White-Box/Black-Box Principle in Expression Manipulation: How Much Can Be Automated?

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White-Box/Black-Box Principle in Expression
Manipulation How Much Can Be Automated?

• Rein Prank
• University of Tartu (Estonia)
• rein.prank_at_ut.ee

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• 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.

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Content

1. White Box/Black Box Principle and expression
   manipulation dialog schemes
2. Results about decidability of equivalence of
   expressions
3. Existence of complete set of rules

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1. White Box/Black Box Principle andexpression
manipulation dialog schemes
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• Prof Buchberger proposed in 1990 White Box/Black
  Box Principle for using symbolic mathematics
  software in teaching/learning of mathematics

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• The Principle divides learning of area X into two
  stages
• In the stage where area X is new to the students
  . Students have to study the area thoroughly,
  i.e. they should study problems, basic concepts,
  theorems, proofs, algorithms based on the
  theorems, examples, hand calculations.
• In the stage where area X has been thoroughly
  studied, when hand calculations for simple
  examples become routine and hand calculations for
  complex examples become intractable, students
  should be allowed and encouraged to use the
  respective algorithms available in the symbolic
  software systems.

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• The paper of 1990 does not speak about creating
  of special programs for computer aided
  teaching/learning
• General-purpose mathematics software (CAS)
  divides the roles between user and computer
  corresponding to Black-Box stage of Principle.
• But the White-Box stage seems to be left in 1
  to traditional technology.
• If we examine the situation today then we
  discover that the expression manipulation
  scenarios for both stages of Principle have been
  implemented in educational software

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Black-Box Stage Rule-based dialog

• In 1990 B.Buchberger had in mind computer algebra
  systems as software for Black-Box. But CASs tend
  to have small number of too powerful rules
• There exists at least one big program that
  implements the Black-Box work better -MathXpert
  Calculus Assistant (M.Beeson).The rules of this
  program are approppriate for building of
  step-by-step solutions

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Work with rule-based program MathXpert

• The student has marked a subexpression and the
  program displays menu with applicable rules.

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• For White-Box learning stage not all the
  activities can be computerized.
• But hand calculations containing expression
  manipulation can be- conversion of expressions
  to required form, - solution of equations etc

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White-Box Stage Input-based dialog

• At White-Box stage the student should do
  everything himself, having also possibility to do
  mistakes.
• This corresponds to Input-based dialog.
• Most well-known input-based learning program is
  probably Aplusix (Nicaud et al, Grenoble
  University)

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Work with input-based program Aplusix

• The student has copied the expression to next
  line and changes now 10x14 to 2(5x7). The
  crossed out sign of equality indicates that the
  expressions are at the moment not yet equivalent.

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Theoretical requirements Input-based dialog
(White-Box )

• White-Box computerization of expression
  manipulation has sense if the computer
  provides feedback about correctness of the steps
• Most important component of correctness is
  equivalence with previous line
• The program should contain an algorithm for
  testing of equivalence (expressions, equations,
  )
• This is possible if equivalence is decidable

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Theoretical requirements for rule based dialog
(Black-Box)

• Assigned problems should be solvable using the
  rules from the menu
• Theoretical requirement existence of complete
  set of rules is necessary

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2. Results about decidability of equivalence of
expressions

• Negative results
• Positive results
• Testing by evaluation

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Definition of equivalence

• 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 )

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Positive results

• P1. First-order theory of structure ?R 0, 1, ,
  -, ?, lt? is decidable (Tarski, 1951)
• White-Box-related questions
• - whether two expressions are equivalent on R,
• whether an equation or equation system has
  solution in 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

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P2. Identity problem is solvable for expressions
in exponential ring ?N 0, 1, , -, ?, ?,lt?

• Richardson 1969,
• Macintyre 1981,
• Gurevic 1985.
• 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.

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P3. If Schanuels Conjecture (for R) is true then
first order theory ?R 0, 1, , -, ?, exp,lt?
isdecidable. (Macintyre, Wilkie 1996)

• Schanuels Conjecture
• 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).

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Negative result

• 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 p, - addition, subtraction,
  multiplication, sin, abs.
• Then equivalence of expressions in F is
  undecidable.

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CorollaryTrigonometry does not fit into White
Box

• We do not use abs very frequently. But abs can be
  expressed by
• x sqrt(x2)
• This means that the White-Box approach as it is
  implemented in Aplusix cannot be generalized to
  whole secondary school mathematics, especially to
  trigonometry.

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Testing the equivalence by evaluation

• It is quite natural to compute for testing of
  equivalence the values of two functions in some
  sample of points, and to compare them
• Some concretization of this approach is described
  in Gonnet, 1984 and used in testeq procedure of
  Maple.
• It is also available a description of educational
  application in University of Nebraska-Lincoln by
  Fisher, 1999.

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Testing equivalence by evaluation (2)

• The key point of checking by evaluation is zero
  testing of values of numerical expressions
• The warning example is the value of quite simple
  expression 3ln(640320)/sqrt(163). It differs
  from ? less than 10-15.

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Testing equivalence by evaluation (3)

• One attempt to estimate the necessary precision
  was made under name Uniformity Conjecture for
  expressions composed from integers using four
  arithmetical operations, roots, exp and log
  Richardson, 2000.
• The conjecture stated that the necessary amount
  of base S precision is proportional to the length
  of expanded expression
• Counterexamples were found in subsequent studies
  Richardson and El-Sonbathy, 2006. They have
  1000 equal decimal digits for expressions of
  length about 100 symbols. The ideas came from
  higher order approximation methods.

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Testing equivalence by evaluation (4)

• Checking by evaluation does not discover the
  differences that occur only inside of a set of
  measure zero
• Fisher warns also about expressions like
  abs(1000-x) and 1000x where the sample values
  can be too small for discovering the difference.

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Equivalence of logical expressions

• 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 consequence of 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.

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3. Existence of complete set of rules
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Polynomials

• 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.

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Tarskis High School Algebra Problem

• Consider the structure ? N 1, , ?, ?? ,
  where N is set of positive natural numbers.
• Already Dedekinds monograph from 1888 Was sind
  und was sollen die Zahlen? contains basic
  identities for this structure

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High School Identities

• xy yx,
• x(yz) (xy)z,
• x?1 x,
• x?y y?x,
• x?(y?z) (x?y)?z,
• x?(yz) x?yx?z,

• 1 x 1,
• x 1 x,
• x yz x y?x z,
• (x?y) z x z?y z,
• (x y) z x y?z.

Tarski asked in sixties whether these identities
allow to prove all valid in N equalities
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• The answer is trivially positive for first six
  basic identities and for equalities without
  exponentiation.
• Following research proved that the answer for
  full system is negative.

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Wilkies identity

• In 1980 A.J.Wilkie built the following identity
  W(x,y) and proved that it cannot be derived from
  (1)-(11)
• ((1x)y (1xx2)y)x ? ((1x3)x (1x2x4)x)y
• ((1x)x (1xx2)x)y ? ((1x3)y
  (1x2x4)y)x
• Wilkie used proof-theoretical methods in his
  proof.
• R. Gurevic 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.

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How to build learning environment for ? N 1,
, ?, ?? ?

• Standard situation input-based design is
  impossible but it is possible to build rule-based
  environment.
• Here we have reversed situation
• R. Gurevic proved in 1990 that there is no
  finite set of identities that axiomatises the
  identities of ? N 1, , ?, ?? .
• But it follows from Macintyre 1981 that the
  identities of ? N 1, , ?, ?? are
  decidable.

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References

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• M.Beeson. The mechanization of mathematics. In
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• R. Di Cosmo, T. Dufour. The Equational Theory of
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• www.cse.unl.edu/sscott/students/tfisher.pdf
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•  

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1. D.Richardson. Some unsolvable problems involving
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