Scientific journal
Modern high technologies
ISSN 1812-7320
"Перечень" ВАК
ИФ РИНЦ = 0,940

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We develop an intellectual computer program solving algebraic equations. In the abstract we analyze different aspects of a procedure of the system creation. The main difference of our system from mathematical systems like MathCad, Maple and Mathematica is that we try to implement a case-based reasoning (CBR) approach (see [1]-[3]) in order to force the system to accumulate a new knowledge from examples rather than from direct formulas. The database of solvable equations should begin with simple types and then be extended in a step by step procedure. Since due to a famous result of Galois, roots of an equation Pn(x) = 0, where Pn(x) is a polynomial of degree n, are not expressible in radicals via its coefficients for n≥ 5, so there is no universal algorithm solving all types of algebraic equations. Below we discuss an application of CBR to decision procedures for algebraic equations. We note first that each algebraic equation is a an expression Tn(x) = 0, where Tn(x) is a term in the alphabet containing symbols of independent variables and symbols for numbers (integers, rationals and reals) under a set of binary arithmetical operations {+, -, *, /} (an operation of rising into an integer power can be expressed via iteration of multiplications while rising in a rational power is not yet considered in our system). Each given expression is transformed into an equivalent (more simple or suitable) form using rules from a database of elementary transformations of terms like the following:

The choice of a better form for a given term is not unique and is ruled by heuristic algorithms. Besides, we create a database of elementary types of equations like the following:

                                         (1)

                                      (2)

                     (3)

Each type (i) containing in the database of elementary types is connected with a procedure Decision_i which obtaining term T expressing the given equation forms an array of answers X. In our system we write down such procedures manually but without principal problems such procedure can be created automatically from given formulas expressing values of variables as arithmetic functions of equation´s coefficients. When the type of an equation is determined as corresponding to case i of this database, procedure Decision_i is started to find the solution of the equation.

Let us consider a CBR approach on the example of square equation . This equation is transformed using an equivalence , that is,   , and the latter is classified as corresponding to a known type and solved by the appropriate procedure. This equation and the sequence of actions leading to the solution is added to the database of cases and in future will be used to decide square equations. We note that such approach allows us to extend easily the class of solvable equations and overcome systems like Maple in many special cases of equations.

 

References

  1. Case-Based Reasoning Technology: From Foundations to Applications, Mario Lenz, Brigitte Bartsch-Sporl, Hans-Dieter Burkhard, Stefan Wess, Lecture Notes in Artificial Intelligence (2005).
  2. Kolodner J. An introduction to case-based reazoning. Artificial Intelligence Review 6(1), 1992, 3-34.
  3. A.Aamondt, E. Plaza. Case-based Reazoning: Foundational Variations, and System Approaches, Published in A.Aamondt, E. Plaza (1994); AICom-Artificial Intelligence Communications, IOS Press. 7:1 (1994) 39-59.
  4. John R.Anderson. Cognitive Psychology and its Applications, Worth Publishers, (russ. trans. Piter Press, 2002).

 

[1] The second author was supported by RFBR grant РФФИ 05-01-00605а