I watched a video on "The first formal definition of computational irreducibility": by Kurt Jaimungal , so I asked ChatGPT a few questions to clarify what the terms meant. Here is the dialogue...
(Gary Gibson & ChatGPT)
Gary:
Would you explain what Turing completeness has to do with computational irreducibility, and define what that is too?
ChatGPT
1. Turing Completeness — What It Means
A system is Turing complete if, with enough time and memory, it can compute anything that any general-purpose computer can compute.
Examples of Turing-complete systems:
- Every modern programming language
- Conway’s Game of Life
- Wolfram’s Rule 110 cellular automaton
- Lambda calculus
- Theoretical Turing machines
The core idea:
A Turing-complete system can simulate any other algorithmic process.
This does not mean it is efficient, just that it is capable of expressing arbitrarily complex computations.
2. Computational Irreducibility — What It Means
Defined most prominently by Stephen Wolfram, computational irreducibility is the idea that:
- For many computational processes, there is no shortcut or “closed-form” method to predict the outcome;
- You must run each step, because the result cannot be reduced to a simpler calculation.
In other words:
- No formula or trick will let you jump ahead.
- The only way to know the state at step N is to compute steps 1 through N.
- The system’s evolution cannot be compressed into a simplified description.
This shows up in:
- Cellular automata
- Weather systems
- Chaotic systems
- Many natural processes (chemical, biological, ecological)
- Some mathematical sequences
If computational irreducibility is present, prediction becomes equivalent to simulation.
3. How Turing Completeness and Computational Irreducibility Relate
They are related but not identical:
Turing completeness is about what a system can compute.
It is a qualitative threshold:
Can the system simulate any computation at all?
Computational irreducibility is about how computations unfold.
It says some computations can’t be shortcut; they inherently require a full step-by-step execution.
The Key Connection
Any Turing-complete system can implement computationally irreducible processes.
Why?
If a system is powerful enough to encode arbitrary algorithms,
it is powerful enough to encode algorithms that have no shortcuts,
therefore it can host irreducible computations.
But the reverse is not true:
A system can exhibit computational irreducibility without being Turing complete.
Example: certain chaotic dynamical systems.
Why They Often Get Mixed Together
Wolfram and others use simple Turing-complete cellular automata (like Rule 110) as examples of irreducibility because:
- They are extremely simple.
- They produce patterns of explosive complexity.
- Some evolutions in these automata have been proven irreducible.
This suggests that even simple rules can produce behaviors that no shortcut can predict.
So Turing completeness + simplicity + emergent complexity = a fertile ground to observe irreducibility.
Putting it Together Simply
- Turing completeness: The system can compute anything computable.
- Computational irreducibility: Some computations cannot be predicted except by running them.
- Relationship: A Turing-complete system contains computations that are irreducible, but is not defined by them.
Computational irreducibility is a property of particular processes, not of a system’s overall capability.
If you want, I can also connect this to:
- Undecidability
- Algorithmic randomness
- Emergence in physics
- The limits of prediction in natural systems
- Or how it relates to your science-fiction computational ideas (photon fields, space-based computing, etc.).
You said:
I like the emergence in physics option, yes in math, number classes that are rational. Every equation, given certain operations to perform within select number classes, is theoretically predictable... wouldn't those that are not rationally constructed operations be the sole class of non-predictable outcomes?
ChatGPT said:
You’re pointing at something subtle and important:
In mathematics, when everything is defined inside a rational number system using well-structured operations, the outcome is deterministic and predictable. So does “unpredictability” only arise when we step outside rational constructions?
The short answer is: No — unpredictability can arise even inside fully rational systems.
But the source of that unpredictability is not “irrational numbers,” but irreducibility, undecidability, and emergent complexity within rational rules.
Let’s unpack this at several levels — physics, math, and the border between them.
1. Emergence in Physics
Physical emergence involves complex macroscopic behaviors arising from simple microscopic laws.
Key point:
Even when the underlying physics is perfectly deterministic and rationally structured (Newtonian mechanics, Schrödinger equation, cellular automata), the large-scale outcomes can be computationally irreducible.
Examples:
- Weather: governed by deterministic fluid dynamics equations, but unpredictable long-term due to chaotic sensitivity and irreducibility.
- Turbulence: Navier–Stokes equations are rational PDEs, but their behavior at scale is irreducible and partially undecidable.
- Quantum many-body systems: Hamiltonians are rational operators; high-level behavior is irreducible.
- Cellular automata (like Rule 110): finite, discrete, rational update rules; emergent patterns are irreducible and Turing-universal.
So in physics:
Rational rules → deterministic → yet irreducibly unpredictable.
Irrationality is not the cause. Complexity is.
2. The Mathematical Analogue: Rational Number Systems and Predictability
You’re absolutely right that:
- In a well-defined number system,
- with well-defined operations,
- the output of any finite expression is predictable.
For specific numeric computations, yes — it’s all reducible.
But unpredictability in mathematics rarely comes from “irrational numbers.”
It comes instead from infinite processes, algorithmic complexity, undecidability, or chaotic dynamics.
A simple example:
Consider the function:f(x)=4x(1−x)
This logistic map uses only rational operations (addition, multiplication).
Yet its iteration produces chaos:
- Small differences explode exponentially.
- Long-term prediction becomes impossible.
- The system is computationally irreducible.
All inside rational arithmetic.
Another example:
Even integer sequences can be irreducible, e.g.:
- Collatz problem
- Busy Beaver numbers
- Many recursively-defined integer sequences
These are perfectly rational, discrete systems — yet their long-term behavior cannot be predicted without brute simulation.
3. Why “Non-Rational Operations” Are NOT the Only Source of Unpredictability
You asked:
Wouldn’t those that are not rationally constructed operations be the sole class of non-predictable outcomes?
No — because:
- a. Irrational numbers themselves are perfectly predictable.
We can compute π, √2, or e to arbitrary precision. There’s nothing inherently chaotic or irreducible about them. - b. The source of unpredictability is the algorithmic structure, not the number class.
A system may involve:- Only rational numbers
- Only integers
- Only binary symbols
Yet still be unpredictable if:
- The evolution encodes a universal Turing machine,
- The dynamics are chaotic,
- The system amplifies small differences,
- The rules implicitly encode undecidable problems,
- Or the computation grows faster than any compression.
In other words:
- Rational systems can generate emergent, irreducible complexity.
4. The Deep Link to Emergence in Physics
Here’s where physics and math meet.
Physical laws are (as far as we know):
- Local
- Algebraic/differential
- Rationally expressible
Yet many physical processes are fundamentally unpredictable in the sense of computational irreducibility:
- Turbulence
- Chaotic orbital dynamics
- Protein folding
- Ecosystems
- Tokamak plasma
- High-energy quantum systems
These are all rationally defined, deterministic, and computable in principle…
…but in practice:
- The only way to know the future state is to simulate every step.
- There is no closed-form shortcut.
Just like in math, irreducibility emerges from structure, not number type.
5. Your Core Insight (Refined)
You are sensing a profound distinction:
Predictability (closed-form, reducible, shortcut)
vs.
Computability (step-by-step simulation, irreducible)
- Rational arithmetic guarantees computability, but not predictability.
- Even the "simplest" rational systems can generate behaviors that are:
- Emergent,
- Irreducible,
- Chaotic,
- Or equivalent to Turing machines.
So the core answer to your question is:
- The dividing line between predictable and unpredictable is not rational vs. irrational operations,
but reducible vs. irreducible computation.
Let me know if you'd like me to explore:
Or how your idea of natural field-based computing might transcend classical computability.
The boundary between physics and computability (is the universe itself computable?)
Whether sub-Planck phenomena require a new form of computation,
Whether the universe has “undefined operations,”
https://www.youtube.com/watch?v=WTus2fOTFNw&t=227s The First Formal Definition of Computational Irreducibility
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