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# Problem Space and Goals

Using the work of Newell and Simon (1972), we can define some of the basic concepts of human problem solving. They postulate that problem solving takes place by search in a problem space. When a problem is first presented, it must be recognized and understood. Then a problem space must be constructed or, if one already exists in LTM, merely evoked. Newell and Simon (1972) define a problem space as consisting of the following:

1.  A set of elements, U, which are symbolic structures, each representing a state of knowledge about the task.
2.  A set of operators, Q, which are information processes, each producing new states of knowledge from existing states of knowledge.
3.  An initial state of knowledge, u*, which is the knowledge about the task that the problem solver has at the start of problem solving.
4.  A problem, which is posed by specifying a set of final, desired states G, to be reached by applying operators from Q.
5.  The total knowledge available to a problem solver when he is in a given state of knowledge, which includes (ordered from most transient to most stable):

a.  Temporary dynamic information created and used exclusively within a single knowledge state.
c. Access information to the additional symbol structures held in LTM or external memory(EM).
d. Path information about how a given knowledge state was arrived at and what other actions were taken in this state if it has already been visited on prior occasions.
e. Access information to other knowledge states that have been reached previously and are now held in LTM or EM.
f. Reference information that is constant over the course of problem solving available in LTM or EM.

With this working definition of problem space, Newell and Simon then give some generalizations of the invariant features of problem spaces:

1.  The set of knowledge states is generated from a finite set of objects, relations, properties, and so on, and can be represented as a closed space of knowledge.
2.  The set of operations is small and finite or at least finitely generated.
3.  The available set of alternative nodes in the space to which the problem solver might return is very small; in fact, it usually contains only one or two nodes.
4.  Problem solving takes place by search in the problem space—-i.e., by considering one knowledge state after another until (if the search is successful) a desired knowledge state is reached. The moves from one state to the next are primarily incremental.
5.  The search involves backup--that is, the individual returns from time to time to old knowledge states and hence abandons present knowledge states.
6.  The knowledge state is typically only moderate in size, containing at most a few hundred symbols and more typically a few dozen.

It is within the problem space that the algorithms utilized during the problem solving sequence are contained. At the outset, the algorithm is followed by reference, step-by—step, to a recipe stored in external memory. The recipe is then memorized, but still has to be executed by a step-by--step interpretation. Next, the memorized recipe is “mechanized”, or compiled in the internal language of programs, so that it can now be executed directly without interpretation. Finally, more or less independently of the previous sequence, an understanding may be acquired of the logical justification for the algorithm-—of why it works.

Note that a high level of mechanization can be achieved in executing the algorithm, without any evidence of understanding; and a high level of understanding can be achieved at a stage where the algorithm still has to be followed from an externally stored recipe (Newell and Simon, 1972).

We can sum this up by saying that the particular memories and processing rates which characterize humans determine that the problem space is a major invariant of problem solving and that all problem solving occurs in some problem space. Problem solving can be effective only if significant information about the environment is encoded in the problem space, where it can be used by the problem solver. Since the function of the program is to search in the problem space, it must make the decisions necessary to operate in such a space. Possible programs can be categorized by the type of information available.

The IPS does have goals, a goal being a symbol structure with certain characteristics. First, a goal carries a test to determine when some state of affairs has been attained, in which case the goal is satisfied. Second, a goal is capable of controlling behavior under appropriate conditions. If there are goals, then the program must contain processes for creating goals, testing them, updating them, and selecting methods for attempting them, discarding them, and so on.

Since directed activity does take place in a problem solving system, the important issue is to distinguish goal behavior from other forms of directed behavior. According to Newell and Simon (1972), we can list a number of criteria for making the distinction, where all depend on the symbolized goal structure that gives the system a way of remembering both that it has a goal and where it is along the way toward attaining that goal. Other systems for directed behavior are more stimulus bound, since they do not retain explicit data about progress towards goals in order to organize behavior. Thus, a key behavioral feature of goals is that they produce correlations of behavior over long periods of time.

Some specific criteria for recognizing a goal—directed IPS are (Newell and Simon, 1972):

1.  Interruptibility. If the IPS is removed or distracted from a situation, it later returns to directed activity at the same point.
2.  Subgoaling. The IPS itself interrupts its activity toward a goal to engage in an activity that is a means to that goal, and then returns to the activity directed toward the original goal, making use of the means produced by the subgoal.
3.  Depth-first subgoaling. When the subgoaling behavior indicated above occurs to a depth of several goals, the evidence is particularly conclusive.
4.  Equifinality. If one method for attaining a goal is attempted and fails, another method towards the same goal, often involving quite different overt behavior, is then attempted.
5.  Avoidance of repetition. More generally, the system operates with a memory of its history of attempts on goals, so as to avoid repetition of behavior.
6.   Consummation. If the goal situation is attained, effort is terminated with respect to the goal.