What is function? What is functional information? Can it be measured?

Let’s try to clarify those points a little.

Function is often a controversial concept. It is one of those things that everybody apparently understands, but nobody dares to define. So it happens that, as soon as you try to use that concept in some reasoning, your kind interlocutor immediately stops you at the beginning, with the following smart request: “Yes, but what is function? How can you define it?

So, I will try to define it.

A premise. As we are not debating philosophy, but empirical science, we need to remain adherent to what can be observed. So, in defining function, we must stick to what can be observed: objects and events, in a word facts.

That’s what I will do.

But as usual I will include, in my list of observables, conscious beings, and in particular humans. And all the observable processes which take place in their consciousness, including the subjective experiences of understanding and purpose. Those things cannot be defined other than as specific experiences which happen in a conscious being, and which we all understand because we observe them in ourselves.

That said, I will try to begin introducing two slightly different, but connected, concepts:

a) A function (for an object)

b) A functionality (in a material object)

I define a function for an object as follows:

a) If a conscious observer connects some observed object to some possible desired result which can be obtained using the object in a context, then we say that the conscious observer conceives of **a function** for that object.

b) If an object can objectively be used by a conscious observer to obtain some specific desired result in a certain context, according to the conceived function, then we say that the object has **objective functionality**, referred to the specific conceived function.

The purpose of this distinction should be clear, but I will state it explicitly just the same: a function is a conception of a conscious being, it does not exist in the material world outside of us, but it does exist in our subjective experience. Objective functionalities, instead, are properties of material objects. But we need a conscious observer to connect an objective functionality to a consciously defined function.

Let’s make an example.

I am a conscious observer. At the beach, I see various stones. In my consciousness, I represent the desire to use a stone as a chopping tool to obtain a specific result (to chop some kind of food). And I choose one particular stone which seems to be good for that.

So we have:

a) The function: chopping food as desired. This is a conscious representation in the observer, connecting a specific stone to the desired result. **The function is not in the stone, but in the observer’s consciousness.**

b) The functionality in the chosen stone: **that stone can be used to obtain the desired result.**

So, what makes that stone “good” to obtain the result? Its properties.

First of all, being a stone. Then, being in some range of dimensions and form and hardness. Not every stone will do. If it is too big, or too small, or with the wrong form, etc., it cannot be used for my purpose.

But many of them will be good.

So, let’s imagine that we have 10^6 stones on that beach, and that we try to use each of them to chop some definite food, and we classify each stone for a binary result: good – not good, defining objectively how much and how well the food must be chopped to give a “good” result. And we count the good stones.

I call the total number of stones: the **Search space**.

I call the total number of good stones: the **Target space**

I call *–log2* of the ratio *Target space/Search space*: **Functionally Specified Information (FSI) **for that function in the system of all the stones I can find in that beach. It is expressed in bits, because we take *-log2* of the number.

So, for example, if 10^4 stones on the beach are good, the FSI for that function in that system is *–log2* of 10^-2, that is 6,64386 bits.

What does that mean? It means that one stone out of 100 is good, in the sense we have defined, and **if we choose randomly one stone in that beach we have a probability to find a good stone of 0.01 (2^-6,64386).**

I hope that is clear.

So, the general definitions:

c) **Specification.** Given a well defined set of objects (the search space), we call “specification”, in relation to that set, any explicit objective rule that can divide the set in two non overlapping subsets: the “specified” subset (target space) and the “non specified” subset. IOWs, **a specification is any well defined rule which generates a binary partition in a well defined set of objects**.

d) **Functional Specification.** It is a special form of specification (in the sense defined above), where the rule that specifies is of the following type: “The specified subset in this well defined set of objects includes all the objects in the set which can implement the following, well defined function…” . IOWs, **a functional specification is any well defined rule which generates a binary partition in a well defined set of objects using a function defined as in a) and verifying if the functionality, defined as in b), is present in each object of the set.**

It should be clear that functional specification is a definite subset of specification. Other properties, different from function, can in principle be used to specify. But for our purposes we will stick to functional specification, as defined here.

e) **The ratio Target space/Search space** expresses the **probability** of getting an object from the search space by one random search attempt, in a system where each object has the same probability of being found by a random search (that is, a system with an **uniform probability** of finding those objects).

f) **The Functionally Specified Information (FSI)** in bits is simply *–log2* of that number. Please, note that I imply no specific meaning of the word “information” here. We could call it any other way. What I mean is exactly what I have defined, and nothing more.

One last step. **FSI is a continuous numerical value, different for each function and system**. But it is possible to categorize the concept in order to have a binary variable (yes/no) for each function in a system.

So, we define a threshold (for some specific system of objects). Let’s say 30 bits. We compute different values of FSI for many different functions which can be conceived for the objects in that system. We say that those functions which have a value of FSI above the threshold we have chosen (for example, more than 30 bits) are **complex**. I will not discuss here how the threshold is chosen, because that is part of the application of these concepts to the design inference, which will be the object of another post.

g) **Functionally Specified Complex Information** is therefore a binary property defined for a function in a system by a threshold. A function, in a specific system, can be “complex” (having FSI above the threshold). In that case, we say that **the function implicates FSCI in that system**, and if an object observed in that system implements that function we say that **the object exhibits FSCI**.

h) Finally, if the function for which we use our objects is linked to a digital sequence which can be read in the object, we simply speak of digital FSCI: **dFSCI**.

So, FSI is a subset of SI, and dFSI is a subset of FSI. Each of these can be expressed in categorical form (complex/non complex).

Some final notes:

1) In this post, I have said nothing about design. I will discuss in a future post how these concepts can be used for a design inference, and why dFSCI is the most useful concept to infer design for biological information.

2) As you can see, I have strictly avoided to discuss what information is or is not. I have used the word for a specific definition, with no general implications at all.

3) Different functionalities for different functions can be defined for the same object or set of objects. Each function will have different values of FSI. For example, a tablet computer can certainly be used as a paperweight. It can also be used to make complex computations. So, the same object has different functionalities. Obviously, the FSI will be very different for the two functions: very low for the paperweight function (any object in that range of dimensions and weight will do), and very high for the computational function (it’s not so easy to find a material object that can work as a computer).

4) Although I have used a conscious observer to define function, there is no subjectivity in the procedures. The conscious observer can define any possible function he likes. He is absolutely free. But he has to define objectively the function, and how to measure the functionality, so that everyone can objectively verify the measurement. So, there is no subjectivity in the measurements, but each measurement is referred to a specific function, objectively defined by a subject.