On the Notion of Algorithm

Abstract

We start from the following intuitive considerations about algorithms:

• 1) An algorithm Γ applied to any “condition” (“initial state”) A from some set G(Γ) (“domain of applicability” of the algorithm Γ) gives a “solution” (“concluding state”) B.

• 2) The algorithmic process may be subdivided into separate steps of apriori bounded complexity; each step consists of an “immediate processing” of the state S (that occurs at this step) into the state S* = Ω_Γ(S).

• 3) The processing of A ⁰ = A into A ¹ = Ω_Γ(A ⁰), A ¹ into A ² = Ω_Γ(A ¹), A ² into A ³ = Ω_Γ(A ²), etc., continues until either a nonresultative stop occurs (if the operator Ω_Γ is not defined for the state that just appeared) or until the signal saying that the “solution” has been obtained occurs. It is not excluded that the process will continue indefinitely (if the signal for the solution’s appearance never occurs).

• 4) Immediate processing of S into S* = Ω_Γ(S) is carried out only on the basis of information on the form of an apriori limited “active part” of the state S and involves only this active part.

Uspekhi Mat. Nauk, 1953, vol. 8,no. 4, pp.175–176.