AI aspects in reasoning, languages, and computation

Preface......Page 6
 Goal and Intended Readers of this Collection of Works......Page 7
Contents......Page 8
 1.1 Background on Type-Theory of Acyclic Recursion......Page 10
 1.2 Introduction to the Type Theory of Acyclic Recursion......Page 12
  1.2.2 Syntax of TTAR......Page 13
  1.2.3 Some Notations and Abbreviations......Page 17
  1.2.4 Denotational Semantics of TTAR......Page 19
  1.2.5 Canonical Forms......Page 26
  1.2.6 Standard Reduction Calculi of TTAR......Page 28
  1.2.7 Some Key Features of the Reduction Calculus of TTAR......Page 30
 1.3 Algorithmic Semantics of TTAR......Page 31
  1.3.1 Recursors and Algorithms: Referential Intensions......Page 32
  1.3.2 Natural Isomorphism and Algorithmic Equivalence......Page 34
  1.3.3 Underspecified Algorithms......Page 37
 1.4 Extended Gamma-Star Reduction Calculi of TTAR......Page 39
 1.5 Sequential Binders......Page 42
 1.6 Algorithmic Computations with Restrictions......Page 51
  1.7.1 Procedural and Declerative Neural Networks......Page 53
  1.7.2 Algorithmic Binding of Functional Neuro-Receptors......Page 54
 1.8 Conclusion and Future Work......Page 55
 References......Page 56
 2.1 Introduction......Page 58
 2.2 Building Hierarchy of Algebraic Structures......Page 59
 2.3 Implementation of Adjectives and Registrations......Page 61
  2.3.2 Clustering Adjectives with Nouns......Page 62
  2.3.3 Existential Registration......Page 63
  2.3.5 Functorial Registration......Page 64
  2.3.6 Synonyms and Antonyms......Page 65
 2.4 Enriching Structures with Attributes......Page 66
 2.5 Ring Homomorphisms......Page 71
 2.6 Extending the Hierarchy......Page 73
  2.6.1 Polynomial Rings......Page 74
  2.6.2 Ordered Fields......Page 77
 2.7 Conclusion and Future Work......Page 81
 References......Page 82
 3.1 Introduction......Page 85
 3.2 The Mizar System......Page 86
 3.3 Topological Preliminaries......Page 88
  3.4.1 Structural Versus Structure-Free Approach......Page 89
  3.4.2 Selectors and Aggregates......Page 92
  3.5.1 Basic Notions......Page 95
  3.5.2 The Closure Operator......Page 97
  3.5.4 Axiomatization of Uniform Spaces......Page 98
  3.6.1 Finite Formal Topological Spaces......Page 101
  3.6.2 Relational Operators......Page 103
 3.7 Attribute Splitting......Page 104
 3.8 Merging Topologies and Rough Sets......Page 106
 3.9 Conclusion and Future Work......Page 110
 References......Page 111