## An Even Closer Integration of Linear Arithmetic into Inductive Theorem Proving (2006)

Venue: | Proc. Calculemus 2005: 12 th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Schmidt-samoa06aneven,

author = {Tobias Schmidt-samoa},

title = {An Even Closer Integration of Linear Arithmetic into Inductive Theorem Proving},

booktitle = {Proc. Calculemus 2005: 12 th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning},

year = {2006},

pages = {3--20},

publisher = {Elsevier}

}

### OpenURL

### Abstract

To broaden the scope of decision procedures for linear arithmetic, they have to be integrated into theorem provers. Successful approaches e.g. in NQTHM or ACL2 suggest a close integration scheme which augments the decision procedures with lemmas about user-defined operators. We propose an even closer integration providing feedback about the state of the decision procedure in terms of entailed formulas for three reasons: First, to provide detailed proof objects for proof checking and archiving. Second, to analyze and improve the interaction between the decision procedure and the theorem prover. Third, to investigate whether the communication of the state of a failed proof attempt to the human user with the comprehensible standard GUI mechanisms of the theorem prover can enhance the speculation of auxiliary lemmas.

### Citations

412 | Simplification by Cooperating Decision Procedures
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- 1979
(Show Context)
Citation Context ...mains; second, the incorporation of decision procedures into heuristic theorem provers using augmentation. Research about the first approach has been initiated by fundamental work from Nelson & Oppen =-=[17]-=- and Shostak [19]. In this paper, we are concerned with the second approach. Seminal work on this topic has been done by Boyer & Moore [7] integrating a decision procedure for rational numbers based o... |

407 |
A Computational Logic Handbook
- Boyer, Moore
- 1988
(Show Context)
Citation Context ... work on this topic has been done by Boyer & Moore [7] integrating a decision procedure for rational numbers based on Hodes [9] (credited to Fourier in [14]) into their inductive theorem prover NQTHM =-=[6]-=-. 1 Email: schmidt@informatik.uni-kl.de This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcssSchmidt... |

280 | Computer-aided reasoning: an approach - Kaufmann, Manolios, et al. - 2000 |

167 | Deciding Combinations of Theories
- Shostak
- 1984
(Show Context)
Citation Context ...e incorporation of decision procedures into heuristic theorem provers using augmentation. Research about the first approach has been initiated by fundamental work from Nelson & Oppen [17] and Shostak =-=[19]-=-. In this paper, we are concerned with the second approach. Seminal work on this topic has been done by Boyer & Moore [7] integrating a decision procedure for rational numbers based on Hodes [9] (cred... |

110 | Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic
- Boyer, Moore
- 1985
(Show Context)
Citation Context ...h has been initiated by fundamental work from Nelson & Oppen [17] and Shostak [19]. In this paper, we are concerned with the second approach. Seminal work on this topic has been done by Boyer & Moore =-=[7]-=- integrating a decision procedure for rational numbers based on Hodes [9] (credited to Fourier in [14]) into their inductive theorem prover NQTHM [6]. 1 Email: schmidt@informatik.uni-kl.de This is a p... |

20 | An online proof-producing decision procedure for mixed-integer linear arithmetic
- Berezin, Ganesh, et al.
- 2003
(Show Context)
Citation Context ...their approach. The soundness of our approach is guaranteed by local properties of our inference rules. Termination properties may be proved by constraining the control in a similar way as in [2]. In =-=[5]-=-, Berezin, Ganesh & Dill also propose an inference system for their integration scheme. Our inference rules are on a higher level. Therefore, they can be easier applied manually. Although their infere... |

19 | Constraint contextual rewriting
- Armando, Ranise
(Show Context)
Citation Context ...quality w.r.t. a fixed wellfounded order. In previous work, the inequalities are stored in an internal state of the decision procedure called linear arithmetic data base in [7] or constraint store in =-=[2,3]-=-. If an unsatisfiable (ground) inequality is derived, the original inequalities are unsatisfiable over rationals, integers and naturals. Otherwise, if the set is closed under variable elimination step... |

14 |
Solving problems by formula manipulation in logic and linear inequalities
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- 1972
(Show Context)
Citation Context ...stak [19]. In this paper, we are concerned with the second approach. Seminal work on this topic has been done by Boyer & Moore [7] integrating a decision procedure for rational numbers based on Hodes =-=[9]-=- (credited to Fourier in [14]) into their inductive theorem prover NQTHM [6]. 1 Email: schmidt@informatik.uni-kl.de This is a preliminary version. The final version will be published in Electronic Not... |

9 | Incorporating decision procedures in implicit induction
- Armando, Rusinowitch, et al.
- 2001
(Show Context)
Citation Context ...rs. [16] and [8] contain proposals for the speculation of rewrite rules. For nonlinear equalities, Armando, Rusinowitch & Stratulat propose the use of Buchberger’s algorithm based on Gröbner basis in =-=[3]-=-. In [1] and [12], approaches are described to extend the integration of linear arithmetic to nonlinear arithmetic. These ex14sSchmidt-Samoa tensions almost only effect the heuristics to choose or spe... |

7 | A practical extension mechanism for decision procedures: the case study of universal presburger arithmetic
- Armando, Ranise
(Show Context)
Citation Context ... successful approaches have been proposed e.g. in [16,8]. But for the automatic speculation of linear rules no general approach has been proposed. There only exist approaches for nonlinear arithmetic =-=[12,1]-=-. As a linear rule contains an estimate, it is more difficult than for rewrite rules to speculate lemmas that are both—valid and useful. In our opinion, lemma speculation 2 For ease of use, we will al... |

5 | Deciding inductive validity of equations
- Giesl, Kapur
(Show Context)
Citation Context ...les are on a higher level. Therefore, they can be easier applied manually. Although their inference rules can be easier checked with an external proof checker this is also possible for ours. [16] and =-=[8]-=- contain proposals for the speculation of rewrite rules. For nonlinear equalities, Armando, Rusinowitch & Stratulat propose the use of Buchberger’s algorithm based on Gröbner basis in [3]. In [1] and ... |

5 | I.: A framework for the flexible integration of a class of decision procedures into theorem provers
- Janicic, Bundy, et al.
- 1999
(Show Context)
Citation Context ...n a huge case distinction. Instead, we use this method only if the intervals that have to be checked are small. Otherwise, we have to use other proof techniques e.g. induction. Janicic, Bundy & Green =-=[11]-=- formalize and generalize the approach from [7]. Their presentation is independent from the theory and the decision procedure to be used. Instead, they assume that the decision procedure can be divide... |

3 |
How to prove inductive theorems
- Avenhaus, Kühler, et al.
- 2003
(Show Context)
Citation Context ...re, the decision procedure only eliminates the heaviest terms. In this paper, we present a new approach to incorporate a decision procedure for PNA closely into our inductive theorem prover QUODLIBET =-=[4]-=-. We strictly distinguish the logic part of the decision procedure from its control aspects: Each elementary step of the decision procedure is represented by a new inference rule providing the state o... |

3 | J.S.: Integrating nonlinear arithmetic into into ACL2
- Hunt, Krug, et al.
- 2004
(Show Context)
Citation Context ... lemma can be applied if the conditions of the lemma can be proved valid. These proofs may be performed by the decision procedure or the theorem prover. Thus, we get mutual dependencies. According to =-=[13]-=-, the integration of linear arithmetic into ACL2 leads to four dependencies between the simplifier and the arithmetic package. Therefore, it is questionable whether the integration of linear arithmeti... |

2 |
A general setting for combining and integrating decision procedures into theorem provers
- Janicic, Bundy
(Show Context)
Citation Context ... binary addition. 2 According to the underlying domain, these theories are called Presburger rational arithmetic (PRA), Presburger integer arithmetic (PIA), and Presburger natural arithmetic (PNA) in =-=[10]-=-. We are interested in a semi-decision procedure for an extended theory of PNA containing additional predicate or function symbols. These symbols are uninterpreted for the decision procedure but may b... |

2 | Automatic generation of simple lemmas from recursive definitions using decision procedures—preliminary report
- Kapur, Subramaniam
- 2003
(Show Context)
Citation Context ...gmentation mechanism was introduced in [7]. The situation gets worse if the required lemmas are not present. To speculate rewrite rules automatically, successful approaches have been proposed e.g. in =-=[16,8]-=-. But for the automatic speculation of linear rules no general approach has been proposed. There only exist approaches for nonlinear arithmetic [12,1]. As a linear rule contains an estimate, it is mor... |

1 |
and nonlinear arithmetic
- Linear
- 2003
(Show Context)
Citation Context ... successful approaches have been proposed e.g. in [16,8]. But for the automatic speculation of linear rules no general approach has been proposed. There only exist approaches for nonlinear arithmetic =-=[12,1]-=-. As a linear rule contains an estimate, it is more difficult than for rewrite rules to speculate lemmas that are both—valid and useful. In our opinion, lemma speculation 2 For ease of use, we will al... |

1 |
Reasoning about numbers
- Kapur, Nie
- 1994
(Show Context)
Citation Context ... are concerned with the second approach. Seminal work on this topic has been done by Boyer & Moore [7] integrating a decision procedure for rational numbers based on Hodes [9] (credited to Fourier in =-=[14]-=-) into their inductive theorem prover NQTHM [6]. 1 Email: schmidt@informatik.uni-kl.de This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Sc... |

1 |
Integration von Entscheidungsverfahren in den induktiven Theorembeweiser QuodLibet
- Rondot
- 2004
(Show Context)
Citation Context ...we present a simplified version of our integration scheme for PNA into the inductive theorem prover QUODLIBET. Due to lack of space, we have to leave out many technical details. These can be found in =-=[18]-=-. Instead, we try to explain the basics of our approach intuitively. To guarantee a consistent integration of PNA, we assume a base specification spec 0 that consists of a sort Nat with constructors 0... |