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It may not be possible to implement a specification. The requirements may be inconsistent, the specification may be meaningless or ill-defined, or, surprisingly, the function specified may not be computable. The concept of an incomputable function comes from computability theory, a branch of mathematical logic with particular relevance to computer science.
Chapter 12 introduces some of the key ideas of computability theory, both as motivation for subsequent study and to provide the background necessary to appreciate the significance of an incomputability claim. In particular, we will prove that some functions are not computable by any C program, explain why such problems are deemed to be algorithmically unsolvable, and list a number of unsolvable problems of practical importance involving context-free grammars.
Additional Reading
Good presentations of elementary computability theory may be found in [RS86, Sip97, Jon97, GH98, LP98, HMU01].
Professor Higgins wants his programmer, Eliza Doolittle, to write a code fragment that is to test for the presence or absence of a value x in an array A. Here is the intended application: A is to contain the student numbers of all the students currently enrolled in his course on computational metaphysics, and x might be the student number of a student who is trying to verify that he or she is properly registered.
Exercise 1.1 Before reading on, pretend that you are Eliza Doolittle and try to write the desired code. Also, try to write an application program that uses the code.
As you will quickly discover if you try to write the desired code, or to write a program to use it, the preceding paragraph is an inadequate specification. Here are some of the questions that must be answered before (or during) the development of the desired code or any associated applications.
What is the range of allowed subscript values for A, and what segment of the array should be searched?
What is the type of variable x (also, presumably, the component type of the array), and how should values of this type be compared?
How should the result be recorded?
Is the array segment sorted in, say, ascending order, to allow use of a more efficient search method?
Autonomous agents are systems capable of autonomous decision-making in real-time environments. Computation is a valuable resource for such decision-making, and yet the amount of computation that an autonomous agent may carry out will be limited. It follows that an agent must be equipped with a mechanism that enables it to make the best possible use of the computational resources at its disposal. In this paper we review three approaches to the control of computation in resource-bounded agents. In addition to a detailed description of each framework, this paper compares and contrasts the approaches, and lists the advantages and disadvantages of each.
Upon assessing probabilities for Bayesian belief networks, the knowledge and practical experience of experts is often the only available source of probabilistic information. It is important to realise that issues concerning human capabilities with respect to making judgements come into play when relying on experts for probability elicitation. A number of methods for the elicitation of probabilities are known from the field of decision analysis. These methods try, to some extent, to deal with those issues. I present here an overview of the issues to consider when relying on expert judgements and describe the methods that are available for expert elicitation, along with their benefits and drawbacks.
This paper focuses upon the contribution which adequate use of the latest development in IT can make to the enhancement, development and improvement of professional expertise in the construction domain. The paper is based on the author's personal expertise and involvement in several UK- and European-funded research projects. First, the paper gives an overview of the benefits that can be gained from improved information and knowledge management in the construction industry. The latest developments in information and communication technologies are then described and presented as an enabler for effective knowledge management. Finally, the paper reviews the most promising techniques for information and knowledge management in the construction domain, highlighting the advantages and disadvantages of each of them, as well as future directions for knowledge management in construction.
There is little doubt that the strength and breadth of UK research into multi-agent systems continues to grow as we move into the new millennium. In the middle of an extremely cold December in 2000, the Third UK Workshop on Multi-Agent Systems (UKMAS 2001) was held at St Catherine's College, Oxford. This was the fifth such meeting in as many years, generously sponsored by EPSRC, FIPA (The Foundation for Intelligent Physical Agents) and Hewlett Packard.
One of the main challenges in question-answering is the potential mismatch between the expressions in questions and the expressions in texts. While humans appear to use inference rules such as ‘X writes Y’ implies ‘X is the author of Y’ in answering questions, such rules are generally unavailable to question-answering systems due to the inherent difficulty in constructing them. In this paper, we present an unsupervised algorithm for discovering inference rules from text. Our algorithm is based on an extended version of Harris’ Distributional Hypothesis, which states that words that occurred in the same contexts tend to be similar. Instead of using this hypothesis on words, we apply it to paths in the dependency trees of a parsed corpus. Essentially, if two paths tend to link the same set of words, we hypothesize that their meanings are similar. We use examples to show that our system discovers many inference rules easily missed by humans.
In recent years, multi-agent systems (MASs) have received increasing attention in the artificial intelligence community. Research in multi-agent systems involves the investigation of autonomous, rational and flexible behaviour of entities such as software programs or robots, and their interaction and coordination in such diverse areas as robotics (Kitano et al., 1997), information retrieval and management (Klusch, 1999), and simulation (Gilbert & Conte, 1995). When designing agent systems, it is impossible to foresee all the potential situations an agent may encounter and specify an agent behaviour optimally in advance. Agents therefore have to learn from, and adapt to, their environment, especially in a multi-agent setting.
On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.
It is shown that the maximum possible chromatic number of the square of a graph with maximum degree d and girth g is (1 +o(1))d2 if g = 3, 4, 5 or 6, and is Θ(d2 / log d) if g [ges ] 7. Extensions to higher powers are considered as well.
The Text REtrieval Conference (TREC) question answering track is an effort to bring the benefits of large-scale evaluation to bear on a question answering (QA) task. The track has run twice so far, first in TREC-8 and again in TREC-9. In each case, the goal was to retrieve small snippets of text that contain the actual answer to a question rather than the document lists traditionally returned by text retrieval systems. The best performing systems were able to answer about 70% of the questions in TREC-8 and about 65% of the questions in TREC-9. While the 65% score is a slightly worse result than the TREC-8 scores in absolute terms, it represents a very significant improvement in question answering systems. The TREC-9 task was considerably harder than the TREC-8 task because TREC-9 used actual users’ questions while TREC-8 used questions constructed for the track. Future tracks will continue to challenge the QA community with more difficult, and more realistic, question answering tasks.
We investigate the problem of complex answers in question answering. Complex answers consist of several simple answers. We describe the online question answering system SHAPAQA, and using data from this system we show that the problem of complex answers is quite common. We define nine types of complex questions, and suggest two approaches, based on answer frequencies, that allow question answering systems to tackle the problem.
We study the asymptotic behaviour of the relative entropy (to stationarity) for a commonly used model for riffle shuffling a deck of n cards m times. Our results establish and were motivated by a prediction in a recent numerical study of Trefethen and Trefethen. Loosely speaking, the relative entropy decays approximately linearly (in m) for m < log2n, and approximately exponentially for m > log2n. The deck becomes random in this information-theoretic sense after m = 3/2 log2n shuffles.
As users struggle to navigate the wealth of on-line information now available, the need for automated question answering systems becomes more urgent. We need systems that allow a user to ask a question in everyday language and receive an answer quickly and succinctly, with sufficient context to validate the answer. Current search engines can return ranked lists of documents, but they do not deliver answers to the user.
Question answering systems address this problem. Recent successes have been reported in a series of question-answering evaluations that started in 1999 as part of the Text Retrieval Conference (TREC). The best systems are now able to answer more than two thirds of factual questions in this evaluation.
The circuit cover problem for mixed graphs (those containing edges and/or arcs) is defined as follows. Given a mixed graph M with a nonnegative integer weight function p on its edges and arcs, decide whether (M, p) has a circuit cover, that is, a list of circuits in M such that every edge (arc) e is contained in exactly p(e) circuits of the list. In the special case when M is a directed graph (contains only arcs), the problem is easy, but when M is an undirected graph not many results are known. For general mixed graphs this problem was shown to be NP-complete by Arkin and Papadimitriou in 1986. We prove that this problem remains NP-complete for planar mixed graphs. Furthermore, we present a good characterization for the existence of a circuit cover when M is series-parallel (a similar result holds for the fractional version). We also describe a polynomial algorithm to find such a circuit cover, when it exists. This is an ellipsoid-based algorithm whose separation problem is the minimum circuit problem on series-parallel mixed graphs, which we show to be polynomially solvable. Results on two well-known combinatorial problems, the problem of detecting negative circuits and the problem of finding shortest paths, are also presented. We prove that both problems are NP-hard for planar mixed graphs.
In this paper we are concerned with the following conjecture.
Conjecture. For any positive integers n and k satisfying k < n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.
We prove this conjecture when 2k [les ] n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t [les ] 32(2r+4)n2+1.