A Calculus of Number Based on Spatial Forms - University of ...

More documents

Recommendations

Info

13 this system uses string rewrite rules, ordering is signicant. The rules on the left <strong>of</strong> Table 2.2 are for doubling and inverse, whereas the rules on the right handle sequencing. In this system, numbers add by concatenation, i.e. spatial collection in one dimension. For example, 3+2 = 5. The result is transformed to a canonical form using the string rewrite rules. 3+2 Given + <strong>Number</strong> Rewrite Addition Rewrite w = w >< = = 5 <strong>Number</strong> Rewrite Multiplication is performed by replacing one expression for the units in the other. For example, 3 2=6. 32 Given <strong>Number</strong> Rewrite i Multiplication Rewrite Multiplication Substitution >< = 6 <strong>Number</strong> Rewrite Subtraction is performed by multiplying by the negative unit and then adding. The result is reduced using the rewrite rules with the negative unit. For example, 8 , 1=7.
14 8 , 1 Given , <strong>Number</strong> Rewrite + Subtraction Replacement Addition Rewrite > = > > = > > = > = = 7 <strong>Number</strong> Rewrite Though this system is limited to the integers, Kauman has dened another boundary as the opposite <strong>of</strong> the magnitude boundary, such that () = : = : A: The string rewrite rules for this system are quite extensive due to the many permutations <strong>of</strong> the characters and are not included here. Alternatively, the extended Kauman numbers can be dened spatially with three rules. The denition below uses his delimiter form <strong>of</strong> the inverse rather than the overbar form shown above [24]. Double AA = : Inverse A[A] = : Half () = : = : A Kauman numbers represent the integers and the system computes basic arithmetic operations except division and powers. Addition by collection is immediate, as is multiplication by substitution. Reduction is done locally by string rewrite rules. 2.3 Bricken <strong>Number</strong>s Kauman's work was given a dierent twist by William Bricken, who interpreted his basic forms as networks [5]. The network form provides singular references and a clear representation <strong>of</strong> the structures being manipulated. Of particular benet, singular reference allows multiplication by stacking, providing an algebraic form. Bricken numbers are summarized in Table 2.3. Bricken numbers are networks whose value is determined by their connectivity. The network includes a context and ground, shown as lines above and below the
Page 1 and 2: A Calculus <strong
Page 3 and 4: University ofWashi
Page 5 and 6: 4.2 The Two-Boundary Calcul
Page 7 and 8: A.2 Functions : : : : : : : : : : :
Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
Page 11 and 12: as an undergraduate at the Universi
Page 14 and 15: Chapter 1 INTRODUCTION Numb
Page 16 and 17: 3 Table 1.1: Simple Calculations in
Page 18 and 19: 5 contents, they are also drawn as
Page 20 and 21: 7 Table 1.4: Axioms of</str
Page 22 and 23: Chapter 2 PRIOR WORK Boundary mathe
Page 24 and 25: 11 3 2 Given Number</stro
Page 28 and 29: 15 Table 2.3: Denition of</
Page 30 and 31: 17 Figure 3.1: The Void. Figure 3
Page 32 and 33: 19 connectivities denes the system
Page 34 and 35: Chapter 4 INSTANCE AND ABSTRACT 4.1
Page 36 and 37: 23 The third rule creates new eleme
Page 38 and 39: 25 Distribution: (A[BC]) : =(A[B])(
Page 40 and 41: 27 Multiplication of</stron
Page 42 and 43: 29 Table 4.1: The Two-Boundary <str
Page 44 and 45: 31 transformations. 5.2.1 Elements
Page 46 and 47: 33 Inversion: A = : Equivalent Expr
Page 48 and 49: 35 Boundary integers add by collect
Page 50 and 51: 37 Multiplication requires manageme
Page 52 and 53: 39 Proof. Rational
Page 54 and 55: 41 For example, a half-cardinality
Page 56 and 57: Chapter 6 PHASE 6.1 Introduction Th
Page 58 and 59: 45 Using J, two concise and useful
Page 60 and 61: 47 The ambiguous sign makes the car
Page 62 and 63: 49 Table 6.1: Transcendentals in th
Page 64 and 65: Chapter 7 FUTURE WORK 7.1 Introduct
Page 66 and 67: 53 ematics problems could be phrase
Page 68 and 69: 55 boundary form, in search <strong
Page 70 and 71: 57 Technology interface sof
Page 72 and 73: 59 such asintegrals and Fourier tra
Page 74 and 75: 61 mulas specify how they transform
Page 76 and 77:
63 dominion A theorem stating that
Page 78 and 79:
BIBLIOGRAPHY [1] B. Banaschewski. O
Page 80 and 81:
67 [28] L. Lamport. LaTeX: A Docume
Page 82 and 83:
69 A.2 Functions The standard algeb
Page 84 and 85:
71 Distribution of
Page 86 and 87:
73 Radians 1+e i =0 ([]) = Euler's
Page 88 and 89:
75 B.2 Square Root The following sq
Page 90 and 91:
77 The a coecient can now be remove
Page 92:
79 f = (([[(([][]) [(([[b]][]))])]]
show all

A Calculus of Number Based on Spatial Forms - University of ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?