## Using Reflection to Build Efficient and Certified Decision Procedures (1997)

Venue: | TACS'97. Springer-Verlag LNCS 1281 |

Citations: | 49 - 0 self |

### BibTeX

@INPROCEEDINGS{Boutin97usingreflection,

author = {Samuel Boutin},

title = {Using Reflection to Build Efficient and Certified Decision Procedures},

booktitle = {TACS'97. Springer-Verlag LNCS 1281},

year = {1997},

pages = {515--529},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we explain how computational reflection can help build efficient certified decision procedure in reduction systems. We have developped a decision procedure on abelian rings in the Coq system but the approach we describe applies to all reduction systems that allow the definition of concrete types (or datatypes). We show that computational reflection is more efficient than an LCF-like approach to implement decision procedures in a reduction system. We discuss the concept of total reflection, which we have investigated in Coq using two facts: the extraction process available in Coq and the fact that the implementation language of the Coq system can be considered as a sublanguage of Coq. Total reflection is not yet implemented in Coq but we can test its performance as the extraction process is effective. Both reflection and total reflection are conservative extensions of the reduction system in which they are used. We also discuss performance and related approaches....

### Citations

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Introduction to HOL
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(Show Context)
Citation Context ...t than ours, which they call the two-level approach. Concerning the topic of defining very efficient decision procedures in the context of theorem provers, Harrisson and Thery [9] propose to link HOL =-=[7]-=- and Maple to perform efficient computations but this also links the reliability of both systems! We believe that complete reflection is sounder than such union between a computer algebra system and a... |

103 |
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Citation Context ... order to extend dynamically the power of such a system, there ought to exist a mechanism to safely add new decision procedures. Such a mechanism exists for instance in the Boyer-Moore theorem prover =-=[3]-=- which belongs to the first category. In the second category are computer aided proof-checkers, inheriting the LCF approach, where any object, proof or decision procedure has to split in a sequence of... |

49 |
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14 | Topics in automated theorem proving and program generation - Hsiang - 1982 |

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Citation Context ...ce a possible canonical form is the elimination of unnecessary occurences of oneA, and systematic association to the right. For more details about canonical forms w.r.t first order theories, refer to =-=[13]-=-. We can write a naive decision procedure on this theory using Coq, call it Monoiddec exactly as it would be possible in HOL: Repeat ((Rewrite -? neutraloneAleft) Orelse (Rewrite -? neutraloneAright) ... |

12 | Interactive theorem proving with Cambridge LCF – a user’s manual - Paulson - 1985 |

4 |
Th' ery Extending the HOL Theorem Prover with a Computer Algebra System to Reason About the Reals
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Citation Context ... the same kind of project than ours, which they call the two-level approach. Concerning the topic of defining very efficient decision procedures in the context of theorem provers, Harrisson and Thery =-=[9]-=- propose to link HOL [7] and Maple to perform efficient computations but this also links the reliability of both systems! We believe that complete reflection is sounder than such union between a compu... |

4 |
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Citation Context ... n) and (R m). What we want to do here, is to use a very nice property of the Coq system: its implementation language is almost a sublanguage of the metalanguage. Coq is implemented in Objective Caml =-=[15]-=- a dialect of ML; and Coq has an extraction process [17] from the Coq object language to Objective Caml. So we can use this extraction process to translate deleteneutral and R from Coq to Objective Ca... |

4 |
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Citation Context ...tion is performed in less than 1 second cpu so that we are not so far from Maple performances on this particular example. 7 Related approaches According to its author Richard Weyrauch, the FOL system =-=[19]-=- was the first theorem prover were reflection was considered as an essential tool for proving theorems. FOL is a proof checker for first order logic so that it was possible for the user to define sign... |

3 | et al.: Implementing - Constable - 1986 |

2 |
Metatheory and Reflection in Theorem Proving: A Survey and Critique
- Harisson
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Citation Context ...he approach described in the next section. 1 if you do not like the word type, call it a set 2 if you do not like the constant, call it an axiom 4 4 Computational reflection in Coq Following Harrison =-=[8], "co-=-mputational reflection principles do not extend the power of the logic, but may make deductions in it more efficient". We can be more precise here as what we do can be summarized as follows: rath... |

1 | et al. The Coq Proof Assistant User's Guide, V6.1, Inria technical report - Barras - 1997 |

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1 |
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Citation Context ...The work about reflection in Nuprl follow the ideas of Weyrauch but now in a sufficient powerful logical system so that the gap between theory and meta-theory is filled in the Nuprl system itself. In =-=[14], Con-=-stable and Knoblock say "The metatheories are tailored for a particular sort of meta-reasoning: representing enough of the proof theory of the previous language in the hierarchy so that proof tac... |

1 |
Paulin-Mohring Inductive Definitions
- unknown authors
- 1993
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Citation Context ...Our general approach to the building of a reflection tactical deciding for a first order theory is as follows (we illustrate the general scheme using the case of monoids): 1. define an inductive type =-=[16]-=- the constructors of which are the constants of the target theory. In the case of monoids this corresponds in Coq to the following inductive definition: Inductive Set monoid := r1 : monoid --- rmult :... |