Computational Irreducibility and Computational Analogy

TL;DR

This paper refines the formal definition of computational irreducibility, introduces the concept of computational analogy to classify functions, and explores their properties related to complexity and irreducibility.

Contribution

It improves the robustness of the formal definition of computational irreducibility and introduces the novel concept of computational analogy for classifying functions.

Findings

Refined the formal definition of computational irreducibility.

Introduced the concept of computational analogy as an equivalence relation.

Established properties of computationally analogous functions.

Abstract

In a previous paper, we provided a formal definition for the concept of computational irreducibility (CIR), i.e. the fact for a function f from N to N that it is impossible to compute f(n) without following approximately the same path than computing successively all the values f(i) from i=1 to n. Our definition is based on the concept of E Turing machines (for Enumerating Turing Machines) and on the concept of approximation of E Turing machines for which we also gave a formal definition. We precise here these definitions through some modifications intended to improve the robustness of the concept. We introduce then a new concept: the Computational Analogy and prove some properties of computationally analog functions. Computational Analogy is an equivalence relation which allows partitioning the set of computable functions in classes whose members have the same properties regarding to their computational irreducibility and their computational complexity.