White-Box/Black-Box Principle in Expression Manipulation: How Much Can Be Automated?. Rein Prank University of T a rtu (Estonia ) r [email protected]
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University of Tartu(Estonia)
The paper analyzes known theoretical results concerning the possibility to construct necessary computational kernels forinput-based and rule-based learning environments in step-by-step expression manipulation.
Prof Buchberger proposed in 1990 possibility to construct necessary computational kernels forWhite Box/Black Box Principle for using symbolic mathematics software in teaching/learning of mathematics
The possibility to construct necessary computational kernels forPrincipledivideslearning of area X into two stages:
The student has marked a subexpression and the program displays menu with applicable rules.
The student has copied the expression to next line and changes now 10x+14 to 2(5x+7). The crossed out sign of equality indicates that the expressions are at the moment not yet equivalent.
A(x1,…,xn) and B(x1,…,xn) are equivalent they represent identical functions i. e. A(x1,…,xn) and B(x1,…,xn) are defined at the same points and are equal wherever they are defined
Some operations do not preserve equivalence (reducing algebraic fractions )
P1. First-order theory of structure R; 0, 1, +, -, , < is decidable(Tarski, 1951)
- whether two expressions are equivalent on R,
- whether two equations/inequalities/equation systems are equivalent,
can be expressed by corresponding first-order formulas are decidable
The original solution algorithm of Tarski is essentially improved - Caviness, Johnson (ed.), 1998
Main idea of testing the equivalence is estimation of upper bound of the number of roots of difference of two expressions
This allows make conclusion about equivalence if the difference is zero in sufficiently many points.
P3. computerized. If Schanuel’s Conjecture (for R) is true then first order theory R; 0, 1, +, -, , exp,< isdecidable. (Macintyre, Wilkie 1996)
If z1,...,zn are real numbers linearly independent
over Q, then the extension field
Q(z1,..., zn, exp(z1),...,exp(zn))
has transcendence degree of at least n (over Q).
N1. (Richardson 1968 + Matiyasevich 1970).
Let F denote the class of functions in one real variable that can be defined by expressionsconstructed from- variable x, - integers and π, - addition, subtraction, multiplication, sin, abs.
Then equivalence of expressions in F is undecidable.
|x| = sqrt(x2)
Equivalence problem for propositional formulas can be solved using truth-tables.
A.Church proved in 1936 that there exists no algorithm for decision of Entscheidungsproblem (question whether a formula of predicate logic is a consequenceof a finite set of axioms). This means also that there is no algorithm for checking of equivalence in predicate logic.
In predicate logic the most well-known class of expressions that has decidable equivalence problem, is monadic logic (where the formulas contain only predicates with one argument).
Corollary. Input-based expression manipulation environment is possible for propositional logic but not for predicate logic.
Usual laws of ring together with numerical calculations are sufficient for transformation of every expression containing rational numbers, variables, plus, minus, multiplication and exponentiation by integer to any equivalent expression.
This follows from the fact that any such expression can be transformed to canonical form.
Consider the structure N+; 1, +, , , where N+is set of positive natural numbers.
Already Dedekind’s monograph from 1888 “Was sind und was sollen die Zahlen?” contains basic identities for this structure:
Tarski asked in sixtieswhether these identities allow
to prove all valid in N+ equalities
In 1980 A.J.Wilkie built the following identity W(x,y) and proved that it cannot be derived from (1)-(11):
((1+x)y + (1+x+x2)y)x ((1+x3)x + (1+x2+x4)x)y =
= ((1+x)x + (1+x+x2)x)y ((1+x3)y + (1+x2+x4)y)x
Wilkie used proof-theoretical methods in his proof.
R. Gurevič constructed in 1985 a finite model of axioms (1)-(11) containing 59 elements where W(x,y) does not hold.
The paper of Burris and Yeats (2004) contains countermodel with only 12 elements.