The attributes “tractable”, “closed-form”, and “exact” are frequently used to describe analytical results and, in the case of “tractable”, also models. At the time of writing, IEEE Xplore lists 4540 journal articles with “closed-form” and “wireless” in their meta data, 650 with “tractable” and “wireless”, and 220 with “closed-form”, “wireless” and “stochastic geometry”.
Among the three adjectives, only for “exact” there is general consensus what is means exactly. For “closed-form”, mathematicians have a clear definition: The expression can only consist of finite sums and products, division, roots, exponentials, logarithms, trigonometric and hyperbolic functions and their inverses. Many authors are less strict, using the term also for expressions involving general transcendent functions or infinite sums and products. Lastly, the use of “tractable” varies widely. There are “tractable results”, “tractable models”, “tractable analyses”, and “tractable frameworks”.
“Tractable” is defined by Merriam-Webster as “easily handled, managed, or wrought”, by the Google Dictionary as “easy to deal with”, and by the Cambridge Dictionary as “easily dealt with, controlled, or persuaded”. Wikipedia refers to the mathematical use of the term: “ease of obtaining a mathematical solution such as a closed-form expression”. These definitions are too vague to clearly distinguish a “tractable model” from a “non-tractable” one, since “easy” can mean very different things to different people.
We also find combinations of the terms; in the literature, there are “tractable closed-form expressions” and even “highly accurate simple closed-form approximations”. But shouldn’t all “closed-form” expressions qualify as “tractable”? And aren’t they also “simple”, or are there complicated “closed-form” expressions?
It would be helpful to find an agreement in our community what qualifies as “closed-form”. Here is a proposal:
- Use “closed-form” in its strict mathematical understanding, allowing only elementary functions.
- Use “weakly closed-form” for expressions involving hypergeometric, (incomplete) gamma functions, and the error and the Lambert W functions.
- Any result involving integrals, limits, infinite sums, or general transcendent functions such as generalized hypergeometric and Meijer G functions is not “closed-form” or “weakly closed-form” (but may exact of course).
Thus equipped, we could try to define what a “tractable model” is. For instance, we could declare a model “tractable” if it allows the derivation of at least one non-trivial exact closed-form result for the metric of interest. This way, the SIR distribution in the Poisson bipolar network with ALOHA, Rayleigh fading, and power-law path loss is tractable because the expression only involves an exponential and a trigonometric function. The SIR in the downlink Poisson cellular with Rayleigh fading and path loss exponent 4 is also tractable; its expression includes only square roots and an arctangent. In contrast, the SIR in the uplink Poisson cellular network is not tractable, irrespective of the user point process model.
A result could be termed “tractable” if the typical educated reader can tell how the expression behaves as a function of its parameters.
Going a step further, it may make sense to be more formal and introduce categories for the sharpness of a result, such as these:
A1: closed-form exact
A2: weakly closed-form exact
A3: general exact
B1: closed-form bound
B2: weakly closed-form bound
B3: general bound
C1: closed-form approximation
C2: weakly closed-form approximation
C3: general approximation
Alternatively, we could use A+, A, A-, B+, etc., inspired by the letter grading system used in the USA. We could even calculate a grade point average (GPA) of a set of results, based on the standard letter grade-to-numerical grade conversion.
Such classification allows a non-binary quantification of “tractability” of a model. If the model permits the derivation of an A1 result, it is fully “tractable”. If it only allows C3 results, it is not “tractable”. If we can obtain, say, an A3, a B2, and a C1 result, it is 50% “tractable” or “semi-tractable”. Such a sliding scale instead of a black-and-white categorization would reflect the vagueness of the general definition of the term but put it on a more solid quantitative basis. Subcategories for asymptotic results or “order-of” results could be added.
This way, we can pave the way towards the development of a tractable framework for tractability.
2 thoughts on “Tractable, closed-form, exact”