# Halting-problem-like paradox due to predictability

Let's assume that the universe and humans are deterministic, and that I can perfectly predict the future based on the laws of physics and the environment. This means I can predict what you will do next. Now let's say you hate the idea, and will do anything to prevent my prediction coming true. For example, if I predict you that you will turn left, you will turn right instead. But this will contradict the initial prediction!

To simplify further, we can reduce this to a scenario with two computers, $$A$$ (me) and $$B$$ (you). $$A$$ is fed the electronic schematic of both machines, as well as the electronic state of each machine. $$A$$ can either output a positive or negative voltage. $$B$$ inverts the voltage of $$A$$ in order to replicate the scenario above.

Now $$A$$ is programmed to simulate the machines, and to output the voltage of $$B$$. This seems to create a halting-problem-like contradiction. If $$A$$ predicts $$B$$ outputs positive, then $$B$$ will actually output negative, and vice versa!

Unlike the algorithm in the halting problem, we already have algorithms to simulate electrical circuits. Yet it still seems yield a paradoxical result. What have I missed? Is the machine analogy equivalent to the first paragraph?

As user341440 has commented, the paradox probably arises due to the false assumption that $$A$$ can simulate $$A$$ and $$B$$.

• For example, to store the memory for the schematic of $$A$$ alone would likely use up all the available memory.
• If we used a more powerful computer $$C$$, we would have to simulate $$C$$ as well, so that wouldn't work.

Possible conclusions:

• $$A$$ cannot definitively predict the future state of systems which it can strongly influence.
• $$A$$ cannot definitively predict its own state in the future. Only some external machine $$A'$$, which does not interact much with $$A$$ (directly or indirectly), is able to do that.
• If your predictions can so easily be avoided, are they really predictions? Or just guesses, or wishes? Certainly the laws of physics are not the relevant factor when you say "you're gonna turn left!" to which I reply "Uh, no". Anyway, this is more a philosophical question than a physics question, yes? Commented Feb 24, 2022 at 18:54
• You are assuming that $A$ can simulate both itself and $B$, but this is clearly impossible - the computational resources required to simulate $A$ and $B$ are necessarily larger than the computational resources of $A$ alone. (You can't simulate a computer with less computations than the computer itself performs!) Commented Feb 24, 2022 at 19:09
• They are paradoxical simply because of your hypothetical situation. Either "participant" $A$ can perfectly predict the deterministic future, in which case "participant" $B$ cannot do anything about that, as they are subject to the same deterministic laws, or they can do something about it in which case either the "laws" were wrong, or the deterministic hypothesis does not hold. Commented Feb 24, 2022 at 19:40
• Regarding your two computers: What you describe is just a somewhat Rube Goldberg way of simulating a dynamical system. Depending on the actual parameters of the simulated system, it may or may not be easy to predict the output of B over time, but in any case, I predict with supreme confidence that if you actually build it, the universe will not implode. Commented Feb 24, 2022 at 19:41
• @aiwl imagine it takes a total time $T$ for $A$ to run the full simulation. Then $B$ reads the prediction and changes its output. This also had to be simulated of course so suppose this part takes some $\Delta T$. But this implies that $A$ had to finish simulating itself running the simulation in less time ($T- \Delta T$) than it takes for it to actually run the full simulation ! Commented Feb 24, 2022 at 20:28

Your requirements of the machines $$A$$ and $$B$$ – that $$A$$'s output is the same as $$B$$'s, and $$B$$'s is different from $$A$$'s – can't be satisfied by any physical objects. The set of machine pairs meeting the requirements is empty, so you can prove anything about members of the set, including $$P$$ and $$\neg P$$ for a $$P$$ of your choice. That isn't a contradiction because the statements you proved don't apply to anything.

It's a variant of the barber paradox.

Your question is nonsensical. You begin by stating that you can perfectly predict what I will do. You then say that I will do the opposite of what you predicted. In other words, you make two starting assumptions that are mutually exclusive, so any logic you build upon that irrational foundation will inevitably lead to irrational conclusions.

The other answers don't really explain this paradox, although this is mostly a philosophy and math question instead of a physics one.

Actually this is a paradox, known as Scriven's paradox, and the solution is the halting problem, as was shown not too long ago in V. Gijsbers. The paradox of predictability. 2023. (PDF link).

The basic conclusion is that determinism does not imply predictability. Predicting something in this case means taking your input data, performing some algorithm on it, and then outputting the answer to some question about a future state. Because of this close link to computation, Turing's result (and Rice's theorem built on top) applies here. This unpredictability is not related to chaotic systems, or quantum mechanics, or anything like that. You just simply can't compute everything.

The truth is that the universe can be deterministic while still being unpredictable, depending on your definition of prediction. Of course, the halting problem says that you can't have one general algorithm to work on all inputs, but presumably your predictor would be trying to use the laws of physics as that one general algorithm. It won't work.