### Characteristics of Problems

Determining the type of problem to be solved is particularly difficult. From the scientific point of view it has not been treated sufficiently yet. It is, nevertheless, of fundamental importance because it covers the whole field of creativity, and the problem solver(s) heuristic behavior is contingent on the type of problem.

What is a problem? This question was asked – and answered – by Karl Duncker (1945). Duncker, who was a Gestalt psychologist, defined a problem in these words: “A problem arises when a living organism has a goal but does not know how this goal is to be reached.” This definition is, no doubt, very useful, because creativity tasks and activities always strive to address a problem. Yet, Duncker’s definition and formulation poses these caveats:

- It is necessary to distinguish between a task and a problem. It is the subject’s level of domain knowledge, including his ability to find pertinent knowledge, if necessary, that makes the difference between the two. A task set by a researcher or experimenter may be a problem to certain subjects and no problem to others.
- A problem may vanish or be resolved if the subject changes his goal.
- A problem does not exist de facto, unless the subject observes discrepancies between his current situation and the goals he pursues.

Reitman (1965) proposed that problems be viewed as three-component entities, having an initial state, a final (goal) state, and a set of processes that facilitate reaching the goal, starting from the initial state.

Minski (1961) proposed a distinction between two types of problems, those that according to the nature of the conditions of acceptability of solutions are either well defined or ill-defined.

A problem satisfying Reitman’s conditions (Reitman, 1965) is a so-called well-defined problem: it can be solved by applying a systematic procedure that makes it possible to decide whether a proposed solution is correct or not. It means that it is totally decidable: all pertinent solutions can be evaluated strictly using one binary variable: right or wrong. The solution can thus be described as an all-or-nothing phenomenon. There are no intermediate solutions between the functional and non-functional ones. In general terms, any tests for which there exists a rigorous method of comparison between what is proposed and what is required is a well-defined problem.

Examples of well-defined problems are board games, problems in mathematics, or problems in logic. They may be very difficult to resolve, nevertheless. Taking man’s limited resources, psychologists face the task of explaining how human beings manage to solve problems in chess, mathematics or geometry within reasonable time.

Ill-defined problems are those that are not well-defined. They result in a multitude of solutions that cannot be classified by using a binary truth-value, but by using a relative qualitative scale. The response to a requirement thus allows grades, the determination of which is left to the referees. The majority of problems occurring in everyday life are ill-defined problems: the improvement on an object or an apparatus, a new use of what already is known, the search for a sales idea or a marketing idea, etc.

Ill-defined problems arise when some components of the problems statement, in the sense of Reitman, are unspecified, or are vague or fuzzy. The definedness of problems varies in degree (Reitman, 1965, Ch. 5). For instance, ‘take a little flower and bake bread for these people’, which is vague in terms of the quantity of flower and the number of people, but specifies clearly the method: bake. Another statement may run like this: ‘Let us overcome the current economic crisis.’ This statement does not specify the method: what should be done to overcome the crisis? ‘Do not just hang around, maximize something’ is an exhortation taken from a cartoon, in which both the initial state, the method and the goal are shrouded in a mental fog. Ill-defined problems are more common than are well-defined problems, but it is all the more difficult to explain how to tackle them.

It is worth noting that Minski’s postulate does not necessarily cover the distinction between problem solving and creativity. For instance, the discovery of a new algorithm, or a new combination of known algorithms, is a creative act. But well-defined problems in the sense of Minsky may lead to an opposition between algorithmic procedures and inferential procedures.

As for the ill-defined problems, Reitman (1964) proposed a typology of six classes of problems comprising the transformation or generation of states, objects, or collections of objects. This taxonomy is not presented as a universal tool covering the whole field of creative situations, but simply as a general structure making it possible to collect the largest number possible of the creative situations. This attempt at systemization has mainly a descriptive value, but it is not unlikely that it could also be used for deducing hypotheses related to the behavior of effectual solutions.

Reitman’s work is based on the introduction of the following three concepts: let A be an initial state or object (one which is expected to undergo transformation, modification, complementing, improvement, etc.) and let B be a final state or object (the solution to be obtained, elimination of problem). Let the symbol î…Œ denote a process, program, or sequence of operations. It is then possible to represent a large number of problematic situations parting from these three symbols by representing them by a general vector [A, B, â‡’]. Using these three concepts, six types of poorly defined problems can be distinguished.

Type I. The initial and terminal states A and B are well specified: the relevant data are known and the requirements to be satisfied are explained precisely. The problem then consists in discovering the process â‡’ that makes it possible to pass from the well-specified state A to a well-specified state B. For instance: how can a given function be incorporated in a specific device? This type seems to cover a large class of problem situations.

Type II. The terminal state B is less precisely specified than in the previous type, while A is left entirely at the discretion of the experimenter. In fact, nothing is said about the state, object or assembly of objects from which to part. The initial material is largely undetermined and admits only one constraint to aid in constituting the one possible solution. For instance: what should be done to make traveling by train more pleasurable? Here, obviously, the current state represents some level of train travel comfort or pleasure, and this should be increased. But what exactly is to be achieved is an open question.

Type III. The initial state A consists in this case of an assembly of constituent parts each of which represents a concrete entity, while B represents a state or object to be achieved which is defined vaguely and is characterized by the fact that one or several of constituent parts of A have lost their separate identities after reaching B. Reitman cites as an example Napoleon’s cook who was charged with the task to “make a good dish” B to celebrate the victory at Marengo using only available ingredients A. This type is undoubtedly less general than the preceding ones.

Type IV. A and B are presented as consisting of sub-components and are rather poorly defined. This type differs from type II in that in the latter case there are no restrictions imposed on search, different analogous paths, and different associative paths the exploration of which can be relatively fruitful. In type IV it is not like that. The distinction between sub-components provides constraints within which the problem solution has to take place. The research is, in other words, more strictly restrained than it is in the problems of type II.

Type V. The initial state A is given by reference to a well defined object, the final state B is given by a set of similarities and dissimilarities with respect to A. An example given by Reitman to illustrate this type is the following: manufacturer Î± of some equipment encounters a serious competition from Î²-company’s product. The first company, Î±, decides to change the design of the product in point to offer a price that is lower for a comparable quality than what its competitor Î² asks. The task thus does not necessarily require an entirely new manufacturing process, because the added cost of the new process would not help to slash the price according to original estimates. Besides, the modification must be implemented fast because the competing product already is in the market while Î±-company’s sales decrease with each passing day. The exigencies of this example illustrate the general type V product as a new device that must be functionally similar to the old version but must be cheaper.

Type VI. In this case, the final state B is well specified while the initial state A remains essentially empty, unstructured and largely undetermined. Characteristic examples cited by Reitman comprise: to explain a new phenomenon, discover an alibi for a criminal deed, etc. This type differs from type II in the degree of precision of the task.

It is thus possible to distinguish among six categories of poorly defined problems resorting to almost formal properties of their application. A research activity the results of which would show that these categories incite heuristically different behavior on the part of individuals and groups still has to be accomplished.

A relevant taxonomy establishes first some ordering, i.e. introduces some logic in the pertinent knowledge field. For this purpose the taxonomy distributes the phenomena or the entities considered according to their relevant characteristics, with no ambiguity involved. It appears that, in general, each taxonomy displays at least two different utility values:

– First of all, the taxonomy presents a reference value that provides a framework for a certain subset of the universe. The information already available about the elements of this subset thus cease to be fragmented and simply accumulated: in the continuation they are ordered with respect to one another. They can be integrated and complemented. Fragmented knowledge thus becomes systematic.

– This knowledge also represents an “operational” or heuristic value of the taxonomy in point. This value becomes apparent when the taxonomy leads to empirical research in order to validate its structure, its principle and its extent, or to uncover which variables of the taxonomy can be expected and unified.

In the case of problem solution and creativity research, one can try to establish some correspondence between certain types of tasks with certain behavioral phenomena, particularly those of psycholinguistic nature.

The first problem differentiation might take into consideration the different objective properties of problems:

The problem is algorithmic: it can be resolved using an ordered sequence of specific operations. It allows, in this sense, a truly coordinated division of labor, and is particularly suitable for groups with the centralized communication structure.

The problem is inferential: it can be visualized by means of trees, but the process of generalization of the trees cannot be decomposed into concatenated elementary operations. A homogeneous structure is, however, more appropriate. It can be seen that groups facing a specific situation adopt spontaneously the optimum organization to respond to this situation.

Most authors, however, have resorted to local dichotomies based on a multitude of imprecise criteria. The straightforward problem typologies are the following:

- Verbal and non-verbal tasks. Verbal tasks are supposed to mobilize important cultural experience and imply the use of specific functions or hypothetic factors. Non-verbal tasks are symbolic, or in other ways dependent on non-verbal perceptions.
- Intellectual and manipulation-dependent tasks. In intellectual tasks, the principal operator is the brain. Manipulation-dependent problems require a coordination of the brain and muscular factors.
- Unique-solution and multiple-solutions tasks. Then there are problems having a unique solution and problems having multiple solutions.

The totality of distinctions pertinent to a particular solution domain cannot be generalized, because their underlying criteria are too coarse and do not allow more than just a very summary control of the situation.

Shaw’s dimensional analysis

In an attempt to present various aspects of group tasks in a systematic manner, Shaw (1963) collected a very eclectic set of 104 statements mostly taken from experimental literature. The statements relate to both ill-defined and well-defined problems, to verbally and non-verbally formulated tasks, etc. These various statements were evaluated according to six a priori defined dimensions, which can be visualized as continuously varying intervals in which each task occupies a point.

The six dimensions are characterized in the following manner:

- Requirements of cooperation. This dimension permits to define the degree to which it is required that members of the group act in a coordinated manner to complete the task successfully. It is thus a measure of dependence between the goal and the coordinated activity of the subjects.
- Verifiability of the decision. It is the degree to which the “rightness” or adequacy of the solution can be proved, either by reference to an authority, or by logical procedures (usually a mathematical proof), or by feedback (for instance by examining the consequences of the decision taken).
- Difficulty. This is defined by Shaw abstractly as the quantity of effort necessary for executing the task. Specifically, an indicator of difficulty can be the time required for solution, the number of errors made, etc.
- Clarity of purpose. This denotes the degree of precision with which the requirements of the task are presented to members of the group, and how the members perceive the requirements.
- Multiplicity of approaches to the goal. This dimension expresses the more or less great possibility to resolve the problem by various procedures. It is thus a matter of possible paths to the solution, i.e. of the number of alternative solutions.
- Relationship between mental and motor requirements. A task that only requires the implementation of intellectual activities will be among the strongest on this dimension. Conversely, tasks requiring only motor abilities will be among the weakest. A task requiring both intellectual and motor activities occupies an intermediate position between the two extremes.
- Intrinsic interest. Problems are not equally attractive, i.e. they do not mobilize the same motivation. This dimension is thus assigned the degree to which a particular task appears interesting to the subjects.
- Operational requirements. This dimension was introduced to evaluate the number of different kinds of operations, knowledge or abilities required for the completion of the task.
- Familiarity within the population. Individuals might have had a previous experience of the task in point, either direct or by means of an analogous task. This dimension thus evaluates the relative “rareness” of a class of problems to a population.
- Multiplicity of solutions. It is the number of different correct solutions for the problem in point. That number can in general be evaluated exactly in a well-defined problem, but not if the estimate is very intuitive.

This family of dimensions is intended to cover the maximum of traits occurring in every heuristic situation. Certain dimensions thus relate to formal properties of the task, for instance numbers 2, 5, or 10, while others, e.g. numbers 7 and 9 refer direct to the consequences of applying a particular semantics (second level of determination).

Forty-nine referees, mostly graduate students of psychology, got the task of distributing the 104 sample tasks according to the 10 dimensions shown above. Eight positions or degrees ordered by their magnitude were defined. The judgments were consistent, except for the dimension “clarity of purpose”.

With these data, Shaw got two factor analyses that resulted in disclosing five significant factors for task analysis:

Difficulty, (factor I), the quantity of required effort displays a close relationship to the number of operations, knowledge, and required abilities for solving the problem. The forth dimension, the “clarity of purpose” is equally an important aspect of difficulty: the less clear the goal is, the more difficult is the task judged to be.

Multiplicity of solutions (factor II) is a complex dimension that relates both to the number of acceptable solutions, to the diversity of paths leading to the solutions, and to the verifiability of a solution. Shaw thinks that the essential aspect is the number of solutions, while the other two merely are its consequences. While there are several solutions available, there also are several ways how to reach them. Proving the adequacy of each solution rigorously is hardly possible.

Cooperation requirements (factor III) correspond exactly to the dimension of the same name. The degree of completing a task successfully implies a coordinated action on the part of group members.

The relationship between intellectual requirements – motor requirements (factor IV) constitutes no doubt an independent dimension. But it only shows a very weak correlation with the familiarity with the task within the population.

Familiarity in the population is considered a separate dimension for the same reason. Nevertheless, it is necessary to point out that the familiarity seems relatively irrelevant, at least under the particular conditions of this work, where the majority of the tasks were somehow familiar to the subjects.

Intrinsic interest (factor V), which corresponds to the intensity of motivation and the attraction exerted by the problem on the group members, too, is a dimension permeated with factor II.

The first three of the six dimensions obtained finally seem to be both the most important and the least ambiguous ones. It is of course possible, as Shaw himself notes, that there are other dimensions, equally important, which continued research could bring forth. This first attempt will make it possible largely to control the principal components of the situation that comes into being as a problem to be solved is given to the subjects. This is the only condition under which accumulation of experimental data in this field can be transformed into scientific knowledge.

Categorization by Roby and Lanzetta

Roby and Lanzetta (1958) proposed a model intended to define and highlight the most important characteristics of a group task. For this purpose, they distinguish four sets of events occurring in the functioning of any group task system:

a. A set Tiof task input data. Here belong, for instance, the formulation of the problem to be solved and of the material it implies.

b. A correlative set Giof initial activities of the group. These comprise, among others, waiting times, observation, data recording, communication associated with input variables, etc.

c. A set Goof outputs produced by the group. In the creative process these comprise the traces of the heuristic process and solution suggestions.

d. A set Toof environmental changes following from the group’s activities.

Roby and Lanzetta define three general types of properties:

– Descriptive aspects, including the qualitative nature of various events, their number, and metric properties.

– Distribution of the events in the space or by relation to other events.

– Functional aspects of events, i.e. their temporal occurrence as a function of foregoing events (sequential analysis).

Each set of events, Ti, Gi, Go and To, can be studied and related to according to these three types of properties. In theory at least, it is possible to characterize any group task, and in particular any creative situation, using a double-entry table for 12 cases.

This is the formal equipment of the descriptive system of group tasks proposed by Roby and Lanzetta. In an abstract analysis, however, this representation does not make the understanding of a truly psychological meaning of a specific task possible. This remark led the authors to propose a complementary notion of “critical exigencies”. This concept was introduced to cover the fact that each task requires certain behavior on the part of the group to be correctly executed, and calls for certain specific types of activities to be carried out. The implementation of these requirements should thus help to reduce the discontinuity mentioned between the structural properties of the task and the psychological or psychosocial phenomena generated by its handling. It is a different manner of contrasting the general and the particular. In a way, this is what was above called the “second level of determination”.

Roby and Lanzetta’s intention was not to put forward a theory permitting to characterize the problems rigorously, but rather to present a table for the analysis of systems of group tasks. Their framework thus permits theoretically to classify any task parting from the values relevant to the task in the 12 boxes of the analysis table, but it does not make it possible to classify the types of tasks using a specific corpus of formal properties. Thus, Roby and Lanzetta did not forge a typological tool, but, rather, a descriptive tool the general purpose of which is found precisely in the fact that the tool is deemed able to adapt itself to any task. The goal of their work was not to distribute the generalized variable “task structure” on an arbitrary scale, but rather to find a set of invariant characteristics that would make it possible to situate the various problems that appear in the life of a working group. Creative problems constitute in this context evidently merely a special case. It follows that the effort to determine the “invariants” of the analysis is probably of utmost importance and should complement any typological effort.

Finally, an adequate taxonomy of poorly defined problems must comprise a meta-linguistic analysis of their formulation in the natural language: it must be possible to establish a rigorous correspondence between a formal type and the multitude of its verbal expressions or concretizations and, in parallel, part from a specific semantics to reach a logical class it illustrates. Roqutte (1975) sketches the first attempt in this respect.

Psychologists studying the ways people solve problem have adopted a reasonable strategy. They study how people handle seemingly well-defined problems, and then apply theprocedure to the study of ill-defined tasks. In some instances shortcuts to solving an ill-defined problem are possible: seek a well-defined version of the same problem and try to solve it, or find a new definition of the problem.

Defininition or interpretation of the problem is as important in tackling well-defined tasks as it is in working with ill-defined tasks.

Adversary and non-adversary problems

This is another distinction between problems. An adversary problem is one in which the problem solver is competing with a thinking opponent, or a seemingly thinking opponent, like a chess-playing computer. In non-adversary problems the battle goes between a thinking problem solver and inert problem features. The latter may be symbolic or real, but they do not react to what the problem solver does, in order to “defeat” him, and they do not care about what the human problem solver feels.

Semantically rich and semantically impoverished problems

This distinction seems to be increasing in importance. It was elaborated by Chi and his coworkers (1982). A problem is semantically rich for the problem solver who brings a significant relevant knowledge to the problem. The opposite is true of semantically impoverished problems. As an example, consider a problem given to two problem solvers. For the domain expert it is a semantically rich problem, for the novice it is a semantically impoverished problem. This distinction thus expresses the problem-solver’s view of the problem situation, or Shaw’s familiarity within the population.

Most puzzles, IQ-tests, and the like, are semantically impoverished for most subjects. Much of psychological research has been focused on solving semantically impoverished puzzles of the non-adversary type. The semantically rich non-adversary tasks are increasing in importance. This category comprises most tasks in computer programming and in physics.