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• W2292809968 abstract "In applied problems, extreme values of continuous real functions specified on a segment should often be found. In considering such problems, we can restrict ourselves to the unit segment [ , ] 0 1 as a domain. From the viewpoint of programmers, some convenient mathematical refinement of the concept of a continuous real function should be taken as a basis in order that algorithms (programs) of processing forms of representation of functions can be constructed for solvable problems. Here, forms of representation of continuous real functions are infinite labeled trees [1–3]. We will consider problems 1a, 1b, and 1c as programming problems. Problem 1. Based on a continuous real function f x ( ) defined on the segment [ , ] 0 1 (in the form of its computation tree), find the following numbers: (a) a number x 0 , where 0 1 0 1 0 0 ≤ ≤ = ≤ ≤ x f x f x x and ( ) ( ) | sup{ }; (b) a pair of numbers ( , ) x y 0 0 , where 0 1 0 ≤ ≤ x and y f x f x x 0 0 0 1 = = ≤ ≤ ( ) ( ) | sup{ }; (c) a number y0 equal to sup{ } f x x ( ) |0 1 ≤ ≤ . The problems formulated above should be refined from the viewpoint of programmers. In all cases, the determination of algorithmic approximating procedures (or a proof of the absence of such procedures) is meant. For problems 1a and 1b, such procedures are absent (i.e., they are not realizable), whereas problem 1c has a favorable solution (it is realizable) and the corresponding approximating program P1 is constructed below. The principle of its operation is described and some programming language is developed, which is convenient for writing approximating programs that represent functionals in the space of continuous real functions defined on the unit segment. The realizability (algorithmic realizability) is understood to be the possibility of construction of the corresponding macrotransducer (algorithmic macrotransducer) [3, 4]. Problems 2 and 3 that are considered at the end of this article are connected with problem 1c. The unrealizability of problem 1b follows from the unrealizability of problem 1a. In this article, the proof of this statement is given in the following more strict form: problem 1a is unrealizable in the class of continuous real functions f :[ , ] [ , ] 0 1 0 1 → specified by finite strict R-transducers. Note that the concept of an R-transducer was introduced in [1]. An R-transducer is called strict if the output symbols 2 or 2 are not used in it. We should develop tools of designing programs of processing of labeled trees. By labeled trees we mean computation trees for continuous real functions from the space C[ , ] 0 1 . Here, we are dealing with programming of computable functionals F C : [ , ] 0 1 → R defined on the space C[ , ] 0 1 or on its subspaces. We use some definitions and notations from [3], where the space P[ , ] 0 1 of real functions is actually found that are defined on the segment [ , ] 0 1 and are specified by R-transducers. This space is wider than C[ , ] 0 1 (it is the so-called G δ-set in the space P[ , ] 0 1 ). Therefore, it is possible and even convenient to specify functionals on the space P[ , ] 0 1 rather than onC[ , ] 0 1 . The concepts of a computation tree and a macrotransducer over computation trees are first described in [2] and then in [3]." @default.
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• W2292809968 date "2003-01-01" @default.
• W2292809968 modified "2023-09-26" @default.
• W2292809968 title "SOME EXTREMUM PROBLEMS FOR CONTINUOUS" @default.
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