The most fundamental common feature of qualitative reasoning systems is the
use of the ``quantity space''. This term derives from Hayes' ``quality
space'', which in naive physics refers to a set of possible values for
properties of an entity, where the properties are meaningful ``independently
of the entities which possess them.'' Distance, therefore, is a quality space
in naive physics, because distance exists independently of particular
entities.^{3.5}

Forbus uses the term quantity space to describe an ordered set of qualitatively differentiated values of a particular quantity. The values that are present in the set are not universal, but apply to the current problem. The boiling point of water, for instance, might appear in the temperature quantity space of some problems, but not in the quantity space of a problem involving oil. The meaning of the term quantity space has been extended to apply to any set resulting from a qualitative division of a continuous value range. It has been used, for example, to describe the three-valued system used by de Kleer in the original roller coaster problem (and by many others since) - positive, negative, and zero.

A number of alternatives to the quantity space have been proposed, using various methods for dividing a continuous value range into discrete qualitatively distinguished regions. These methods include the ``fuzzy logic'' of Zadeh [Zad79], which has been used for qualitative reasoning by D'Ambrosio [D'A87], Simmons' ``Commonsense Arithmetic'' [Sim86], and Raiman's ``Order of Magnitude'' reasoning [Rai86]. The most essential feature for qualitative reasoning, whichever of these techniques are used, is that the ``discretisation'' is done with reference to landmark values in the problem itself, rather than with respect to arbitrary ranges.

Another fundamental concept in qualitative reasoning is ``envisionment'', which refers to the process of predicting and analysing changes of qualitative state. Important programs which perform envisionment on discrete systems are ENVISION, by de Kleer and Brown [dKB82] and QSIM, by Kuipers [Kui82]. The transitions between states are determined by causal relationships between components of the system.

In discrete systems, the future state of the system can be analysed in terms of individual component behaviour. Envisionment involves predicting the series of qualitative states that will result from any perturbation to the system. The envisionment depends on system topology, and on component properties, because the perturbation propagates between discrete components of the system. The results of envisionment can be used to make conclusions about function, and to determine stable states (in which any small perturbation will tend to return the system to its current state).

As well as detecting stable states, the envisionment method can predict instability. This has lead to a number of explanatory programs that use oscillators as test examples - both mechanical, electrical, and pneumatic oscillators (see Falkenhainer et. al. [FFG86]). It has also encouraged the use of qualitative reasoning methods to analyse feedback systems, since the stability of a feedback system is one of its most important properties.

The envisionment technique can be extended either by the use of ``independent experts'' (as in de Kleer's roller coaster, where a mathematical analysis could be used to define more precisely an envisioned qualitative state), or by more sophisticated envisionment (as in Kuipers' QSIM algorithm, or Forbus' qualitative process theory, which make use of derivative information as part of the system state in addition to magnitude of quantities).

Recent developments in qualitative reasoning applied to discrete domains have largely retained the de Kleer and Brown envisionment structure, and have concentrated on more sophisticated representation of causality through state transitions, or on more sophisticated organisation of the quantity space. (See [Wil86], [IS86a], [dKB86], and [IS86b], for example). On the other hand, a number of qualitative reasoning workers have become aware that these programs share a lack of power in spatial reasoning, and they have begun concentrating on applying qualitative techniques to spatially complex domains that cannot be adequately represented as networks of discrete components.