## Logic for Learning: Learning Comprehensible Theories from Structured DataThis book is concerned with the rich and fruitful interplay between the fields of computational logic and machine learning. The intended audience is senior undergraduates, graduate students, and researchers in either of those fields. For those in computational logic, no previous knowledge of machine learning is assumed and, for those in machine learning, no previous knowledge of computational logic is assumed. The logic used throughout the book is a higher-order one. Higher-order logic is already heavily used in some parts of computer science, for example, theoretical computer science, functional programming, and hardware verifica tion, mainly because of its great expressive power. Similar motivations apply here as well: higher-order functions can have other functions as arguments and this capability can be exploited to provide abstractions for knowledge representation, methods for constructing predicates, and a foundation for logic-based computation. The book should be of interest to researchers in machine learning, espe cially those who study learning methods for structured data. Machine learn ing applications are becoming increasingly concerned with applications for which the individuals that are the subject of learning have complex struc ture. Typical applications include text learning for the World Wide Web and bioinformatics. Traditional methods for such applications usually involve the extraction of features to reduce the problem to one of attribute-value learning. |

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### Contents

Introduction | 1 |

12 Setting the Scene | 5 |

13 Introduction to Learning | 10 |

14 Introduction to Logic | 16 |

Bibliographical Notes | 27 |

Exercises | 28 |

Logic | 31 |

22 Type Substitutions | 35 |

Bibliographical Notes | 127 |

Exercises | 128 |

Predicates | 131 |

42 Standard Predicates | 139 |

43 Regular Predicates | 146 |

44 Predicate Rewrite Systems | 151 |

45 The Implication Preorder | 158 |

46 Efficient Construction of Predicates | 163 |

23 Terms | 38 |

24 Subterms | 45 |

25 Term Substitutions | 55 |

26 AConversion | 64 |

27 Model Theory | 72 |

28 Proof Theory | 76 |

Bibliographical Notes | 79 |

Exercises | 80 |

Individuals | 83 |

32 Normal Terms | 89 |

33 An Equivalence Relation on Normal Terms | 93 |

34 A Total Order on Normal Terms | 95 |

35 Basic Terms | 97 |

36 Metrics on Basic Terms | 105 |

37 Kernels on Basic Terms | 115 |

Bibliographical Notes | 175 |

Exercises | 176 |

Computation | 183 |

52 Definitions of Some Basic Functions | 188 |

53 Programming with Abstractions | 193 |

Bibliographical Notes | 203 |

Learning | 207 |

62 Illustrations | 214 |

Bibliographical Notes | 240 |

Exercises | 241 |

A Appendix | 243 |

245 | |

Notation | 251 |

253 | |

### Common terms and phrases

A4 projMake Abloy abstractions algorithm Alkemy append applications argument arity AtomType background theory basic terms binary BNode Null BTree Chubb computation concat constant constructor of arity declarative programming languages default data constructor default term defined definition denoted domCard edges equations exists final predicate first-order logic follows free occurrence free variable free with relative function functional programming grounding type substitution Hence higher-order logic hypothesis language idempotent induction hypothesis initial predicate kernel knowledge representation List Int logic programming machine learning Medium metric Molecule multiset normal terms parameters predicate derivation predicate rewrite system problem projLength projNumProngs projWidth Proof Proposition redex regular predicate regularisation relative type result satisfies Conditions setExists setExistsi signature standard predicate subderivation subterm Suppose switchable term of type term substitution tn then sn top top total order transformations Triangle type List unifiable well-founded sets Xx.if Xx.t