Idempotent mathematics: A correspondence principle and its applications to computing. (English. Russian original) Zbl 0916.49019

Litvinov, G. L.; Maslov, V. P.

Russ. Math. Surv. 51, No. 6, 1210-1211 (1996); translation from Usp. Mat. Nauk 51, No. 6, 209-210 (1996).

In this short communication the authors are stressing the possible efficiency of using the so-called “idempotent mathematics” presented in some detail in previous works [e.g., V. P. Maslov and V. N. Kolokol’tsov, “Idempotent analysis and its applications to optimal control” (1994; Zbl 0857.49022)]; the main argument seems to be the fact that essentially nonlinear mathematical objects in traditional mathematics over fields become “linear” when translated into analogous objects over a semiring \((A,\oplus ,\odot)\) with commutative idempotent addition \(\oplus \), multiplication \(\odot\) with identity element \(1\) and standard partial order: \(a<b\) iff \(a\oplus b=b\).
Besides certain problems (e.g., idempotent measure and integration, Hamilton-Jacobi and Bellman equations, standard problems in dynamic programming, etc.) already discussed in previous works, the authors are pointing out here the possibility to use idempotent analogues of standard numerical algorithms and their computer realizations, a problem that seems to be discussed in more detail in [G. V. Litvinov and V. P. Maslov, “The correspondence principle for idempotent calculus and some computer applications” (1998; Zbl 0897.68050)].

Reviewer: S.Mirica (Bucureşti)

MSC:

49L20	Dynamic programming in optimal control and differential games
16Y60	Semirings
68W30	Symbolic computation and algebraic computation
65Y99	Computer aspects of numerical algorithms

Keywords:

idempotent semiring; semimodule of functions; idempotent measure; idempotent integral; Bellman equation; semiring algorithms