The problem of abduction can be characterized as finding the best explanation of a set of data. In this chapter we focus on one type of abduction in which the best explanation is the most plausible combination of hypotheses that explains all the data. We then present several computational complexity results demonstrating that this type of abduction is intractable (NP-hard) in general. In particular, choosing between incompatible hypotheses, reasoning about cancellation effects among hypotheses, and satisfying the maximum plausibility requirement are major factors leading to intractability. We also identify a tractable, but restricted, class of abduction problems.

Introduction

What kinds of abduction problems can be solved efficiently? To answer this question, we must formalize the problem and then consider its computational complexity. However, it is not possible to prescribe a specific complexity threshold for all abduction problems. If the problem is “small,” then exponential time might be fast enough. If the problem is sufficiently large, then even O(n2) might be too slow. However, for the purposes of analysis, the traditional threshold of intractability, NP-hard, provides a rough measure of what problems are impractical (Garey & Johnson, 1979). Clearly, NP-hard problems will not scale up to larger, more complex domains.

Our approach is the following. First, we formally characterize abduction as a problem of finding the most plausible composite hypothesis that explains all the data. Then we consider several classes of problems of this type, the classes being differentiated by additional constraints on how hypotheses interact.

TIPS (Task Integrated Problem Solver) and PATHEX/LIVER were built using the PEIRCE tool. Both are examples of third-generation abduction machines. PEIRCE is not specialized for diagnoses and might be used as a shell for any abductive-assembly system. TIPS and PATHEX/LIVER, however, are diagnostic systems. They are complicated systems that are similar in organization and capabilities. Despite their similarities, in the following descriptions we emphasize TIPS's ability to dynamically integrate multiple problem-solving methods and PATHEX/LIVER's proposed ability to combine structure-function models – for causal reasoning – with compiled diagnostic knowledge. First we describe TIPS, and then PATHEX/LIVER.

TIPS

TIPS is a preliminary framework that implements the idea (described in chapter 4) of making alternative problem-solving methods available for a task. Method invocation depends on the problem state and the capabilities of the method, not on a preset sequence of invocations. TIPS presents a general mechanism for the dynamic integration of multiple methods in diagnosis.

One can describe diagnosis not only in terms of the overall goal (say, explaining symptoms in terms of malfunctions), but also in terms of the rich structure of subgoals that arise as part of diagnostic reasoning and in terms of the methods used to achieve those goals. We call such a description a task-structure analysis. A diagnostic system explicitly realized in these terms has a number of advantages:

1. a. Such a system has multiple approaches available for solving a problem. Thus the failure of one method does not mean failure for the whole problem solver.

2. b. Such a system can potentially use more kinds of knowledge.

3. c. Such a system can potentially solve a broader range of diagnostic problems.

Tractable abduction

Abduction can be described as “inference to the best explanation,” which includes the generation, criticism, and possible acceptance of explanatory hypotheses. What makes one explanatory hypothesis better than another are such considerations as explanatory power, plausibility, parsimony, and internal consistency. In general a hypothesis should be accepted only if it surpasses other explanations for the same data by a distinct margin and only if a thorough search was conducted for other plausible explanations.

Abduction seems to be an especially appropriate and insightful way to describe the evidence-combining characteristics of a variety of cognitive and perceptual processes, such as diagnosis, scientific theory formation, comprehension of written and spoken language, visual object recognition, and inferring intentions from behavior. Thus abductive inference appears to be ubiquitous in cognition. Moreover, humans can often interpret images, understand sentences, form causal theories of everyday events, and so on, apparently making complex abductive inferences in fractions of a second.

Yet the abstract task of inferring the best explanation for a given set of data, as the task was characterized in chapter 7, has been proved to be computationally intractable under ordinary circumstances. Clearly there is a basic tension among the intractability of the abduction task, the ubiquity of abductive processes, and the rapidity with which humans seem to make abductive inferences. An adequate model of abduction must explain how cognitive agents can make complex abductive inferences routinely and rapidly.

In chapters 1 and 2, we describe abduction, design science, and the generictask approach to building knowledge-based systems. In this chapter we examine the first two of our abductive systems, which we call here RED-1 and RED-2. RED-2 extended RED-1 in several dimensions, the most important being a more sophisticated strategy for assembling composite hypotheses. RED-2 was widely demonstrated and served as a paradigm for our subsequent work on abduction. The RED systems show that abduction can be described precisely enough so that it can be programmed on a digital computer. Moreover, the RED systems do not use methods that are explicitly or recognizably deductive or probabilistic, and thus the RED systems demonstrate evidence-combining inference that apparently goes beyond those classical frameworks.

The red-cell antibody identification task

The RED systems are medical test-interpretation systems that operate in the knowledge domain of hospital blood banks. Our domain experts for these two RED systems were Patricia L. Strohm, MT (ASCP) SBB and John Svirbely, MD. The blood bank is a medical laboratory responsible for providing safe blood and blood products for transfusion. The major activities required are A-B-O and Rh blood typing, red-cell antibody screening, redcell antibody identification, and compatibility testing. The RED systems provide decision support for red-cell antibody identification.

Blood cells have chemical structures on their surfaces called red-cell antigens. When a donor's cells are transfused into a patient, these antigens can be recognized as foreign by the patient's immune system.

The science of AI

The field of artificial intelligence (AI) seems scattered and disunited with several competing paradigms. One major controversy is between proponents of symbolic AI (which represents information as discrete codes) and proponents of connectionism (which represents information as weighted connections between simple processing units in a network). Even within each of these approaches there is no clear orthodoxy. Another concern is whether AI is an engineering discipline or a science. This expresses an uncertainty about the basic nature of AI as well as an uncertainty about methodology. If AI is a science like physics, then an AI program is an experiment. As experiments, perhaps AI programs should be judged by the standards of experiments. They should be clearly helpful in confirming and falsifying theories, in determining specific constants, or in uncovering new facts. However, if AI is fundamentally engineering, AI programs are artifacts, technologies to be used. In this case, there is no such reason for programs to have clear confirming or falsifying relationships to theories. A result in AI would then be something practical, a technique that could be exported to a real-world domain and used. Thus, there is confusion about how results in AI should be judged, what the role of a program is, and what counts as progress in AI.

It has often been said that the plurality of approaches and standards in AI is the result of the extreme youth of AI as an intellectual discipline.

In chapter 7 abduction stumbled. Our powerful all-purpose inference pattern, maybe the basis for all knowledge from experience, was mathematically proved to be impossible (or anyway deeply impractical under ordinary circumstances). How can this be? Apparently we do make abductions all the time in ordinary life and science. Successfully. Explanation-seeking processes not only finish in reasonable time, they get right answers. Correct diagnosis is possible, even practical. (Or maybe skepticism is right after all, knowledge is impossible, correct diagnosis is an illusion.)

Maybe there is no deep question raised by those mathematical results. Perhaps all they are telling us is that we do not always get the right answer. Sometimes our best explanation is not the “true cause” (ways this can occur are systematically described in chapter 1). Sometimes we cannot find a best explanation in reasonable time, or we find one but do not have enough time to determine whether it is unique. Maybe knowledge is possible after all, but it is a kind of hit or miss affair. Yet if knowledge is possible, how can we succeed in making abductions without being defeated by incompatible hypotheses, cancellation effects, and too-close confidence values?

Whether or not knowledge is possible, we can build diagnostic systems able to achieve good performance in complex domains. This chapter presents two such systems and also includes a special section on how a kind of learning can be fruitfully treated as abduction. A fuller response to the complexity results is given in chapter 9.

This chapter develops the hypothesis that perception is abduction in layers and that understanding spoken language is a special case. These rather grand hypotheses are rich with implications: philosophical, technological, and physiological.

We present here a layered-abduction computational model of perception that unifies bottom-up and top-down processing in a single logical and information-processing framework. In this model the processes of interpretation are broken down into discrete layers where at each layer a best-explanation composite hypothesis is formed of the data presented by the layer or layers below, with the help of information from above. The formation of such a hypothesis is an abductive inference process, similar to diagnosis and scientific theory formation. The model treats perception as a kind of frozen or “compiled” deliberation. It applies in particular to speech recognition and understanding, and is a model both of how people process spoken language input, and of how a machine can be organized to do it.

Perception is abduction in layers

There is a long tradition of belief in philosophy and psychology that perception relies on some form of inference (Kant, 1787; Helmholtz; Bruner, 1957; Rock, 1983; Gregory, 1987; Fodor, 1983). But this form of inference has been typically thought of as some form of deduction, or simple recognition, or feature based classification, not as abduction. In recent times researchers have occasionally proposed that perception, or at least language understanding, involves some form of abduction or explanation-based inference (Charniak & McDermott, 1985, p. 557; Charniak, 1986; Dasigi, 1988; Josephson, 1982, pp. 87-94; Fodor, 1983, pp. 88, 104; Hobbs, Stickel, Appelt and Martin, 1993).

Abduction machines

In this book, we described six generations of abduction machines. Each generation's story was told by describing an abstract machine and experiments with realizations of the machine as actual computer programs. Each realization was approximate, partial, something less than a full realization of the abstract machine. Each realization was also more than the abstract machine: an actual chunk of software, a knowledge-based expert system constructed to do a job, with an abundance of insights, domain-specific solutions, and engineering shortcuts to get the job done. The abstract machines are simplified idealizations of actual software.

An abstract abduction machine is a design for a programming language for building knowledge systems. It is also a design for a tool for constructing these systems (a partial design, since a tool also has a programming environment).

Each of the six machines has a strategy for finding and accepting best explanations. Machine 6 inherits all the abilities of the earlier machines. Suppose that we connect it to abstract machines for the subtasks of hypothesis matching, hierarchical classification, and knowledge-directed data retrieval (see chapter 2). Then we conjoin abstract machines able to derive knowledge for various subtasks from certain forms of causal and structural knowledge (see chapters 5 and 8). Then we implement the whole abductive device as a program written for a problem-solving architecture, which is an abstract device of a different sort that provides control for generalized, flexible, goal-pursuing behavior (see chapter 4).

The PEIRCE tool

In the first generation of work on generic tasks the invocation of a generictask problem solver was pre-programmed, hard-wired during system programming. In RED-2 we wanted to test many variations of the algorithm empirically, but a significant amount of work was required to reprogram the system each time a change was desired. Also, the RED-2 hypothesis-assembly module seemed too “algorithmic,” too much like a rule follower and not enough like “fluid intelligence.” So we decided to analyze the system in terms of the goals and subgoals of abductive problem solving and to identify the different methods that were used to achieve the various goals. This analysis allowed us to reorganize the program in a way that made adding new methods and modifying the goal structure easy, so that we could better experiment with new strategies for abduction. For multiple methods to be available to achieve problem-solving goals, method selection would occur at run time based on the state of the problem and the availability of needed forms of knowledge. These efforts resulted in a programming tool called “PEIRCE” for building systems that perform abductive assembly and criticism in this more flexible manner.

Systems built with PEIRCE are examples of third generation abduction machines (where RED-1 is taken as the first generation and RED-2 as the second). PEIRCE allows for specifying strategies for dynamically integrating various hypothesis-improvement tactics, or operators, during the course of problem solving. The idea was to capture domain-independent aspects of abductive assembly and criticism and embed them in the tool, leaving domain-specific aspects to be programmed when building particular knowledge-based systems.

Throughout this book we have depended in numerous ways on the idea of plausibility. It appears as “confidence value,” “symbolic plausibility value,” “degree of certainty,” “plausibilities of elementary hypotheses,” and the like.

Just what is this idea of plausibility? Can it be made precise? Is a plausibility a kind of likelihood, a kind of probability? What are the semantics? How much plausibility should be set for a given hypothesis under given circumstances? Is an objective standard possible? Does it even matter for a theory of intelligence, or for epistemology? Unfortunately, we cannot yet give definite answers to these questions.

The plausibility talk in this book is, in the first place, naturalistic. People mention plausibility when they describe their reasoning processes and when they write discussions of results in scientific papers. Plausibility talk seems to come naturally to us and is reflected in our ordinary language. Common usage allows both categorical judgments that something is or is not plausible, and graded comparative judgments, such as that one thing is “a lot” or “a little bit” more plausible than another. Throughout the work that we describe in this book we used our best judgments of when hypotheses were plausible, of how to evaluate evidence, and of how to weigh hypotheses. We were encouraged in our judgments by other people being likewise persuaded by appeals to “because it is the only plausible explanation,” and “more plausible than any other explanation for the data,” and the like. We were encouraged by the systems we built acting mostly according to our expectations, and producing correct answers.

Abduction machines – summary of progress

All six generations of abduction machines described in this book are attempts to answer the question of how to organize knowledge and control processing to make abductive problem solving computationally feasible. How can an intelligent agent form good composite explanatory hypotheses without getting lost in the large number of potentially applicable concepts and the numerical vastness of their combinations? What general strategies can be used? Furthermore, it is not enough simply to form the best explanation, which already appears to be difficult, but an agent needs to be reasonably sure that the explanation chosen is significantly better than alternative explanations, even though generating all possible explanations so that they can be compared is usually not feasible.

Thus it seems that we are in deep trouble. Logic demands that an explanation be compared with alternatives before it can be confidently accepted, but bounded computational resources make it impossible to generate all of the alternatives. So it seems, tragically, that knowledge is impossible! Yet we are saved after all by a clever trick; and that trick is implicit comparison. A hypothesis is compared with alternatives without explicitly generating them all. One way to do this, as we have seen, is by comparing parts of hypotheses. By comparing hypothesis-part h1 with hypothesis-part h2, all composite hypotheses containing h1 are implicitly compared with all composites containing h2. Another way to implicitly compare hypotheses is to rely on a hypothesis generator that generates hypotheses in approximate order of most plausible first.

Exponential sums are important tools in number theory for solving problems involving integers—and real numbers in general—that are often intractable by other means. Analogous sums can be considered in the framework of finite fields and turn out to be useful in various applications of finite fields.

A basic role in setting up exponential sums for finite fields is played by special group homomorphisms called characters. It is necessary to distinguish between two types of characters—namely, additive and multiplicative characters—depending on whether reference is made to the additive or the multiplicative group of the finite field. Exponential sums are formed by using the values of one or more characters and possibly combining them with weights or with other function values. If we only sum the values of a single character, we speak of a character sum.

In Section 1 we lay the foundation by first discussing characters of finite abelian groups and then specializing to finite fields. Explicit formulas for additive and multiplicative characters of finite fields can be given. Both types of characters satisfy important orthogonality relations.

Section 2 is devoted to Gaussian sums, which are arguably the most important types of exponential sums for finite fields as they govern the transition from the additive to the multiplicative structure and vice versa. They also appear in many other contexts in algebra and number theory. As an illustration of their usefulness in number theory, we present a proof of the law of quadratic reciprocity based on properties of Gaussian sums.