Information is a Counterfactual Property

I received a small grant from the ACX Grants, allocated by the Long-Term Future Fund to write about Constructor Theory, the kind of problems that it attempts to solve, and whether it can give any insights into the AI alignment problem. This is the first of a few posts which will be on the paper Constructor Theory of Information by David Deutsch and Chiara Marletto. The next few posts will be mostly based on this paper, the paper Constructor Theory by Deutsch and The Science of Can and Can’t, Marletto’s popular book on this subject.

Imagine traveling through a field with your friend. It is late at night, dark, and foggy. You are both lost, but you suspect that on the other side of the field there is a road which will lead back home. You wish to walk across the field to see if the road is there, but your friend is tired and has hurt his foot so does not wish to walk across the field unless he is certain that the road really is at the other end. To solve this dilemma, you agree on the following plan: you will walk to the end of the field and your friend will stay put. If you find the road at the end of the field, at 11pm, you will turn on the strong lamp that you are carrying. Your friend will see the signal and follow you. If not, you will leave the lamp off, and will return back the way you came, having saved your friend an unnecessary journey. Having agreed this plan and synchronized your watches, you head off into the darkness, leaving your friend. After walking through the field for ten minutes, you find the road, turn on your lamp and send the signal to you friend, who soon joins you.

It seems reasonable to say that in this situation that you are using the lamp to transmit information to your friend. The information starts in your brain and is transferred to the switch on the lamp. Whether the lamp is on or off affects whether or not the fog in the field is illuminated, which affects what your friend sees.

We may be tempted to say that the information is encoded in each physical state at each stage of the process: the information starts as a pattern of neuron firings in your brain, is transferred into a set of electromagnetic waves traveling through the fog and then is transformed again into patterns of neurons in your friend’s brain. The information seems to be a factual feature of each stage of the procedure.

For now, let us ignore the people at either end of this chain of communication and consider only the light, as it travels from your lamp, through the fog and onto your friends eyes. What is it about this light that means that it carries information? The light itself consists of waves of electromagnetic radiation which described physically using Maxwell’s equations. We can deduce whether your friend sees the light by finding out the initial conditions of the electromagnetic field at 11pm (which are determined by whether your lamp was switched on or off) and then using Maxwell’s equations to determine how the field evolves in the subsequent time that it takes for the light to traverse the distance to your friend. If the lamp is off, the electromagnetic field will remain in its original state and the fog will remain dark. If the lamp is on, it will create oscillations in the electromagnetic field, illuminating the fog.

Is the information that is sent to your friend a factual feature of the electromagnetic field between you? In other words, is the information encoded as a property similar to, say, the amplitude or frequency of the electromagnetic waves? It might seem like it is, after all, the state of the electromagnetic field seems inextricably linked to the message that your friend receives. However, on closer inspection, we find that this intuition is mistaken. Information is not a factual property of the electromagnetic field, as we can see by considering a slightly different scenario.

Suppose that you and your friend are in the same situation, but while agreeing your plan, you both test the lamp and find that it becomes stuck in the ‘on’ position. Assuming that you cannot turn the lamp off or cover it in any way, what happens if you continue with your plan anyway? You trek to the other side of the field, find the road, and at 11pm … then what? The lamp is already on and there is nothing you can do. Notice that in this case, the electromagnetic field between you and your friend is exactly the same as in the case where the lamp worked, yet now it does not carry any information. Your friend looks out at 11pm and sees the light from the lamp, but this does not give him any information about whether the road is at the end of the field, because he knew that he would see the light regardless of what lay at the end of the field. The information encoded in the light cannot be a property of the light itself, otherwise it would be present in both scenarios. Instead, the information is a counterfactual property: it is only meaningful to say that the electromagnetic field carries information if it could have been in a different state. (We can reach a similar conclusion by considering the case where the lamp is stuck in the ‘off’ position.)

Why should we care about this distinction between factual and counterfactual properties? One reason might be if we are trying to link information with physics. In a lot of theories in physics, systems are described using what Deutsch and Marletto call the Prevailing Conception of fundamental physics (PC). Under the PC, physical systems are described in terms of initial conditions and laws of evolution. The PC has been very successful and most of our theories of physics operate under it. But the PC is not particularly well-suited to describing counterfactuals. Once the initial conditions are set, it can describe what will happen, but it cannot describe why the light in the first scenario carries information, and the identical evolution in the second scenario does not. To do this, we would need to appeal to deeper principles. We would need a way of expressing what physical transformations are possible or impossible; whether the lamp could have been switched off, or whether it couldn’t.

Constructor theory is a framework which seeks do just this (express physics in terms of which transformations are possible or impossible). It is hoped that this will allow for solutions to some of the problems which are inherent to the prevailing conception of physics while opening up new avenues of investigation and allowing us to talk about concepts like information. In future posts, I’ll explain how it does this in more detail.

One might argue that we do not need to appeal to counterfactual properties to explain the behaviour of any physical system. After all, the PC can perfectly describe the evolution of a system without reference to counterfactuals. In our two imaginary scenarios, we would be able to perfectly describe the evolution of the electromagnetic field under the PC by using the initial conditions, along with Maxwell’s equations (which provide dynamical laws describing how the field evolves). If we can do this, maybe it doesn’t matter if it cannot describe things like information? Alternatively, it might be the case that things that we think of as counterfactuals emerge naturally from the PC. Maybe introducing counterfactual properties separately into physics is superfluous conceptual baggage? In later posts, I will address some these issues, and make the case for introducing counterfactuals into physics and explain more formally how constructor theory hopes to do this.

As it has been proposed, constructor theory is not just a theory of information, it is a theory of all of physics, but one of its main avenues has been investigating properties of information and it is on this which I will mostly be focusing. The next post will be about a different counterfactual property of information: the distinguishability of signals.