## Translating Higher-Order Clauses to First-Order Clauses

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Citations: | 26 - 2 self |

### BibTeX

@MISC{Meng_translatinghigher-order,

author = {Jia Meng and Lawrence C. Paulson},

title = {Translating Higher-Order Clauses to First-Order Clauses},

year = {}

}

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### Abstract

Abstract. Interactive provers typically use higher-order logic, while automatic provers typically use first-order logic. In order to integrate interactive provers with automatic ones, it is necessary to translate higher-order formulae to first-order form. The translation should ideally be both sound and practical. We have investigated several methods of translating function applications, types and λ-abstractions. Omitting some type information improves the success rate, but can be unsound, so the interactive prover must verify the proofs. This paper presents experimental data that compares the translations in respect of their success rates for three automatic provers. 1.

### Citations

718 |
Isabelle/HOL — A Proof Assistant for HigherOrder Logic
- Nipkow, Paulson, et al.
- 2002
(Show Context)
Citation Context ... paper presents experimental data that compares the translations in respect of their success rates for three automatic provers. 1. Introduction Interactive theorem provers, such as HOL4 [5], Isabelle =-=[14]-=- and PVS [15] are widely used for formal specification and verification. They provide expressive formalisms and tools for managing large scale proof projects. However, a weakness of interactive prover... |

704 |
Types and Programming Languages
- Pierce
- 2002
(Show Context)
Citation Context ...TP proofs from an unsound translation to a sound one. The constant-typed and partially-typed translations contain enough information to reconstruct full types using standard type inference techniques =-=[18]-=-. Failure of type inference would indicate that the proof was unsound. Success would not necessarily produce a correct TSTP proof, as the reconstructed term could contain new type variables, so a fina... |

536 |
The Implementation of Functional Programming Languages
- Jones
- 1987
(Show Context)
Citation Context ... limit below 300 seconds 19 translations.tex; 7/09/2007; 15:18; p.19s20 that is exponential in the number of nested λ-abstractions. Even with all five combinators, the size of the output is quadratic =-=[17]-=-. They are inefficient: during a β-reduction, namely the evaluation of a function application, numerous occurrences of combinators must be expanded in the function body. In an attempt to improve the s... |

500 |
T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge
- Melham
- 1993
(Show Context)
Citation Context ...e proofs. This paper presents experimental data that compares the translations in respect of their success rates for three automatic provers. 1. Introduction Interactive theorem provers, such as HOL4 =-=[5]-=-, Isabelle [14] and PVS [15] are widely used for formal specification and verification. They provide expressive formalisms and tools for managing large scale proof projects. However, a weakness of int... |

206 | PVS: Combining specification, proof checking, and model checking
- Owre, Rajan, et al.
- 1996
(Show Context)
Citation Context ...ts experimental data that compares the translations in respect of their success rates for three automatic provers. 1. Introduction Interactive theorem provers, such as HOL4 [5], Isabelle [14] and PVS =-=[15]-=- are widely used for formal specification and verification. They provide expressive formalisms and tools for managing large scale proof projects. However, a weakness of interactive provers is their la... |

191 |
A new implementation technique for applicative languages
- Turner
- 1979
(Show Context)
Citation Context ...ring types again, this corresponds to the following two clauses: translations.tex; 7/09/2007; 15:18; p.4s-B(@(@(fequal,X),Y)) | X = Y B(@(@(fequal,X),X)) Following Hurd [8] (based of course on Turner =-=[25]-=-), we remove λ-abstractions by translating them to Curry’s combinators S, K, I, B and C. These are easily defined in higher-order logic by the usual combinator reduction equations. This and alternativ... |

124 | Unification under a mixed prefix
- Miller
- 1992
(Show Context)
Citation Context ...olymorphism. An example may be illustrative. Consider proving that map id, the function that applies the identity function to every element of a list, equals the identity function for lists: 1 Miller =-=[13]-=- notes that providing Skolem functions as constants yields the effect of the axiom of choice. Isabelle/HOL includes this axiom anyway, so we see this as no danger. 5 translations.tex; 7/09/2007; 15:18... |

88 | Combining superposition, sorts and splitting
- Weidenbach
- 2001
(Show Context)
Citation Context ... λ-abstractions by newly-defined functions [6], and two different combinator translations [24]. We have carried out extensive experiments on all of these translations, using the provers E [20], SPASS =-=[26]-=- and Vampire 8.1 [19]. One compact translation significantly outperforms the basic, fully-typed one. translations.tex; 7/09/2007; 15:18; p.2sCompared with our previous work [10], we test more translat... |

49 | First-order proof tactics in higher-order logic theorem provers
- Hurd
- 2003
(Show Context)
Citation Context ...the soundness of proofs when importing them into Isabelle (Sect. 2.8). We build upon the work of Hurd. He has integrated Metis, his own first-order prover, with the HOL4 interactive proof environment =-=[8]-=-. We consider alternatives to Hurd’s treatments of function applications, types and λ-abstractions, backing up our choices with extensive experimentation. A translation should preserve type informatio... |

43 | System description: E 0.81
- Schulz
- 2004
(Show Context)
Citation Context ...ich replaces λ-abstractions by newly-defined functions [6], and two different combinator translations [24]. We have carried out extensive experiments on all of these translations, using the provers E =-=[20]-=-, SPASS [26] and Vampire 8.1 [19]. One compact translation significantly outperforms the basic, fully-typed one. translations.tex; 7/09/2007; 15:18; p.2sCompared with our previous work [10], we test m... |

41 |
Super Combinators: A New Implementation Method for Applicative Languages
- Hughes
- 1982
(Show Context)
Citation Context ...l translations. We have also addressed the question of how to remove λ-abstractions from the HOL problems. We have implemented both λ-lifting, which replaces λ-abstractions by newly-defined functions =-=[6]-=-, and two different combinator translations [24]. We have carried out extensive experiments on all of these translations, using the provers E [20], SPASS [26] and Vampire 8.1 [19]. One compact transla... |

37 |
Vampire 1.1 (system description
- Riazanov, Voronkov
- 2001
(Show Context)
Citation Context ...wly-defined functions [6], and two different combinator translations [24]. We have carried out extensive experiments on all of these translations, using the provers E [20], SPASS [26] and Vampire 8.1 =-=[19]-=-. One compact translation significantly outperforms the basic, fully-typed one. translations.tex; 7/09/2007; 15:18; p.2sCompared with our previous work [10], we test more translations, we use a larger... |

32 | An LCF-style interface between HOL and first-order logic
- Hurd
- 2002
(Show Context)
Citation Context ... fully-typed, partially-typed and constant-typed translations. Finally, we say more about soundness and proof reconstruction. 2.4. The Fully-Typed Translation The fully-typed translation, due to Hurd =-=[7]-=-, is sound. A special function ti pairs each term with its type. For instance, the term X ≤ Y is translated to ti(@(ti(@(ti(le, T_a => T_a => bool), ti(X, T_a)), T_a => bool), ti(Y, T_a)), bool). tran... |

32 | Lightweight relevance filtering for machine-generated resolution problems
- Meng, Paulson
(Show Context)
Citation Context ...theorems, to be supplied as axiom clauses to assist the proof. Our integration, however, allows all known theorems to be considered as lemmas. Given a conjecture, we apply our simple relevance filter =-=[11]-=- to reduce the number of clauses from thousands to hundreds, and then call an ATP such as Vampire. From the resulting proof, we discover which lemmas were actually used, finally generating a Metis cal... |

30 | TSTP Data-Exchange Formats for Automated Theorem Proving Tools
- Sutcliffe, Zimmer, et al.
- 2004
(Show Context)
Citation Context ...king the problem small enough for Metis to prove it. Some five percent of problems are too difficult for Metis even with this reduction [16]. However, with a prover such as E that outputs TSTP format =-=[23]-=-, we can use Metis 13 translations.tex; 7/09/2007; 15:18; p.13s14 to reconstruct each proof line individually. Each clause is translated to the corresponding Isabelle/HOL assertion; it is proved by a ... |

28 | Automation for interactive proof: First prototype - Meng, Quigley, et al. |

27 |
The TPTP problem library for automated theorem proving
- Sutcliffe, Suttner
- 2005
(Show Context)
Citation Context ...the widest spread of success rates, depending on the translation used. To obtain a quantitative picture of the differences between the three translations, we chose one of the problems and used tptp2X =-=[22]-=- to summarize its syntactic features. This problem is of median size in our problem set. It has 329 clauses after relevance filtering, of which 310 are non-trivial; the remaining 19 constitute a monad... |

22 |
Another algorithm for bracket abstraction
- Turner
- 1979
(Show Context)
Citation Context ...tion of how to remove λ-abstractions from the HOL problems. We have implemented both λ-lifting, which replaces λ-abstractions by newly-defined functions [6], and two different combinator translations =-=[24]-=-. We have carried out extensive experiments on all of these translations, using the provers E [20], SPASS [26] and Vampire 8.1 [19]. One compact translation significantly outperforms the basic, fully-... |

21 | Using first-order theorem provers in the Jahob data structure verification system
- Bouillaguet, Kuncak, et al.
- 2007
(Show Context)
Citation Context ...t. 2.2). Our translations permit the use of high-performance ATPs, unmodified. ATP technology is developing rapidly, and we do not wish to be tied to a single system such as Otter. Bouillaguet et al. =-=[4]-=- have developed a translation from higher-order logic to first-order logic. They eliminate type information and have proved this to be sound and complete. They even use Isabelle/HOL. Their work is imp... |

17 |
Director strings as combinators
- Kennaway, Sleep
- 1988
(Show Context)
Citation Context ... reductions in the size of the output. Despite Turner’s claim [24, p. 269] that “the sizes of the successive terms now [form] only a linear progression,” the translation as a whole is still quadratic =-=[9]-=-. We observed reductions of approximately 30%, in the number of combinators produced. Set against this modest reduction is the larger number of combinator equations that must be used in proofs. transl... |

17 | Translating higher-order problems to first-order clauses
- Meng, Paulson
- 2008
(Show Context)
Citation Context ... provers E [20], SPASS [26] and Vampire 8.1 [19]. One compact translation significantly outperforms the basic, fully-typed one. translations.tex; 7/09/2007; 15:18; p.2sCompared with our previous work =-=[10]-=-, we test more translations, we use a larger problem set, and we take strong measures to ensure that only sound proofs are counted. Paper outline. We first describe three approaches to translating typ... |

11 | Can a higher-order and a first-order theorem prover cooperate
- Benzmüller, Sorge, et al.
- 2005
(Show Context)
Citation Context .... . . . While there probably are interesting connections to typed logics, some of the questions about those relationships are open at present . . . [2, p. 313] Even with an ATP for higher-order logic =-=[3]-=-, we would somehow need to formalize Isabelle/HOL’s unusual type system (Sect. 2.2). Our translations permit the use of high-performance ATPs, unmodified. ATP technology is developing rapidly, and we ... |

7 |
and Kong Woei Susanto. Source-level proof reconstruction for interactive theorem proving
- Paulson
- 2007
(Show Context)
Citation Context ... because all proofs are reconstructed in Isabelle. We use Hurd’s Metis prover, which generates proof objects specifically to assist reconstruction [8]. Metis has now been integrated with Isabelle/HOL =-=[16]-=-. Hurd envisaged users calling Metis with a list of hand-chosen theorems, to be supplied as axiom clauses to assist the proof. Our integration, however, allows all known theorems to be considered as l... |

5 |
Integrated proof transformation services
- Zimmer, Meier, et al.
- 2004
(Show Context)
Citation Context ...ding Isabelle/HOL assertion; it is proved by a Metis call whose arguments refer to the proof lines justifying that inference. The idea is similar to that of the Otterfier proof transformation service =-=[27]-=-, which pushes arbitrary resolution proofs through Otter. Our implementation of proof reconstruction [16] is an instance of a general approach to converting TSTP proofs from an unsound translation to ... |

3 | Mathematical induction in Otter-Lambda
- Beeson
- 2006
(Show Context)
Citation Context ...fs. We outline an algorithm to translate proofs from an unsound translation into a sound one, which can be used to test the soundness of the original proofs. A contrasting approach is Otter-λ: Beeson =-=[2]-=- has modified the source code of the Otter theorem prover, in particular its unification algorithm, to provide limited higher-order features. Beeson notes, however, that Otter-λ does not implement hig... |