## Efficiently Checking Propositional Resolution Proofs in Isabelle/HOL

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@MISC{Weber_efficientlychecking,

author = {Tjark Weber},

title = {Efficiently Checking Propositional Resolution Proofs in Isabelle/HOL},

year = {}

}

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### Abstract

This paper describes the integration of zChaff and MiniSat, currently two leading SAT solvers, with Isabelle/HOL. Both SAT solvers generate resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem prover. The presented approach significantly improves Isabelle’s performance on propositional problems, and exhibits counterexamples for unprovable conjectures. It is shown that an LCF-style theorem prover can serve as a viable proof checker even for large SAT problems. An efficient representation of the propositional problem in the theorem prover turns out to be crucial; several possible solutions are discussed. 1

### Citations

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220 |
CVC Lite: A new implementation of the cooperating validity checker
- Barrett, Berezin
- 2004
(Show Context)
Citation Context ...es), and we use a SAT solver, rather than a first-order prover. Other work on combining proof and model search includes [dNM06]. A custom-built SAT solver has been integrated with the CVC Lite system =-=[BB04]-=- by Clark Barrett et al. [BBD03]. While this solver produces proofs that can be checked independently, our work shows that it is possible to integrate existing, highly efficient solvers with an LCF-st... |

200 |
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(Show Context)
Citation Context ...t representation of the propositional problem in the theorem prover turns out to be crucial; several possible solutions are discussed. 1 Introduction Interactive theorem provers like PVS [ORS92], HOL =-=[GM93]-=- or Isabelle [Pau94] traditionally support rich specification logics. Proof search and automation for these logics however is difficult, and proving a non-trivial theorem usually requires manual guida... |

81 | A SAT based approach for solving formulas over boolean and linear mathematical propositions - Audemard, Bertoli, et al. - 2002 |

43 | Integrating Gandalf and HOL
- Hurd
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(Show Context)
Citation Context ...cently further explored by Joe Hurd [Hur99, Hur02], Jia Meng [Men03], and Lawrence Paulson [MP04, MP06]. Proofs found by the automated system are either verified by the interactive prover immediately =-=[Hur99]-=-, or translated into a proof script that can be executed later [MP04]. Also Andreas Meier’s TRAMP system [Mei00] transforms the output of various automated first-order provers into natural deduction p... |

41 |
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(Show Context)
Citation Context ...TP library have decreased in a similar fashion and are well below one second each now. This enables us to evaluate the performance on some significantly larger problems, taken from the SATLIB library =-=[HS00]-=-. These problems do not only push Isabelle’s inference kernel to its limits, but also other parts of the prover. While the smaller TPTP problems were converted to Isabelle’s input syntax by a Perl scr... |

37 |
TRAMP: Transformation of machine-found proofs into natural deduction proofs at the assertion level (system description
- Meier
- 2000
(Show Context)
Citation Context ...found by the automated system are either verified by the interactive prover immediately [Hur99], or translated into a proof script that can be executed later [MP04]. Also Andreas Meier’s TRAMP system =-=[Mei00]-=- transforms the output of various automated first-order provers into natural deduction proofs. The main focus of their work however is on the necessary translation from the interactive prover’s specif... |

34 | An LCF-Style Interface between HOL and First-Order Logic - Hurd - 2002 |

28 | Experiments on supporting interactive proof using resolution
- Meng, Paulson
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(Show Context)
Citation Context ...and Lawrence Paulson [MP04, MP06]. Proofs found by the automated system are either verified by the interactive prover immediately [Hur99], or translated into a proof script that can be executed later =-=[MP04]-=-. Also Andreas Meier’s TRAMP system [Mei00] transforms the output of various automated first-order provers into natural deduction proofs. The main focus of their work however is on the necessary trans... |

25 | From LCF to HOL: a short history
- Gordon
- 2000
(Show Context)
Citation Context ...es back at least to the early nineties [KKS91]. However, to ensure that a potential bug in the automated prover does not render the whole system unsound, theorems in Isabelle, like in other LCF-style =-=[Gor00]-=- provers, can be derived only through a fixed set of core inference rules. Therefore it is not sufficient for the automated prover to return whether a formula is provable, but it must also generate th... |

13 | A proof-producing boolean search engine
- Barrett, Berezin
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(Show Context)
Citation Context ...ather than a first-order prover. Other work on combining proof and model search includes [dNM06]. A custom-built SAT solver has been integrated with the CVC Lite system [BB04] by Clark Barrett et al. =-=[BBD03]-=-. While this solver produces proofs that can be checked independently, our work shows that it is possible to integrate existing, highly efficient solvers with an LCF-style prover: the information prov... |

13 | Integrating a first-order automatic prover in the HOL environment
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(Show Context)
Citation Context ...eractive systems, we can preserve the richness of our specification logic and at the same time increase the degree of automation [Sha01]. This is an idea that goes back at least to the early nineties =-=[KKS91]-=-. However, to ensure that a potential bug in the automated prover does not render the whole system unsound, theorems in Isabelle, like in other LCF-style [Gor00] provers, can be derived only through a... |

8 | Geometric resolution: A proof procedure based on finite model search
- Nivelle, Meng
- 2006
(Show Context)
Citation Context ...s, but we have focused on performance (rather than on difficult translation issues), and we use a SAT solver, rather than a first-order prover. Other work on combining proof and model search includes =-=[dNM06]-=-. A custom-built SAT solver has been integrated with the CVC Lite system [BB04] by Clark Barrett et al. [BBD03]. While this solver produces proofs that can be checked independently, our work shows tha... |

7 | Prensa Nieto, and Alwen Tiu. Expressiveness + automation + soundness: Towards combining smt solvers and interactive proof assistants - Fontaine, Marion, et al. - 2006 |

6 | Compressing propositional refutations
- Amjad
- 2006
(Show Context)
Citation Context ...orting resolutions may help to derive shorter clauses during the proof, which should improve the performance of individual resolution steps. Some preliminary results along these lines are reported in =-=[Amj06a]-=-. The approach presented in this paper has applications beyond propositional reasoning. The decision problem for richer logics (or fragments thereof) can be reduced to SAT [ABC + 02, Str02, MS05, RH06... |

5 | Integration of interactive and automatic provers
- Meng
(Show Context)
Citation Context ...consistency in HOL 4. Perhaps closer related to our work is the integration of automated first-order provers, in the context of Isabelle recently further explored by Joe Hurd [Hur99, Hur02], Jia Meng =-=[Men03]-=-, and Lawrence Paulson [MP04, MP06]. Proofs found by the automated system are either verified by the interactive prover immediately [Hur99], or translated into a proof script that can be executed late... |

3 |
HolSatLib documentation, version 1.0b
- Gordon
- 2001
(Show Context)
Citation Context ... implemented: a naive encoding that may cause an exponential blowup of the formula, and a Tseitin-style encoding [Tse83] that may introduce (existentially quantified) auxiliary Boolean variables, cf. =-=[Gor01]-=-. The technical details can be found in [Web05a]. More sophisticated CNF conversions, e.g. from [NRW98], could be implemented as well. The main focus of our work however is on efficient proof reconstr... |

2 | Proof reconstruction for first-order logic and set-theoretical constructions
- Hurlin
- 2006
(Show Context)
Citation Context ...rate the more powerful, yet SAT-based decision procedure with an LCF-style theorem prover. This has already been used to integrate haRVey, a Satisfiability Modulo Theories (SMT) prover, with Isabelle =-=[Hur06]-=-. haRVey, like other SMT systems, uses various decision procedures (e.g. congruence closure for uninterpreted functions) on top of a SAT solver. Acknowledgments. The author would like to thank Markus ... |

1 |
A HOL/MiniSat interface
- Amjad
- 2006
(Show Context)
Citation Context ... proof object in a theorem prover, no custom-built solver is necessary. An earlier version of this work was presented in [Web05a], and improved by Alwen Tiu et al. [FMM + 06]. Furthermore Hasan Amjad =-=[Amj06b]-=- has recently integrated a proof-generating version of the MiniSat solver with HOL 4 in a similar fashion. In this paper we have discussed our most recent implementation, which is based on a novel cla... |

1 |
ASCII proof traces for MiniSat
- Matthews
- 2006
(Show Context)
Citation Context ...unsat answers in 2005.) More recently, a proof-logging version of MiniSat has been released [ES06], and John Matthews has extended this version to produce human-readable proofs that are easy to parse =-=[Mat06]-=-, similar to those produced by zChaff. Hence our main task boils down to using Isabelle/HOL as an independent checker for the resolution proofssInput formula Isabelle Counterexample Preprocessing Proo... |