View source for Idempotent correspondence principle

A heuristic principle in [[Idempotent analysis|idempotent analysis]]. It states that there is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semi-rings (cf. [[Idempotent semi-ring|Idempotent semi-ring]]) in the spirit of N. Bohr's correspondence principle in quantum mechanics. The field of real numbers and other fields can be treated as  "quantum objects"  with respect to idempotent semi-rings; so, idempotent semi-rings can be treated as  "classical"  or  "semi-classical"  objects and as results of a  "dequantization"  of these quantum objects. The correspondence principle is a powerful tool for applying unexpected analogies and ideas borrowed from different areas of mathematics and theoretical physics [[#References|[a1]]], [[#References|[a2]]].

For example, the well-known superposition principle in quantum theory corresponds to the idempotent correspondence principle in [[Idempotent analysis|idempotent analysis]]; this means that the Hamilton–Jacobi equation (cf. also [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) and the [[Bellman equation|Bellman equation]] (and its generalizations) are linear over appropriate semi-rings; see, e.g., [[#References|[a3]]]. There is a natural analogy between idempotent measures and probability measures (cf. also [[Probability measure|Probability measure]]). This analogy leads to a parallelism between [[Probability theory|probability theory]] and stochastic processes (cf. [[Stochastic process|Stochastic process]]) on the one hand, and optimization theory and decision processes on the other hand (cf. also [[Decision problem|Decision problem]]). That is why it is possible to develop optimization theory at the same level of generality as probability theory and the theory of stochastic processes. In particular, the Markov causality principle corresponds to the Bellman optimality principle; so the Bellman equation is an idempotent version of the [[Kolmogorov–Chapman equation|Kolmogorov–Chapman equation]] for Markov stochastic processes (see, e.g., [[#References|[a4]]] and [[Markov process|Markov process]]). The well-known [[Legendre transform|Legendre transform]] is an idempotent version of the usual [[Fourier transform|Fourier transform]]. There are many other examples of this type (see [[#References|[a1]]]–[[#References|[a3]]] and [[Idempotent analysis|Idempotent analysis]]).

The idempotent correspondence principle is also valid for algorithms and their software and hardware implementations (cf. [[Idempotent algorithm|Idempotent algorithm]]).

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> G.L. Litvinov, V.P. Maslov, "Correspondence principle for idempotent calculus and some computer applications"  J. Gunawardena (ed.) , ''Idempotency'', ''Publ. Isaac Newton Institute'' '''11''', Cambridge Univ. Press  (1998) {{ISBN|0-521-55344-X}} pp.420-443 {{ZBL|0897.68050}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> G.L. Litvinov, V.P. Maslov, "Idempotent mathematics: correspondence principle and applications"  ''Russian J. Math. Phys.'' , '''4''' :  4  (1996)  (In Russian)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> V.N. Kolokoltsov, V.P. Maslov, "Idempotent analysis and applications" , Kluwer Acad. Publ.  (1996)  (In Russian)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> P. Del Moral, "A survey of Maslov optimization theory"  V.N. Kolokoltsov (ed.)  V.P. Maslov (ed.) , ''Idempotent analysis and applications'' , Kluwer Acad. Publ.  (1996)  pp. Appendix  (In Russian)</TD></TR>
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