# What does the term "closed expression" mean?

In the field of logic systems there is a property for reasoning algorithms called incompleteness or incompletion. In this context the phrase "any closed expression that is not derivable inside the same system" appeared. My question is what means "closed expression that is not derivable".

A closed expression refers to a formula which has no free variables [1]. This is also called sentence. In a logic system you have a set of axioms which are sentences and rules which state how to derive a sentence from this [2]. If a sentence can be derived from the axioms, this means that the axioms entail this sentence. If a sentence is not derivable, it is not entailed by the axioms.

The term closed-form expression, with the word form in it, is a mathematical expression that can be evaluated in a finite number of operations. In the formal discipline of first-order logic, the technicality of no free variables is a requirement, a technicality that surrounds the use and meaning of the operators

• $$\forall$$ (for all) and
• $$\exists$$ (exists).

The most common use of the term is in calculus and other higher mathematics, where direct evaluation presents challenges. If you bind $$x$$ in the following calculus expression, solving for a closed-form for $$y$$ is possible, but one cannot bind $$y$$ and solve for $$x$$ to produce a closed-form.

$$e^x = \int \, \sin (y + \log x)$$

Few integrals reduce to a closed-form expression. The integrals in high school honours math or undergraduate calculus are chosen specifically because they submit to the technique being covered. We couldn't learn integration by parts if it was impossible. Those examples are useful but not representative. Most mathematical models of natural distributions of probability or natural phenomena that can be modeled by systems of differential equations are unusable to the vast majority of people who would benefit from using those models.

Not everyone in science and technology has experience with various techniques of evaluating those forms that cannot be expressed in closed form, which is where search algorithms come in. One can search for values of $$x$$ when $$y$$ is bound. Depending on the problem, we can download finite element analysis frameworks, machine learning frameworks, or numerical evaluation libraries.

In AI, someone with more mathematical experience will, write a paper or advise the person writing it, and papers that contain an algorithm for evaluation has become a common practice. The author or someone else will write code to realize the algorithm. That's largely how the machine learning libraries formed.

Until the representations of the models developed through scientific research are converted to either

• A closed-form expression (such as a relation or formula) or
• An algorithm (perhaps with an open-source implementation,

most people won't ever find the time to learn how to use them. Closed forms are the most usable and widely used, and algorithms are next in line. Thus those developing the theory behind the technology, upstream from engineers and other users of the technology, convert the theoretical forms into closed-form expressions where possible. Those are the forms that appear in engineering textbooks as formulas or relations, with abbreviated proofs, some examples, and problems to practice for tests.

In AI, as with other areas of technology, we may study and develop the mathematical expressions, write code that implements the published algorithms or just download what we hope is already tested code and apply it. Rarely all three. Most of us focus primarily on one part of the stream from ontological inception to the consumer.

$$\text{pioneering investigation} \\ \Downarrow \\ \text{development of theory} \\ \Downarrow \\ \text{conversion in to practical forms} \\ \Downarrow \\ \text{initial users of those forms} \\ \Downarrow \\ \text{writers of technical publications} \\ \Downarrow \\ \text{experts in using the technology}$$

Closed-form expressions (along with algorithms) are critical to the communication between those working in the middle two roles in this stream. This is one of the many reasons why science and technology are collaborative endeavours.