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We observe that on occasion expressions in some language are put forward that purport to state "a problem.” In response a method (or algorithm) is advanced that claims to solve the problem. That is, if input data are given that meet all the specifications of the problem statement, the method produces another expression in the language that is the solution to the problem. If there is a challenge as to whether the method actually provides a general solution to the problem (i.e., for all admissible inputs), a proof may be forthcoming that it does. If there is a challenge to whether the problem statement is well defined, additional formalization of the problem statement may occur. In the extreme this can reach tock to formalization of the language used to state the problem, until a formal logical calculus is used.

We also observe that for other problems that people have and solve there seems to be no such formalized statement and formalized method. Although usually definite in some respects problems of this type seem incurably fuzzy in others. That there should be ill-defined problems around is not very surprising. That is, that there should be expressions that have some characteristics of problem statements but are fragmentary seems not surprising. However, that there should exist systems (i.e., men) that can solve these problems without the eventual intervention of formal statements and formal methods does pose an issue. Perhaps there are two domains of problems, the well structured and the ill structured. Formalization always implies the first. Men can deal with both kinds. By virtue of their capacity for working with ill-structured problems, they can transmute some of these into well-structured (or formalized) problems. But the study of formalized problems has nothing to say about the domain of ill-structured problems. In particular, there can never be a formalization of ill-structured problems, hence never a theory (in a strict sense) about them. All that is possible is the conversion of particular problems from ill structured to well structured via the one transducer that exists, namely, man.

Perhaps an analog is useful: well-structured problems are to illstructured problems as linear systems are to nonlinear systems, or as

I wish to acknowledge the help of J. Moore in clarifying the nature of the methods of artificial intelligence and also my discussions with my colleague H. A. Simon. The research was supported by the Advanced Rescarch Projects Agency of the Office of the Secretary of Defense (SD-146).

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stable systems are to unstable systems, or as rational behavior is to nonrational behavior. In cach case, it is not that the world has been neatly divided into two parts, each with a theory proper to it. Rather, one member of the pair is a very special case about which much can be said, whereas the other member consists of all the rest of the world—uncharted, lacking uniform approach, inchoate, and incoherent.

This is not the only view, of course. Alternatively, one can assert that all problems can be formulated in the same way. The formalization exists because there is some symbolic system, whether on paper or in the head, that holds the specific, quite definite information in the problem statement. The fragmentation of problem statement that occurs when an attempt is made to explicate the problem only shows that there are serious (perhaps even profound) communication problems. But it does not say that ill-structured problems are somehow different in nature.

Thus we have an issue—somewhat ill structured, to be sure—but still an issue. What are ill-structured problems and are they a breed apart from well-structured ones? This chapter is essentially an essay devoted to exploring the issue, as it stands in 1968.

We have an issue defined. What gives life to it are two concerns, one broad, one narrow. At root, there is the long-standing concern over the rationalization of human life and action. More sharply stated, this is the challenge of art by science. In terms of encounters long resolved, it is whether photography will displace painting, or whether biology and physiology can contribute to the practice of medicine. In terms of encounters now in twilight, it is whether new products come from applied science or from lone inventors. In terms of encounters still active, it is whether the holistic diagnostic judgment of the clinical psychologist is better than the judgments of a regression equation (12]. For the purpose of this essay, of course, it is to what extent management science can extend into the domain of business judgment.

When put in this context, the issue has a charge. The concern flows partly from economics, where it is now usually labeled the problem of automation. Concern also flows from a problem of identity, in which some are compelled to ask what attributes man can uniquely call his own. As has been pointed out, probably most thoroughly by Ellul (3), it makes no sense to separate hardware and software, that is, to separate machines from procedures, programs, and formalized rules. They are all expressions of the rationalization of life, in which human beings become simply the agents or carriers of a universalistic system of orderly relations of means to ends.

Thus, viewed broadly, the issue is emotionally toned. However, this

Heuristic Programming: Ill-Structured Problems

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fact neither eliminates nor affects the scientific questions involved, although it provides reasons for attending to them. Our aim in this essay is essentially scientific, a fact which leads to the second, more narrow context.

Within management science the nature of rationalization has varied somewhat over time. In the early days, those of Frederick Taylor, it was expressed in explicit work procedures, but since World War II it has been expressed in the development of mathematical models and quantitative methods. In 1958 we put it as follows:

A problem is well structured to the extent that it satisfies the following criteria:

1. It can be described in terms of numerical variables, scalar and vector quantities.

2. The goals to be attained can be specified in terms of a well-defined objective function-for example, the maximization of profit or the minimization of cost.

3. There exist computational routines (algorithms) that permit the solution to be found and stated in actual numerical terms. Common examples of such algorithms, which have played an important role in operations research, are maximization procedures in the calculus and calculus of variations, linear-programming algorithms like the stepping-stone and simplex methods, Monte Carlo techniques, and so on (21, pp. 4-5).

Ill-structured problems were defined, as in the introduction of this essay, in the negative: all problems that are not well structured. Now the point of the 1958 paper, and the reason that it contrasted well- and ill-structured problems, was to introduce heuristic programming as relevant to the issue:

With recent developments in our understanding of heuristic processes and their simulation by digital computers, the way is open to deal scientifically with illstructured problems—to make the computer co-extensive with the human mind [21, p. 9).

That is, before the development of heuristic programming (more generally, artificial intelligence) the domain of ill-structured problems had been the exclusive preserve of human problem solvers. Now we had other systems that also could work on ill-structured problems and that could be studied and used.

This 1958 paper is a convenient marker for the narrow concern of the present essay. It can symbolize the radical transformation brought by the computer to the larger, almost philosophical concern over the nature and possibilities for rationalization. The issuc has become almost technical, although now it involves three terms, where before it involved only two:

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What is the nature of problems solved by formal algorithms?
What is the nature of problems solved by computers?
What is the nature of problems solved by men?

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We have called the first well-structured problems; the last remains the residual keeper of ill-structured problems; and the middle term offers the opportunity for clarification.

Our course will be to review the 1958 paper a little more carefully, leading to a discussion of the nature of problem solving. From this will emerge an hypothesis about the nature of generality in problem solving, which will generate a corresponding hypothesis about the nature of illstructured problems. With theses in hand, we first consider some implications of the hypotheses, proceed to explore these a little, and finally bring out some deficiencies.

The 1958 paper asserted that computers (more precisely, computers appropriately programmed) could deal with ill-structured problems, where the latter was defined negatively. The basis of this assertion was twofold. First, there had just come into existence the first successful heuristic programs, that is to say, programs that performed tasks requiring intelligence when performed by human beings. They included a theoremproving program in logic (15), a checker-playing program (19), and a pattern recognition program [20]. These were tasks for which algorithms either did not exist or were so immensely expensive as to preclude their use. Thus, there existed some instances of programs successfully solving interesting ill-structured problems. The second basis was the connection between these programs and the nature of human problem solving (16). Insofar as these programs reflected the same problem-solving processes as human beings used, there was additional reason to believe that the programs dealt with ill-structured problems. The data base for the assertion was fairly small, but there followed in the next few years additional heuristic programs that provided support. There was one that proved theorems in plane geometry, one that did symbolic indefinite integration, a couple of chess programs, a program for balancing assembly lines, and several pattern recognition programs [5].

The 1958 paper provided no positive characterization of ill-structured problems. Although it could be said that some ill-structured problems were being handled, these might constitute a small and particularly "wellformed” subsct. This was essentially the position taken by Reitman, in one of the few existing direct contributions to the question of ill-formed problems [17, 18). He observed, as have others, that all of the heuristic programs, although lacking well-specified algorithms, were otherwise quite precisely defined. In particular, the test whereby one determined whether the problem was solved was well specified, as was the initial data base from which the problem started. Thus, he asserted that all existing heuristic programs were in a special class by virtue of certain aspects being well defined, and thus shed little light on the more general case.

Stating this another way, it is not enough for a problem to become ill structured only with respect to the methods of solution. It is required also to become ill structured with respect to both the initial data and the criteria for solution. To the complaint that one would not then really know what the problem was, the rejoinder is that almost all problems dealt with by human beings are ill structured in these respects. To use an example discussed by Reitman, in the problem of making a silk purse from a sow's ear, neither "silk purse” nor "sow's ear” is defined beyond cavil. To attempt really to solve such a problem, for instance, would be to search for some ways to stretch the implicit definitions to force acceptance of the criteria, for example, chemical decomposition and resynthesis.

Reitman attempted a positive characterization of problems by setting out the possible forms of uncertainty in the specification of a problem: the ways in which the givens, the sought-for transformation, or the goal could be ill defined. This course has the virtue, if successful, of defining “ill structured” independently of problem solving and thus providing a firm base on which to consider how such problems might be tackled. I will not follow him in this approach, however. It seems more fruitful here to start with the activity of problem solving.

1. THE NATURE OF PROBLEM SOLVING

A rather general diagram, shown in Fig. 10.1, will serve to convey a view of problem solving that captures a good deal of what is known, both casually and scientifically. A problem solver exists in a task environment, some small part of which is the immediate stimulus for evoking the problem and which thus serves as the initial problem statement. This external representation is translated into some internal representation (a condition, if you please, for assimilation and acceptance of the problem by the problem solver). There is located within the memory of the problem solver a collection of methods. A method is some organ

* Its statement form is clear when given linguistically, as in “Where do we locate the new warehouse?” Otherwise, "statement” is to be taken metaphorically as comprising those clues in the environment attended to by the problem solver that indicate to him the existence of the problem.

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