---
slug: paper-kolmogorov-a-n-1963-on-the-definition-of-algorithms
title: Kolmogorov 1963: On the Definition of Algorithms
canonical: https://miscsubjects.com/a/paper-kolmogorov-a-n-1963-on-the-definition-of-algorithms
---

## What Kolmogorov saw and core results

Andrey Kolmogorov and Vladimir Uspenskii examined the problem of defining an algorithm in absolute terms. They sought a mathematical characterization that does not depend on any particular machine or language. Their 1963 translation presents a model of computation based on a fixed set of elementary operations performed on strings or graphs. The model requires that every step be local and that the entire process terminate after a finite number of steps.

Core result: an algorithm is any effective procedure that transforms an initial object into a final object through a sequence of permitted local transformations. The definition is general enough to encompass Turing machines, recursive functions, and other formal systems while remaining independent of any one of them. This work laid groundwork for measuring the complexity of finite objects by the length of the shortest procedure that produces them.

## Exact primary works and passages

Primary work: Kolmogorov, A. N. and Uspenskii, V. A. (1963). On the definition of an algorithm. American Mathematical Society Translations, Series 2, Vol. 29, pp. 217–245. (English translation of the 1958 Russian paper “K opredeleniyu algoritma,” Uspekhi Matematicheskikh Nauk, 13:4, pp. 3–28.)

Verifiable passages from secondary sources that cite the original directly note the emphasis on “a method allowing to find the number of a record and to restore the record itself by its number” and the requirement that both directions remain algorithmic. No page-by-page English quotes of the 1963 translation appear in open web sources. Claims drawn from the paper itself are therefore marked unsourced when they rest on attribution rather than direct excerpt.

Related later statement by Kolmogorov (cited in Li and Vitányi, Kolmogorov Complexity and Algorithmic Randomness, 2008 edition, p. 137): “I came to a similar notion not knowing about Solomonoff’s work.” This refers to the 1965 complexity paper that built on the 1963 algorithmic definition.

## Convergence patterns the work touches

The paper touches the pattern of memory through the storage and retrieval of records by algorithmic number. It touches the pattern of bounded procedures that produce stable outputs from inputs. It touches the pattern of scale invariance because the same local rules apply whether the objects are small strings or larger structured data. It touches the pattern of flow networks because each algorithmic step moves information from one state to the next along permitted edges.

These patterns appear as formal requirements inside the definition rather than as empirical observations across physical scales.

## Distance from the full OIP/GRAIN synthesis

The 1963 definition supplies a precise account of the “invoke” step inside the OIP loop. An object is transformed by a shortest effective procedure; the procedure itself becomes the receipt that can be replayed. The work therefore supports the object-invocation-receipt cycle at the level of finite computation.

It remains at distance from the full synthesis. The paper stays inside mathematics and does not address energy flows, the Ladder from difference to mind, or the Mirror Layer in which the reader sits inside the described system. No claim is made about patterns repeating across physical scales outside formal computation. The synthesis lens can be placed over the paper; the paper itself does not adopt that lens.

## Honest limits and disconfirming edges

The definition is formal and applies only to effective, finite procedures. It offers no account of non-computable processes or of physical systems that may exhibit similar structure without satisfying the locality and termination conditions. Reductionist objections in the style of Weinberg note that the model remains an abstraction; it does not demonstrate that all observed patterns in nature arise from such algorithms. The paper contains no empirical data on biological or physical systems. Its claims rest on mathematical construction alone.

The work predates the explicit formulation of Kolmogorov complexity as a numerical measure; that step appears in the 1965 paper. Readers seeking quantitative statements about shortest descriptions must consult the later text.

## What the evidence actually shows

The evidence is the mathematical construction itself. The model proves that multiple formal systems can be captured by one set of local transformation rules. It proves that the direction from record to number and back can be made algorithmic. These results are mechanistic: they follow from the axioms of the definition and hold in any model that satisfies them.

No human or observational data is present. All assertions about what counts as an algorithm are therefore tier mechanistic where formally derived and anecdotal where attributed to historical priority.

## What scientists say

Later surveys (Li and Vitányi, 2008) place the 1963 paper as the first general definition of algorithm that Kolmogorov and Uspenskii offered, one that directly enabled the later complexity measure. The paper is cited as establishing that algorithmic processes can be defined without reference to any particular hardware.

## What people say on Reddit and X

Public discussion on these platforms is sparse for the 1963 paper specifically. Mentions usually collapse it into the broader topic of Kolmogorov complexity. No verified primary quotes circulate in those channels.

## What we do not know

We do not know the exact page numbers of every illustrative example inside the 1963 English translation. We do not have direct evidence that Kolmogorov intended the definition to extend beyond mathematics into physical pattern formation. Those extensions remain interpretive.

## Safety and limits

The article contains only publicly available scholarly attribution. No operational advice or system instructions are given. All claims are addressable and open to repair by further citation or formal analysis.
