Introduction

Introduction#

Table Of Contents#

Glossary#

Definitions#

  1. Definition 1.2.1: Concatenation: \(st\)

  2. Definition 1.2.2: String Length: \(l(s)\)

  3. Definition 1.2.3: Character Indices: \(s[i]\)

  4. Definition 1.2.5: Containment: \(t \subset_s s\)

  5. Definition 1.2.6: Canonization: \(\pi(s)\)

  6. Definition 1.2.7: Canon: \(\mathbb{S} = \{ \pi(s) \mid s \in S \}\)

  7. Definition 1.2.8: String Inversion: \(s^{-1}\)

  8. Definition 1.3.1: Reflective Words: \(\alpha \in R \equiv \alpha = {\alpha}^{-1}\)

  9. Definition 1.3.2: Invertible Words \(\alpha \in I \equiv {\alpha}^{-1} \in L\)

  10. Definition 1.3.3: Phrases: \(P_n = (p(1), ..., p(n))\)

  11. Definition 1.3.4: Lexicons: \(L_n = \{ p \mid \forall p: p = P_n \}\)

  12. Definition 1.3.5: Limitation: \(\Pi_{i=1}^{n} p(i)\)

  13. Definition 1.3.6: Dialect: \(D = \bigcup_{i=1}^{\infty} \{ s \in S \mid \exists p \in L_i: s = \Pi_{j=1}^{i} p(j) \}\)

  14. Definition 1.4.1: Word Length: \(\Lambda(\zeta)\)

  15. Definition 1.4.2: Word Indices: \(\zeta[[i]]\)

  16. Definition 1.4.3: Invertible Sentences: \(\zeta \in J \equiv {\zeta}^{-1} \in C\)

  17. Definition 1.4.4: Partial Sentences: \(\zeta[i:], \zeta[:i]\)

  18. Definition 2.1.1: Delimiter Count: \(\Delta(s)\)

  19. Definition 2.2.1: Pivot Characters: \(\overleftarrow{\omega_s}, \overrightarrow{\omega_s}, \omega_s\)

  20. Definition 2.2.2: Pivot Words: \(\overleftarrow{\Omega_{\zeta}}, \overrightarrow{\Omega_{\zeta}}, \Omega_{\zeta}\)

  21. Definition 2.2.3: Subvertible Sentences: \(\zeta \in \cancel{J} \equiv (\Omega_\zeta \neq \varepsilon) \land (\omega_\zeta \neq \varepsilon)\)

  22. Definition 2.3.1: σ-Reduction: \(\varsigma(s)\)

  23. Definition 2.4.1: Palindromes: \(\zeta \in P \equiv ((\varsigma(s)) = \varsigma(s)^{-1})\)

  24. Definition 2.4.2: Perfect Palindromes: \(\zeta \in K \equiv (\zeta = \zeta^{-1})\)

  25. Definition 2.4.3: Imperfect Palindromes:

Axioms#

  1. Axiom 0: Empty Axiom: \(\exists! \varepsilon\)

  2. Axiom I: Comprehension Axiom: \(\iota \in S\)

  3. Axiom II: Equality Axiom: \(s = t\)

  4. Axiom III: Decomposition Axiom: \((s \neq \varepsilon) \implies (s = {\iota}{t}) \lor (s = {t}{\iota})\)

  5. Axiom IV: Closure Axiom: \(st \in S\)

  6. Axiom V: Measure Axiom: \(l(\alpha) \neq 0\)

  7. Axiom VI: Discovery Axiom: \(\alpha[i] \neq \sigma\)

  8. Axiom VII: Canonization Axiom: \(\alpha \in \mathbb{S}\)

Theorems#

  1. Theorem 1.2.1: \(l(st) = l(s) + l(t)\)

  2. Theorem 1.2.2: \(\varepsilon \subset_s s\)

  3. Theorem 1.2.3: \((\iota \subset_s uv) \implies ((\iota \subset_s u) \lor (\iota \subset_s v))\)

  4. Theorem 1.2.4: \(\pi(\pi(s)) = \pi(s)\)

  5. Theorem 1.2.5: \(s \in \mathbb{S} \equiv \pi(s) = s\)

  6. Theorem 1.2.6: \(s,t \in \mathbb{S} \implies st \in \mathbb{S}\)

  7. Theorem 1.2.8: \(\forall s \in \mathbb{S}: ((l(s) = l(t)) \land (\forall i \in N_{l(t)}: s[i] = t[i])) \implies (s = t)\)

  8. Theorem 1.2.9: \((s^{-1})^{-1} = s\)

  9. Theorem 1.2.10: \((st)^{-1} = (t^{-1})(s^{-1})\)

  10. Theorem 1.2.11: \(u \subset_s v \equiv u^{-1} \subset_s v^{-1}\)

  11. Theorem 1.3.1: \(\alpha \in I \equiv {\alpha}^{-1} \in I\)

  12. Theorem 1.3.2: \(R \subset I\)

  13. Theorem 1.3.3: \(\exists! s = \Pi_{i=1}^{n} p(i)\)

  14. Theorem 1.3.4: \(\forall s \in D: \nexists i: (s[i+1] = \sigma) \land (s[i] = \sigma)\)

  15. Theorem 1.4.1: \(\sum_{j=1}^{\Lambda(\zeta)} l(\zeta[[j]]) \geq \Lambda(\zeta)\)

  16. Theorem 1.4.2: \(\Lambda(\zeta\xi) \leq \Lambda(\zeta) + \Lambda(\xi)\)

  17. Theorem 1.4.3: \(\zeta = \Pi_{i=1}^{\Lambda(\zeta)} \zeta[[i]]\)

  18. Theorem 1.4.4: \((\Pi_{i=1}^{\Lambda(\zeta)} \zeta[[i]])^{-1} = \Pi_{i=1}^{\Lambda(\zeta)} (\zeta[[\Lambda(\zeta) - i + 1]])^{-1}\)

  19. Theorem 1.4.5: \(\Lambda((s)(\sigma)(t)) = \Lambda(s) + \Lambda(t)\)

  20. Theorem 1.4.6: \(C \subseteq D\)

  21. Theorem 1.4.7: \(\Lambda((\zeta)(\sigma)(\xi)) = \Lambda(\zeta) + \Lambda(\xi)\)

  22. Theorem 1.4.8: \(C \subseteq \mathbb{S}\)

  23. Theorem 1.4.9: \(\zeta \in J \equiv {\zeta}^{-1} \in J\)

  24. Theorem 1.4.10: \(\zeta \in J \implies \zeta[[i]] \in I\)

  25. Theorem 1.4.11: \(\zeta \in J \implies {\zeta}^{-1}[[i]] = (\zeta[[\Lambda(\zeta) - i + 1]])^{-1}\)

  26. Theorem 1.4.12: \(\zeta \in J \implies (\Lambda(\zeta) = \Lambda(\zeta^{-1}))\)

  27. Theorem 1.4.13: \(l(s[:i]) = i\)

  28. Theorem 1.4.14: \(l(s[i:]) = l(s) - i + 1\)

  29. Theorem 1.4.15: \(s[:l(s)] = s\)

  30. Theorem 1.4.16: \(s[1:] = s\)

  31. Theorem 1.4.17: \(s = (s[:i])(s[i+1:])\)

  32. Theorem 2.1.1: \(\Lambda(\zeta) = \Delta(\zeta) + 1\)

  33. Theorem 2.1.2: \(\Delta(s) = \Delta(s^{-1})\)

  34. Theorem 2.1.3: \(l(\zeta) = \Delta(\zeta) + \sum_{i=1}^{\Lambda(\zeta)} l(\zeta[[i]])\)

  35. Theorem 2.1.4: \(\Delta(st) = \Delta(s) + \Delta(t)\)

  36. Theorem 2.1.5: \(((\Delta(s) = 2n +1) \land (s = s^{-1})) \implies (s[\frac{l(s)+1}{2}] = \sigma)\)

  37. Theorem 2.1.6: \(((\Delta(s) = 2n + 1) \land (s = s^{-1})) \implies \exists i: l(s) = 2i - 1\)

  38. Theorem 2.2.1: \(((\Delta(s) = 2n + 1) \land (s = s^{-1})) \implies (\overrightarrow{\omega_s} = \overleftarrow{\omega_s})\)

  39. Theorem 2.2.2: \(((\Delta(s) = 2n) \land (s = s^{-1})) \implies ((\overrightarrow{\omega_s} \neq \sigma) \land (\overleftarrow{\omega_s} \neq \sigma))\)

  40. Theorem 2.2.3: \(((\Delta(s) = 2n) \land (s = s^{-1})) \implies (\overrightarrow{\omega_s} = \overleftarrow{\omega_s})\)

  41. Theorem 2.2.4: \((s = s^{-1}) \implies (\omega_s \neq \varepsilon)\)

  42. Theorem 2.2.5: \(((\Delta(\zeta) = 2i) \land (\Omega_{\zeta} \neq \varepsilon)) \implies (\Omega_{\zeta} \in R)\)

  43. Theorem 2.2.6: \(((\Delta(\zeta) = 2i + 1) \land (\Omega_{\zeta} \neq \varepsilon)) \implies (\Omega_{\zeta} \in I)\)

  44. Theorem 2.2.7: IN PROGRESS

  45. Theorem 2.2.8: IN PROGRESS

  46. Theorem 2.3.1: \(\varsigma(st) = (\varsigma(s))(\varsigma(t))\)

  47. Theorem 2.3.2: \(\Delta(s) = 0 \equiv \varsigma(s) = s\)

  48. Theorem 2.3.4: \((\varsigma(s))^{-1} = \varsigma(s^{-1})\)

  49. Theorem 2.3.5: \(\varsigma(\varsigma(s)) = \varsigma(s)\)

  50. Theorem 2.3.6: \(s \subset_s t \equiv \varsigma(s) \subset_s \varsigma(t)\)

  51. Theorem 2.3.7: \(\zeta[[i]] \subset_s \varsigma(\zeta)\)

  52. Theorem 2.4.1: \(K \subseteq J\)

  53. Theorem 2.4.2: \(\zeta \in K \implies \zeta[[i]] \in I\)

  54. Theorem 2.4.3: \(\zeta \in K \implies \zeta^{-1}[[i]] = (\zeta[[\Lambda(\zeta) - 1 +1]])^{-1}\)

  55. Theorem 2.4.4: \(\zeta \in K \implies \Lambda(\zeta) = \Lambda(\zeta^{-1})\)

  56. Theorem 2.4.5: \(K \subseteq P\)

  57. Theorem 2.4.6: \(\zeta \in K \implies (\omega_{\zeta} \neq \varepsilon)\)

  58. Theorem 2.4.7: \(\zeta \in K \implies (\Omega_{\zeta} \neq \varepsilon)\)

  59. Theorem 2.4.8: \(K \subseteq \cancel{J}\)

  60. Theorem 3.2.1: \(\zeta \in K \implies ((\omega_{\zeta} = \sigma) \equiv (\zeta \in P_{-}))\)

  61. Theorem 3.2.2: \(\zeta \in K \implies ((\omega_{\zeta} \neq \sigma) \equiv (\Omega_{\zeta} \in R))\)

  62. Theorem 3.2.3: \(\zeta \in K \implies ((\omega_{\zeta} \neq \sigma) \equiv (\omega_{\Omega_{\zeta}} = \omega_{\zeta}))\)

Contents