## Higher-Order and Modal Logic as a Framework for Explanation-Based Generalization (1989)

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Citations: | 16 - 6 self |

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@TECHREPORT{Dietzen89higher-orderand,

author = {Scott Dietzen and Frank Pfenning},

title = {Higher-Order and Modal Logic as a Framework for Explanation-Based Generalization},

institution = {},

year = {1989}

}

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### Abstract

Logic programming provides a uniform framework in which all aspects of explanation-based generalization and learning may be defined and carried out, but first-order Horn logic is not well suited to application domains such as theorem proving or program synthesis where functions and predicates are the objects of computation. We explore the use of a higher-order representation language and extend EBG to a higher-order logic programming language. Variables may now range over functions and predicates, which leads to an expansion of the space of possible generalizations. We address this problem by extending the logic with the modal ⊔ ⊓ operator (indicating necessary truth) which leads to the language λ ⊔ ⊓ Prolog. We develop a meta-interpreter realizing EBG for λ ⊔ ⊓ Prolog and give some examples in an expanded version of this extended abstract which is available as a technical report [2]. 1

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Citation Context ...ly involve name binding constructs, and are thus best represented in a higher-order language: logics (when viewed as an object language to be manipulated), programming languages, and natural language =-=[39, 32, 29]-=-. This same lack of adequate representation also arises when one wants to reason ‘at the meta-level’ — that is, about control strategies for logic programming, theorem proving, or the EBG algorithm it... |

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Citation Context ...proach incorporating modal logic into the logic programming framework which is independent of EBG [6, 7]. For treatments of automated theorem proving in modal logics outside of logic programming, see =-=[45, 44]-=-. λ⊔⊓ Prolog. As has already been suggested, the motivation for the extended language, λ⊔⊓ Prolog, is that higher-order EBG may be realized within its underlying architecture. λ⊔⊓ Prolog is currently ... |

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Citation Context ...ly involve name binding constructs, and are thus best represented in a higher-order language: logics (when viewed as an object language to be manipulated), programming languages, and natural language =-=[39, 32, 29]-=-. This same lack of adequate representation also arises when one wants to reason ‘at the meta-level’ — that is, about control strategies for logic programming, theorem proving, or the EBG algorithm it... |

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Citation Context ...roach to learning relies primarily on explanation-based generalization (EBG) as its central mechanism [8, 1]. Recently, the logic programming paradigm has been used as a foundation for supporting EBG =-=[7, 11, 5]-=-. One argument put forward in favor of the logic programming framework is that it admits a uniform representation for all aspects of EBG: domain theory, training instance, goal, goal concept, operatio... |

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Citation Context ... is available as a technical report [2]. 1 Introduction To date, work on the explanation-based approach to learning relies primarily on explanation-based generalization (EBG) as its central mechanism =-=[8, 1]-=-. Recently, the logic programming paradigm has been used as a foundation for supporting EBG [7, 11, 5]. One argument put forward in favor of the logic programming framework is that it admits a uniform... |

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Citation Context ...the inclusion of an implication within the preconditions of a derived rule. Del Cerro takes another approach incorporating modal logic into the logic programming framework which is independent of EBG =-=[6, 7]-=-. For treatments of automated theorem proving in modal logics outside of logic programming, see [45, 44]. λ⊔⊓ Prolog. As has already been suggested, the motivation for the extended language, λ⊔⊓ Prolo... |

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Citation Context ...cknowledgments. Our meta-interpreter depends on the eLP, the implementation of λProlog developed by Conal Elliott and Frank Pfenning in the framework of the Ergo project at Carnegie Mellon University =-=[12]-=-. We thank Conal Elliott, Masami Hagiya, Haym Hirsh, Dale Miller, Tom Mitchell, and William Scherlis for their thoughtful comments on our presentation. 2 First-order EBG in the Logic Programming Frame... |

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Citation Context ...roach to learning relies primarily on explanation-based generalization (EBG) as its central mechanism [8, 1]. Recently, the logic programming paradigm has been used as a foundation for supporting EBG =-=[7, 11, 5]-=-. One argument put forward in favor of the logic programming framework is that it admits a uniform representation for all aspects of EBG: domain theory, training instance, goal, goal concept, operatio... |

2 |
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Citation Context ...is, EBG in which functions and predicates as well as firstorder constants may be abstracted, or replaced with variables. Recently, the logic programming paradigm has been used as a foundation for EBG =-=[27, 40, 22, 2]-=-. One argument put forward in favor of the logic programming framework is that it admits a uniform representation for all aspects of EBG: domain theory, training instance, goal, derived rule, operatio... |

2 |
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Citation Context ...the inclusion of an implication within the preconditions of a derived rule. Del Cerro takes another approach incorporating modal logic into the logic programming framework which is independent of EBG =-=[6, 7]-=-. For treatments of automated theorem proving in modal logics outside of logic programming, see [45, 44]. λ⊔⊓ Prolog. As has already been suggested, the motivation for the extended language, λ⊔⊓ Prolo... |

2 |
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Citation Context ...nt, rather than the architecture, is in a position to control assimilation. This approach stands in sharp contrast to systems such as SOAR in which learning is confined to the underlying architecture =-=[28, 41]-=-. Interaction and EBG. Explanation-based generalization is often labeled ‘speed-up’ learning in that EBG extends the domain theory by constructing new rules in the deductive closure of that domain the... |

1 |
Dietterich et al. Learning and inductive inference
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- 1982
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Citation Context ...rning and generalization (SBG), on the other hand, rely upon multiple training examples (often both positive and negative) to arrive at an articulation of the sharing among those (positive) instances =-=[1, 8]-=-. While the proof-based generalizations of EBG are necessarily valid, similarity-based generalizations are guaranteed only to the extent that they cover the given examples. Hybrids of these approaches... |