Computational irreducibility

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

Principle of Computational EquivalencedisagreeNKS
Computational irreducibility is one of the main ideas proposed by Stephen Wolfram in his book A New Kind of Science.
For instance, he argues that the concept of computational irreducibility (that some complex computations are not amenable to short-cuts and cannot be "reduced"), is ultimately the reason why computational models of nature must be considered in addition to traditional mathematical models.

Stephen Wolfram

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

System

systemssubsystemsubsystems
Wolfram terms the inability to shortcut a program (e.g., a system), or otherwise describe its behavior in a simple way, "computational irreducibility".

Empiricism

empiricistempiricalempirically
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.

Fact

factsscientific factaccurate
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.

Behavior

behaviourbehavioralbehaviors
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.

Undecidable problem

undecidableundecidabilitysemi-decidable
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.

Prediction

predictpredictionspredictive
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.

Phenomenon

phenomenaphenomenalphysical phenomena
Wolfram states several phenomena are normally computationally irreducible.

Free will

freedomfreewillfreedom of the will
Computational irreducibility may also provide a scientifically-based resolution for free will.

Approximation

estimateapproximate
Israeli and Goldenfeld found that some less complex systems behaved simply and predictably (thus, they allowed approximations).

Gödel's incompleteness theorems

Gödel's incompleteness theoremincompleteness theoremincompleteness theorems

Emergence

Holism in science

holisticHolistic scienceHolism
Natural systems can produce surprisingly unexpected behavior, and it is suspected that behavior of such systems might be computationally irreducible, which means it would not be possible to even approximate the system state without a full simulation of all the events occurring in the system.

Cellular automaton

cellular automataCACell games (cellular automaton)
Additionally, during this period Wolfram formulated the concepts of intrinsic randomness and computational irreducibility, and suggested that rule 110 may be universal—a fact proved later by Wolfram's research assistant Matthew Cook in the 1990s.