# Computational irreducibility

Computational irreducibility is one of the main ideas proposed by Stephen Wolfram in his book A New Kind of Science.

## The idea

Wolfram terms the inability to shortcut a program (e.g., a system), or otherwise describe its behavior in a simple way, "computational irreducibility". The empirical fact is that the world of simple programs contains a great diversity of behavior, but, because of undecidability, it is impossible to predict what they will do before essentially running them. The idea demonstrates that there are occurrences where theory's predictions are effectively not possible. Wolfram states several phenomena are normally computationally irreducible.

Computational irreducibility explains observed limitations of existing mainstream science. In cases of computational irreducibility, only observation and experiment can be used. Computational irreducibility may also provide a scientifically-based resolution for free will.

## Implications

• There is no easy theory for any behavior that seems complex.
• Complex behavior features can be captured with models that have simple underlying structures.
• An overall system's behavior based on simple structures can still exhibit behavior indescribable by reasonably "simple" laws.

## Analysis

Israeli and Goldenfeld found that some less complex systems behaved simply and predictably (thus, they allowed approximations). However, more complex systems were still computationally irreducible and unpredictable. It is unknown what conditions would allow complex phenomena to be described simply and predictably.

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