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

More documents

Recommendations

Info

31 transformations. 5.2.1 Elements <strong>of</strong> the <strong>Calculus</strong> Elements <strong>of</strong> the calculus are constructed out <strong>of</strong> the void using three boundary distinctions: instance, abstract, and inverse. Denition. Let B denote the collection <strong>of</strong> all well-formed boundary expressions composed <strong>of</strong> three boundaries, denoted as (); []; and : B is dened recursively by four rules: 1. The void belongs to B. 2. If a belongs to B, then (a) and [a] each belong to B. 3. If b belongs to B and b6=2; then belongs to B. 4. If c 1 ;c 2 ;:::;c n (n 1) belong to B, then the unordered collection c 1 c 2 :::c n belongs to B. Rules 1, 2 and 4 are identical to those in Chapter 4. The rst rule establishes the void as the foundational element inB. The second rule creates elements by nesting elements in the instance and abstract boundaries. The fourth rule creates elements by collecting other elements. The new, third rule creates elements by nesting other elements in the inverse boundary. It works identically to the second rule, except that the black hole is excluded. In other words, is undened. The rst elementintroduced by this rule is the inverse boundary with no content: 4: The distinction rules (2 and 3) introduce the following elements into B from the void: (4);[4];;;();[];;;;;: Between the four rules, deep and wide expressions are introduced into B, including the forms: ; (([[]])):
32 5.2.2 Equivalence Axioms Elements in the three-boundary calculus are related by three equivalence axioms: involution, distribution, and inversion. The involution and distribution axioms are dened as before. Axiom 1 (Involution) Instance and abstract are functional inverses: ([A]) : =A : =[(A)]: Axiom 2 (Distribution) An instance around abstract distributes over the contents <strong>of</strong> the abstract: (A[BC]) : =(A[B])(A[C]): The new axiom <strong>of</strong> inversion denes transformations with the inverse boundary. Axiom 3 (Inversion) An element and its inverse are equivalent to void: A : = Inversion matches if the complete contents <strong>of</strong> an inverse boundary are found in the context <strong>of</strong> that boundary. Examples <strong>of</strong> inversion are given in Figure 5.1. Inversion uses void substitution. An element and its inverse can be introduced anywhere in a boundary expression, so long as both elements are put into the context. Likewise, an element and its inverse can be removed from a boundary expression, so long as both reside in the same context. These three axioms dene all <strong>of</strong> the transformations <strong>of</strong> the calculus. 5.3 The Additive Inverse The inverse boundary serves directly as the additive inverse. The additive inverse extends the natural numbers built in Chapter 4 to the integers. Adding an element to its additiveinverse reduces to the additive identity, the void. For example, 3 , 3 = 0 translates to = : The additive inverse appears generally in standard and boundary numbers as: x +(,x)=0 versus x= :
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 26 and 27: 13 this system uses string rewrite
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 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]][]))])]]

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?