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

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19 connectivities denes the system <strong>of</strong> mathematics. 3.3.1 Match and Substitute Equivalences are described by equivalence rules which transform by algebraic match and substitute. The rules given to dene a system are its equivalence axioms. An equivalence rule consists <strong>of</strong> two or more boundary templates which include template variables. To apply a rule to a target expression, replace each template variable with an expression so that one <strong>of</strong> the lled templates matches some part <strong>of</strong> the target expression. Another lled template from the rule can then substitute for the matched template in the target expression. For example, the rule ((A)) = A includes two templates which use the template variable, A. The target expression ((())) can be matched by the left template by replacing the A with to produce (())=. Substituting the right template for the left template in the target expression transforms it into (), essentially erasing two boundaries. 3.3.2 Properties Transformations in boundary mathematics have some special properties. Match and substitute can operate in parallel and at all levels <strong>of</strong> granularity in an expression. For example, the rule ((A)) = A applies to the form (( (()) () )) twice; both applications can occur simultaneously without conict to reduce it to (). In these boundary numbers, the equivalence rules are bi-directional. The rule itself states no direction <strong>of</strong> preference, though one may be imposed for reduction purposes. Therefore, a reduced expression is mathematically equivalent to many more complicated forms. Because functions are constructed with the same boundaries as are numbers, the specication <strong>of</strong> a calculation is equivalent to a reduced result. The specication is the result|transformation just changes the form that it is in. 3.4 Conclusions Boundary mathematics is a representational and computational paradigm based on spatial forms. Boundary notation looks and acts dierently than standard notation, using spatial constraints and spatial properties rather than rote rules.
20 Spatial forms dier fundamentally from standard forms. Space does not impose ordering constraints; associativity and commutativity are not necessary as no order or arity has been imposed in the rst place. Non-representation has meaning and contributes to computation in ways unparalleled in standard systems. The void acts as a built-in identity that is always available. Spatial match and substitute operates at all parts <strong>of</strong> an expression simultaneously, an inherently parallel computation mechanism. These properties make boundary mathematics a unique and worthwhile paradigm <strong>of</strong> representation and computation. Many concepts generally attributed to standard mathematics are notably absent from boundary mathematics and vice-versa. Boundary notation makes no distinction between objects and operations upon them, as both are built out <strong>of</strong> the same forms. Standard mathematics does not allow void substitution but boundary mathematics does. Boundary rules tend to embody symmetry <strong>of</strong> function whereas standard mathematics addresses identity elements and rearrangement properties. The tradeos are many. Boundary mathematics is exotic enough to seem obscure and <strong>of</strong> unclear advantage. This thesis will demonstrate the advantage <strong>of</strong> the boundary mathematics paradigm by dening in it a calculus that simplies number representation and calculation.
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Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
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Page 16 and 17: 3 Table 1.1: Simple Calculations in
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Page 22 and 23: Chapter 2 PRIOR WORK Boundary mathe
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Page 28 and 29: 15 Table 2.3: Denition of</
Page 30 and 31: 17 Figure 3.1: The Void. Figure 3
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
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Page 80 and 81: 67 [28] L. Lamport. LaTeX: A Docume
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69 A.2 Functions The standard algeb
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71 Distribution of
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73 Radians 1+e i =0 ([]) = Euler's
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75 B.2 Square Root The following sq
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77 The a coecient can now be remove
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79 f = (([[(([][]) [(([[b]][]))])]]
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