Henri Poincare (1854 - 1912), arguing on the future of mathematics, specifies (see [7] page 294): "The best method for a prediction of the future development of mathematical sciences consists in studying a history and a present condition of these sciences".
The retrospective sight should convince us that consistently carried
out positional way of designations having huge advantages before literal,
should not stop on numbers, but should be continued also on functions:
you see concept of function are the following after number fundamental
concept of mathematics. And if it was not made, we have all bases to assert
that or the concept of positionality was not realized up to the end or
there are not learnt lessons from a history of our mathematical science.
Really, in modern mathematics in representation of function f (x)
it is used positionality only for x but for f we continue
to use not positional symbols. Really, those problems of discrete mathematics
and an artificial intellect (logic transformations, a logic conclusion,
logic and functional programming and many different things) which are considered
in the theory of NP-completeness use the text in not positional record.
Really, symbols of logic operations: &
(conjunction), v (disjunction), O
(addition on the module two), !
(function Webb), / (stroke Sheffer) and differents do not change the sense
from a position. The same expansions of system of operations which are
undertaken by increase àðíîñòè too keep the literal character that is why
do not relieve and can not relieve of "combinational explosions".
The unique way is necessity of expansion of positionality principle from arguments of function on symbols of these functions that is in realization full arithmetization of functions.
Let's notice that arithmetization by K.Godel (1906 - 1978) carried out by him in 1931 at the proof of the theorem of incompleteness as it is far from the purpose formulated above as systems Archimedes and Apollony, stated in them "Psammit" and "Qiuck-calculation" from positional representation of numbers, but these analogies are rather useful meaning that they specify new opportunities which should be still realized.
Necessity full arithmetization of functions on the basis of positionality principle becomes an actual problem not only in applied (computing) mathematics, but also for information technologies and for developers of new architecture of computers, in particular with approach limiting values of physical restrictions (speed, miniaturization). Let's illustrate told on the following facts of a recent history.
As is known [8], the overall aim of the Japanese project was to develop computers more productive, more flexible, more competent, more intellectual. Decisions of problems and reception of logic conclusions is an overall objective of system of the fifth generation which, among different things, should allow to reach "friendliness" in relation to the user.
Now we know that the formulated purposes were not achieved to the full (term of input in build the first prototype of system it was assumed in 1991) and could not be achieved without the appropriate fundamental discovery (about it below).
After publication of the project (1981) it was subjected to the critical analysis [9, 10]. For example, the author of a well-known method of resolutions John Robinson in the public lecture read in institute ICOT in Tokyo in 1983 has specified (see [10]) in very soft form that from planned can be achieved and that is it problematic.
Criticism has resulted to that character of the project was specified and, the most important, the ultimate goal was changed, namely, it was specified [11] that the project at all does not assume creation after end of a ten years' stage of any commercial product: its purpose is performance of the basic researches connected to development of new technology of computers.
Criticism had strong reasons. To time of creation of the Japanese project mathematics know that for a way of creation of effective methods of the decision of discrete problems there is a central theoretical-methodological problem of all discrete mathematics: Whether it is possible to exclude reassemble at the decision of discrete problems? In different words, the question is a basic opportunity to find the necessary decision not reassemble all or nearly so all variants in a problem. This problem has not only only mathematical but also deep cognitive value. The applied party (we shall notice that the problem of a logic conclusion is a problem of discrete mathematics) this problem is those: in reassemble problems, as a rule, there is a final set of variants among which it is necessary to find the decision. For example, binary vectors of dimension n is present 2n and having reassembled it exponentiak set of vectors we can find those of a vector which satisfy to the given property. But with growth n the number of vectors quickly grows and the problem becomes "hard-solved" that is practically unsoluble. Began standard to consider reassemble a problem solved effectively if there is an algorithm solving it in time limited a polynom from dimension of a problem.
Thus, in the mentioned above problem the main objects of the theory are: class NP of all reassemble problems and class P of reassemble problems solved for polynomial time. Concerning classes P and NP there is a lot of researchers among which here we shall note only results of two winners of premium of Turing [12]: S.Cook and R.Karp.
In S.Cook 1971 has shown in the basic work [13] that the problem of satisfiability is complete in class NP be relative of polynomial reduction, or is shorter, NP-complete. Roughly speaking, NP-complete problems have the maximal difficulty among all problems of reassemble. From here follows that if the problem of satisfiability is easy, any problem of reassemble is easy (that is belongs to class P). From more accurate information of theorem of Cook stronger statement, namely such follows even: the fast algorithm for the decision of a problem of satisfiability quite mechanically would result in fast allowing procedure for any effectively given problem of reassemble. Such algorithm would serve as a master key to problems of reassemble from all areas of mathematics.
In 1972 R.Karp has considerably expanded the list of NP-complete problems (see [14]). To the present moment the majority of naturally arising problems of reassemble are classified either as P or as NP-complete (see [15]).
In [15] (on page 28) it is readable: "the question on, whether really NP-complete problems are hard-solved, now are considered one of the basic open questions of modern mathematics and theoretical cybernetics. Contrary to readiness of the majority of experts to consider that all NP-complete problems are hard-solved, progress both in the proof and in a refutation of this far-reaching assumption is rather insignificant. However, despite lacking the proof of that from NP-completeness follows hard-solved, NP-completeness of a problem means that its decision by polynomial algorithm needs at least large opening".
And nevertheless, there are all bases to assert that that NP-complete problems had the decision by polynomial algorithm, it is necessary to distribute a positionality principle to representations of functions with which help ñonditions of a NP-complete problem enter the name. In case of a problem of satisfiability to which are reduced hard-solvrd problems, it means that the positionality principle should be distributed on representations of functions of algebra of logic.
Arithmetization of functions of algebra of logic on the basis of positionality principle that is development of positional notation of functions and systems of calculation that is system of operating with functions in their positional representation, began in first half of 1978. The first publications on this theme have appeared only in 1981. The list [16-21] does not cover all volume of work: the beginning of researches is reflected in the brief form in article [17]. In more developed kind it is made in publications [16, 18]. Expansion of positionality principle and its development is reflected in [19-22]. Later it was shown that the positionality principle is similarly distributed and to functions k-unit and k * m-unit logics. But nevertheless the most important is the judgement and a system available to this time of results, make a material stated in this volume.