Mathematical Methods in Artificial Intelligence

Rating:
Summary: Excellent
Review: Although using only elementary mathematics, and not at all addressing new areas of artificial intelligence, such as inductive logic programming, this book gives an excellent overview of how mathematics is used in artificial intelligence. Mathematics at all levels is used in this field, both in the algorithms and in discussing its foundations, and this book serves as a good introduction to its application in A.I. Only elementary algebra and calculus are used in the book, making it very accessible to the beginning student in computer science. Readers with more sophisticated background in mathematics can then extend the results in the book to more advanced mathematical contexts. The author's writing style is very informal, and in many places in the book he encourages the reader to "stop and think" before continuing in the reading. Exercises, some simple and some very challenging, are found at the end of most chapter sections.

The author gives a brief overview of the history of A.I. in chapter one, including a discussion of the issues of computational complexity in A.I. algorithms, a discussion of expert systems (with examples), and a few biographical sketches.

Chapter 2 is a fairly detailed overview of search algorithms, and the author introduces some notions from the mathematical field of combinatorics, namely directed graphs and ordered trees. Induction and recursion are then reviewed as tools for search algorithms. The recursive formulation of algorithms in A.I. is of course very powerful, and one that students need to master early on. Fields such as bioinformatics and data mining are becoming increasingly dependent on search algorithms from A.I., and the author reviews these in detail, including 'simple' search methods such as breadth-first, depth-first, and iterative-deepening, along with 'heuristic' methods.

The reader gets introduced to first-order predicate calculus in chapter 3. This topic could be said to be one of the most important ones in A.I., and it is discussed in this chapter using the (declarative) programming language Prolog. One could easily use the language Lisp, but Prolog makes more apparent the head/body clause structure of predicate logic. In addition, if a reader wants to move on to more modern developments in A.I., such as inductive logic programming, which can be viewed essentially as predicate logic but with inductive reasoning, a mastery of the content of this chapter is essential.

Chapter 4 introduces the reader to the proof theory, namely the technique of resolution, which is discussed for propositional calculus, where it is very simple, and for predicate logic, in the latter wherein some specialized techniques must be brought in, such as Skolemization. The author also discussed proof in the context of Prolog, and introduces the cut operator, which inhibits Prolog from fully implementing resolution. He also gives an interesting discussion on the problem of negation in Prolog and the closed-world assumption.

The author has been careful to not write a purely theoretical book in computer science, and evidence of this is given in chapter 5, which discusses how to implement first-order logic (FOL) into real-world applications. It is one thing to discuss the properties of logic, quite another to actually use it productively to solve problems of interest. The author discusses the limitations of FOL in these applications, and how they can be resolved through alternative reasoning tools, such as nonmonotonic logics, Bayesian networks, and fuzzy sets.

One of these alternatives, nonmonotonic reasoning, is discussed in the next chapter, wherein the author gives a fairly detailed overview of default reasoning and how it is implemented in Prolog. Rule sets and semantic nets are also discussed, along with defeasible reasoning. Applications of these techniques are stymied by their computational complexity, and the author gives references for discussions of this.

After a review of probability theory in chapter 7, the author discusses Bayesian networks in chapter 8. These have been extremely important in recent applications of A.I., and the author gives a fine review of their properties, especially their ability to incorporate causality by imposing a directed graph structure on the event space. The author gives a few examples of Bayesian networks, including a medical diagnosis, wherein he introduces a very important concept in A.I., namely that of abductive inference. Detailed discussion (with proofs) is given for the Kim-Pearl algorithm for singly connected networks.

Chapter 9 is an introduction to fuzzy logic and belief theory. The author motivates nicely the reasons for considering fuzzy reasoning instead of probabilistic methods. The Dempster-Shafer belief theory, which has become popular in recent years, is also discussed in some detail.

So as to motivate the discussion of neural networks, the next chapter overviews automatic pattern classification. Contrasting between supervised and unsupervised learning of patterns, the author then outlines the types of automatic classifiers, such as decision trees and neural networks. The chapter on neural networks is a good introduction considering the vastness of the subject. Indeed, an enormous amount of research has been done on neural networks, and their use in applications of A.I. has finally been achieving success in recent years.

Concepts from information theory are of course very important in A.I. and these are discussed in chapter 12, along with more advanced topics in probability and statistics that were not treated earlier in the book. These ideas are used in the next chapter wherein neural networks and decisions trees are discussed in more detail. The most interesting part of this discussion is the idea that noise can improve the generalization capabilities of neural networks. This strategy will be obvious to the physicist reader who has studied the effects of noise on dynamical systems governed by potentials with local minima.

The last chapter of the book discusses some additional topics that should be included in a study of A.I., such as genetic algorithms and more discussion of optimization, such as simulated annealing. Hidden Markov models are also briefly discussed, and this is somewhat disappointing given their importance in current applications. The reader is also introduced to robotics, certainly the most exciting of all topics in 21st century A.I.

Rating:
Summary: It is a useful book for research oriented readers.
Review: Most AI books do not emphasize the mathematical issues. Consequently, the readers face difficulty to read journals. This is a highly recommended book for those research oriented readers. It requires no formal background of mathematics beyond high school level. I read the book several times. It helped me a lot to understand many difficult papers. Among the chapters the most useful are chapter 6 on nonmonotonic reasoning and chapter 8 on Bayesian networks. The beginners will find chapter 3 and 4 on predicate logic and the theory of resolution highly useful. I strongly feel that the book should be read by all people working in the domain of AI.