi-completion as introduced by Kurihara and Kondo (1999) is used to keep track of multiple orientations in parallel while exploiting sharing to boost efficiency. In this paper we describe the inference system, give the full proof of its correctness and comment on completeness issues. Critical pair criteria and isomorphisms are presented as refinements together with all proofs. We furthermore describe the implementation of our approach in the tool $\mathsf{mkbTT}$ , present extensive experimental results and report on new completions.

An inductive theorem is an equation over terms which holds on recursively-defined data structure. In the field of formal verification of information system, inductive theorem proving plays an important role. Rewriting induction(RI) is a method for inductive theorem proving proposed by Reddy. Multi-context rewriting induction with termination checker (MRIt) proposed by Sato is a variant of RI which tries to find a suitable context for induction automatically. However, MRIt should perform a large amount of termination checking of term rewriting systems which is becoming efficiency bottleneck. In this paper, we propose a method of parallelizing the termination checker used in MRIt based on the lexicographic path order method to improve its efficiency. For implementation, we used the functional programming language Erlang. We discuss the efficiency of our implementation based on the experiments.

Self-organizing neural network which is an unsupervised learning algorithm is to discover the inherent relationships of data. Such technique has become an important tool for data mining, machine learning and pattern recognition. Most self-organizing neural networks have a difficulty in reflecting data distributions precisely if data distributions are very complex. And meanwhile, it is also hard for these algorithms to learn new data incrementally without destroying the previous learnt data. In this paper, we propose a robust energy artificial neuron based incremental self-organizing neural network with a dynamic structure (REISOD). It can adjust the scale of network automatically to adapt the scale of the data set and learn new data incrementally with preserving the former learnt results. Moreover, several experiments show that our algorithm can reflect data distributions precisely.

In this paper, we demonstrate through a case study the effectiveness of a description method that we developed as a system analysis tool to describe the structure of the security of systems. We apply the description method in design assessment of a digital right management system which has to meet complex security requirements. This method, based on the assumption of the use of standard encryption technologies and existing cryptographic protocols, reveals hidden security threats and vulnerabilities of systems. It extracts only security elements that constitute the trust relationship of system components, describing the relation between the elements, and analyzing the relation. This method provides a valuable assistance tool for frontline engineers in system development fields to build secure systems, and an efficient communication paradigm between stakeholders of a system to help them in understanding the security of the system and confirming that their security requirements are fulfilled.

Consensus formation models are mathematical models for obtaining potential candidates for a group when all group members pursue their individual preferences. In such models, the members often change their preferences based on the group's global preference which summarizes the overall individual preferences. Based on the theory of partially-filled reachability matrices, the Flexible Interpretive Structural Modeling (FISM) provides a method for developing structural models of complex systems and is expected to be applied to decision-making support systems. In this paper, we propose a method for simulating consensus formation in FISM and study its effectiveness through the experiments with several consensus strategies.

There are a lot of works for software development to mine snippets of code from a large corpus. A new technique, keyword programming is very different from those related works. The input keywords are free from type restriction and this technique generates new code. Because of not requiring a large corpus, it is useful for small projects just started. Our main contributions for this keyword programming are improvement of algorithm so that it outputs multiple candidates rather than one, sorted in the order based on their evaluation taking a frequency of use into account. We have implemented these ideas as Eclipse' s plug-in. We have experimented with this tool and the results show that this tool is more useful than the existing tool.

Model Checking is an automatic technique for verifying finite-state concurrent systems such as communication protocols and sequential circuit designs. It has a number of advantages over traditional approaches to this problem that are based on simulation, testing, and deductive reasoning. Model checking tools are, however, not widely introduced into industry, and one of the reasons is that they are tricky and difficult to use for engineers. In this paper, we extend the functions of Java PathFinder, a software model checker to verify executable Java bytecode programs, and propose a graphical user interface with a high degree of usability. Our GUI depicts the state transition graphs of not only a whole program but also each thread as a result of verification. Users can get much information through the GUI, for example, the internal states of a program and the correspondence relation between the graphs, with interactive mouse operation. One of the key features of our GUI is a variable-value-based graph abstraction that allows users to focus upon an aspect they are interested in. Our GUI also has an intuitively easy-to-use interface for users to input linear temporal logic (LTL) formulae as a program specification based on the Specification Pattern System.

Consensus formation models are useful for obtaining harmonious consensus in a group when all group members pursue their individual preferences. In the process of consensus formation, each agent accepts macro-information and adapts its preference to the overall group preference. Flexible Interpretive Structural Modeling (FISM) is a method for developing structural models of complex systems. In this paper, we propose consensus formation models based on FISM and analyze some quantitative measures of the difference between individual preferences and the agreed preference.

In order to group web pages, vector space models for analyzing documents similarities have been often used variously. However, they are not used for analyzing link structures, partly because they are complex and links do not necessarily satisfy the similarity relation. If we can devise vector space models for link structures, we can combine them with those models for document similarity in order to develop the unified basis for grouping web pages. In this paper, we present a vector space model for link structures, based on the notion of link vectors, the specifically designed characteristic vectors for link structures. We also discuss the extension of this model to the model called content-link vector space model, which can treat document information and link information of web pages in a unified way. The preliminary experiments show that the models show good performance even when document information is ignored.

In equational theorem proving, convergent term rewriting systems play a crucial role. In order to compute convergent term rewriting systems, the standard completion procedure (KB) was proposed by Knuth and Bendix and has been improved in a various way. The multi-completion system MKB developed by Kurihara and Kondo accepts as input a set of reduction orders in addition to equations and efficiently simulates parallel processes each of which executes the KB procedure with one of the given orderings. Wehrman and Stump also developed a new variant of completion procedure, constraint-based completion, in which reduction orders need not be given by using automated modern termination checker. As a result, the constraint-based procedures simulate the execution of parallel KB processes in a sequential way, but the naive breadth-first search sometimes causes serious inefficiency when the number of the potential reduction orders is large. We present a new procedure, called a constraint-based multi-completion procedure MKBcs, by augmenting the constraint-based completion with the framework of the multi-completion suppressing the combinatorial explosion by sharing inferences among the processes.

A fuzzy constraint satisfaction problem is an extension of the classical CSP, a powerful tool for modeling various problems based on constraints among variables, and a dynamic CSP is a framework for modelling the transformation of problems. These schemes are the technique to formulate real world problems as CSPs more easily. The CSP model that combines these is already has splendid researches. The fuzzy local change algorithm is practicable enough in small-scale problems, but larger problems require the use of approximate methods. The algorithms for solving CSPs are classified into two categories: systematic searches (complete methods based on search trees), and local searches (approximate methods based on iterative improvement). Both have advantages and disadvantages. In the work reported in this paper we tested a hybrid approximate method, called the spread-repair-shrink algorithm, on dynamic, large-scale problems. The algorithm repairs local constraints by repeatedly spreading and shrinking a set of search trees until the degree to which the worst constraints (the roots of the trees) are satisfied is improved. In this process, the ldquostabilityrdquo of solutions can be maintained because the reassignment is locally limited. Additionaly, we innovate SRSD filter as after filtering. We empirically show that spread-repair-shrink and SRSD algorithm keep the stability of solutions rather than other algorithms. It is able to quickly get a good-quality approximate and stabile solution to a large problem.

A fuzzy constraint satisfaction problem (FCSP) is an extension of the classical CSP, a powerful tool for modeling various problems based on constraints among variables. Basically, the algorithms for solving CSPs are classified into two categories: the systematic search (complete methods based on search trees) and the local search (approximate methods based on iterative improvement). Both have merits and demerits. Recently, much attention has been paid to hybrid methods for integrating both merits to solve CSPs efficiently, but almost no attempt has been made so far for solving FCSPs. In this paper, we present a hybrid, approximate method for solving FCSPs. The method, called the spread-repair shrink (SRS) algorithm, combines a systematic .search with the spread-repair (SR) algorithm, a local search method recently developed by the authors. SRS repeats spreading and shrinking a set of search trees in order to repair local constraints until the satisfaction degree of the worst constraints (which arc the roots of the trees) is improved. We empirically show that SRS outperforms SR and other well-known methods such as Forward Checking and Fuzzy GENET, when we want to quickly get a good-quality approximate solution of sufficiently large size of problems.

We consider the connectivity of a class of random graph models called graphs defined by the fixed radius model . The model has a clear relationship with state-of-the-art wireless communication networks, often called ad-hoc networks. In this model, each random graph is defined by nodes placed in a Euclidean plane randomly according to some distribution; each pair of nodes is connected by an edge if and only if the distance between the nodes is within the common radius . Hence the model can be naturally interpreted as a mathematical model of wireless communication networks in which every mobile node can communicate with other nodes within the distance In this paper, we present some analytical results concerning the probability that such random graphs are connected (i.e., there is a path between every pair of nodes), assuming that the fixed number of nodes are distributed in one-dimensional space according to the uniform distribution. Related results have been obtained in our previous paper only implicitly in the form of recursive equations. On the other hand, the results of this paper are significant in that they are closed form solutions of the recursive equations thus present the probability explicitly.

For user-preferred interfaces, we propose a service architecture, the interface-client/logic-server (ICLS). It separates specific interfaces from services. An ICLS-based service consists of some interface clients and a logic server. The target of this work is the services with intermediary computers in various scenes of our daily activities. Nowadays, against various users' characteristics, the services offer different UIs individually, and they are mostly single GUI. In ICLS, users can switch interfaces of services to suit their preferences and to customize flexibly. Interface clients and logic servers work together independently with common descriptions of interaction. Those descriptions are written in the abstract interaction description language (AIDL), we propose. AIDL is an application of semantic web technologies, and describes interactions as graphs. These graphs represent a specification of interfaces and the current state of interactions constantly. We propose a framework of ICLS-based service design. Developers can use it like GUI toolkits. With this framework, we developed three clients and two servers.

Reinforcement learning is a framing of enabling agents to learn from interaction with environments. It has focused generally on Markov decision process (MDP) domains, but a domain may be non-Markovian in the real world. In this paper, we introduce a new description of macro-actions with tree structure in reinforcement learning. The macro-action is an action control structure which provides an agent with control which applies a collection of related microscopic actions as a single action unit. And we propose a simple method for dynamically acquiring macro-actions from the experiences of agents during reinforcement learning process.

We introduce a class of computational problems called partial-order constraint satisfaction problems (POCSPs) for formulating the general problem of determining partial orders that satisfy the given constraints. Then we present some heuristic methods for encoding and solving these problems efficiently by using the binary decision diagram (BDD) data structure. Finally, we show that these methods can be successfully applied to expert systems in the subdomain of software verification.

Assumption-based truth maintenance systems (ATMS) have become powerful and widely used tools in artificial intelligence problem solvers. In this paper, we apply ATMS to verification of termination of computer programs written as a set of rewrite rules. Compared with the traditional methods based on the ordinary backtracking, our method can greatly improve the overall efficiency by virtue of the ATMS's ability to avoid futile backtracking, rediscovering inferences, and rediscovering contradictions. The originality of our work lies in the practical use of the ATMS in a software engineering problem and in the communication protocol between the termination verifier and the ATMS.

A formal, algebraic specification is presented for a functional programming language, called Brown, with reflective computation mechanisms. The emphasis is put on how it is possible to specify the reflection abstractly (in the sense of abstract data types) in equational logic. The major part of the specification consists of a set of equations (axioms ) based on a denotational semantics which model environment, continuation and metacontinuation. The specification is executable by term rewriting, because the equations can be regarded as the rewrite rules which replace instances of the left-hand sides with the corresponding instances of the right-hand sides.>

A reachability matrix M is a binary matrix with the reflexive and transitive property, i.e., M + I = M, and M2 = M, where I is the identity matrix. The entries of the matrix M are shown to form a multilevel implication structure derived using the transitivity property. The fundamental implication matrix P that defines this structure is derived. The matrix Q of the transitive closure of P, the complete implication matrix, is defined. It is proved that Q = p2. The problem of efficiently filling the partially filled reachability matrix is considered. An algorithm for determining all of the implied values of the unknown elements of the partially filled reachability matrix M derived from a supplied value is proposed. The algorithm requires 0(n2) computer time and 0(n2) storage, where n is the size of the matrix M. Use of the algorithm to the interpretive structural modeling (ISM) process makes it possible to do a flexible and an efficient transitive embedding.

A control algorithm and a probability-theoretical traffic analysis of a grade-crossed intersection of the computer-controlled vehicle system, a computer-controlled rapid transit system developed in Japan, are presented. The control algorithm is simple but realistic and optimal in the sense that delay is minimized under the following constraints: the first-come first-served rule, the maneuver length limitation, and the collision-freeness condition. The analysis is completely based on this detailed control algorithm; random arrival of vehicles is assumed, and the behaviour of the algorithm is formulated as a simple Markov chain known as a one-dimensional random walk. Exact and explicit expressions for some performance measures such as delay, abort rate, and throughput are obtained.

Transitive coupling is a procedure encountered in structuring complex systems using interpretive structural modeling (ISM). It involves the interconnection of two subsystems defined on the same contextual relation that is transitive. An efficient procedure or transitive coupling is proposed. The procedure presented is highlighted by savings in computer time and storage and by ease of programming. For a given value of one or zero for an unknownxijoryij, all inferences can be drawn with O(mn) computer time and θ((m+n)2) storage. Sufficient results obtained by a computer experiment using randomly generated problems are in close agreement with the results of the theoretical analysis.

Interconnecting two multilevel systems which are defined on the same transitive contextual relation is a problem encountered in structuring complex systems using the interpretive structural modelling process. A theorem that states that the complete implication matrix can be obtained by squaring the adjacency implication matrix is proved. Then a procedure for calculating the complete implication matrix in O((mn)2) time by means of the results obtained through the theorem is presented, wheremandnare the number of element of each system. A simple expression of the complete implication matrix for transitive bordering is also given.