## SETHEO and E-SETHEO - The CADE-13 Systems (1997)

Venue: | Journal of Automated Reasoning |

Citations: | 10 - 0 self |

### BibTeX

@ARTICLE{Moser97setheoand,

author = {M. Moser and O. Ibens and R. Letz and J. Steinbach and C. Goller and J. Schumann and K. Mayr},

title = {SETHEO and E-SETHEO - The CADE-13 Systems},

journal = {Journal of Automated Reasoning},

year = {1997},

volume = {18},

pages = {237--246}

}

### OpenURL

### Abstract

. The model elimination theorem prover SETHEO (version V3.3) and its equational extension E-SETHEO are presented. SETHEO employs sophisticated mechanisms of subgoal selection, elaborate iterative deepening techniques, and local failure caching methods. Its equational counterpart E-SETHEO transforms formulae containing equality (using a variant of Brand's modification method) and processes the output with the standard SETHEO system. The paper gives an overview of the theoretical background, the system architecture, and the performance of both systems. Key words: Automated theorem proving, competition, SETHEO, E-SETHEO, first-order logic, model elimination, equality. 1. Introduction In this paper we describe the theorem provers SETHEO and E-SETHEO. SETHEO is based on the model elimination calculus [13] and performs proof search using iterative deepening. The proof procedure is implemented as an extension of the Warren Abstract Machine. The system is being continuously extended and enh...

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Citation Context ...ll-known that this approach suffers from the severe deficiency that specific redundancy elimination techniques of equality cannot be applied. On the other hand, most of the variants of paramodulation =-=[18, 2]-=- cannot easily be incorporated into model elimination because of their incompatibility with goaldirectedness. 2.4.1. Lazy Basic Paramodulation In a goal-directed setting, lazy paramodulation [6] can b... |

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Citation Context ...s approach is the necessity of generating bottom-up consequences to ensure completeness under the ordering restrictions. This can be realized by saturation procedures based on Knuth-Bendix completion =-=[10]-=-. In order to provide an efficient equality handling that overcomes the disadvantages of lazy basic paramodulation, in [17, 15] a new calculus for clause logic with equality was developed. This calcul... |

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Citation Context ... SETHEO, E-SETHEO, first-order logic, model elimination, equality. 1. Introduction In this paper we describe the theorem provers SETHEO and E-SETHEO. SETHEO is based on the model elimination calculus =-=[13]-=- and performs proof search using iterative deepening. The proof procedure is implemented as an extension of the Warren Abstract Machine. The system is being continuously extended and enhanced with add... |

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Citation Context ...m is being continuously extended and enhanced with additional inference mechanisms and reduction techniques. The main advances of the current version V3.3 over the last version, which is described in =-=[12, 7]-=-, are the following. Firstly, a more flexible iterative-deepening bound was developed, the so-called weighted-depth bound. Secondly, we have integrated a new backtracking strategy which permits the al... |

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Citation Context ...d by imposing bounds on the sizes and structures of the permitted tableaux. The two bounds used traditionally in SETHEO are the number of inferences needed to generate the current tableau (as used in =-=[23]-=-) and the depth of the current tableau (see [12]). Both bounds, however, have severe deficiencies (see [7, 8, 9]). In order to overcome the limitations of the above bounds, the weighteddepth bound has... |

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Citation Context ...e of certain clauses. In most cases some clauses are known as belonging to the (negated) conjecture and not to the theory axioms or hypotheses. Such relevance information is also provided in the TPTP =-=[25]-=-. A goal-directed system can profit from the relevance information by considering only those clauses as start clauses that stem from the conjecture. When this information is being employed, however, a... |

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Citation Context ... of the model elimination type is the handling of equality. Since the most successful paradigms for treating equality in saturation-based theorem proving, paramodulation [19] and its refined versions =-=[2]-=-, are not complete in a goal-directed setting, the Work funded within the projects `SETHEO II' (Je112/5) and `Erweiterte Logikprogrammierung' (Je112/3) of the Deutsche Forschungsgemeinschaft. 2 M. MOS... |

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Citation Context ... problem library. We conclude with general comments on the systems and with an outlook on future improvements. 2. Architecture 2.1. The Calculus SETHEO is based on the calculus of connection tableaux =-=[11]-=-, which is just a generalized reformulation of weak model elimination [13] in the tableau format. A connection tableau for a set of clauses is generated by first applying the start rule and then repea... |

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Citation Context ...rmulae with equality axioms into formulae without equality, which are then processed by the standard SETHEO. Interestingly, this transformation is a variant of the STE-modification developed by Brand =-=[4]-=-. The paper is organized as follows. The next section sets out the architecture of SETHEO and E-SETHEO, including descriptions of the calculus and the control strategies. Section 3 contains informatio... |

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Citation Context ... 2 (d 0 ; n) = d 0 + 1 \Gamma 1:2 \Delta n, w 3 = d a =1:2 \Delta \Deltai. The same parameters were also used in the competition. 2.2.3. Local Failure Caching Global failure caching was introduced in =-=[1]-=-, in two variants: solution caching and path caching. Both methods, however, are not successful in practice, since solution caching is not compatible with regularity and incomplete in the non-Horn cas... |

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Citation Context ...ithin the projects `SETHEO II' (Je112/5) and `Erweiterte Logikprogrammierung' (Je112/3) of the Deutsche Forschungsgemeinschaft. 2 M. MOSER ET AL. prover E-SETHEO is based on lazy basic paramodulation =-=[6]-=-, a variant of paramodulation which is compatible with model elimination. This method is implemented by means of a transformation from input formulae with equality axioms into formulae without equalit... |

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Citation Context ...ent of K. The extension rule implements the goal-directedness of the calculus. As an additional inference rule, the folding up operation [11] is integrated, which corresponds to Shostak's c-reduction =-=[21]-=-. This rule provides a restricted but highly efficient kind of lemma generation. It can also be seen as a controlled integration of the atomic cut rule [11]. 2.2. Control Since the pure calculus of co... |

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Citation Context ...ly in SETHEO are the number of inferences needed to generate the current tableau (as used in [23]) and the depth of the current tableau (see [12]). Both bounds, however, have severe deficiencies (see =-=[7, 8, 9]-=-). In order to overcome the limitations of the above bounds, the weighteddepth bound has been developed and integrated into SETHEO. The main idea of the weighted-depth bound is to take the inferences ... |

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Citation Context ...raditional weakness of calculi of the model elimination type is the handling of equality. Since the most successful paradigms for treating equality in saturation-based theorem proving, paramodulation =-=[19]-=- and its refined versions [2], are not complete in a goal-directed setting, the Work funded within the projects `SETHEO II' (Je112/5) and `Erweiterte Logikprogrammierung' (Je112/3) of the Deutsche For... |

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Citation Context ...is can be realized by saturation procedures based on Knuth-Bendix completion [10]. In order to provide an efficient equality handling that overcomes the disadvantages of lazy basic paramodulation, in =-=[17, 15]-=- a new calculus for clause logic with equality was developed. This calculus combines goal-directed tableau-based equational reasoning with restricted bottom-up inferences, and thus allows for the orde... |

5 |
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(Show Context)
Citation Context ...ly in SETHEO are the number of inferences needed to generate the current tableau (as used in [23]) and the depth of the current tableau (see [12]). Both bounds, however, have severe deficiencies (see =-=[7, 8, 9]-=-). In order to overcome the limitations of the above bounds, the weighteddepth bound has been developed and integrated into SETHEO. The main idea of the weighted-depth bound is to take the inferences ... |

4 |
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Citation Context ... avoided, for example, the uncontrolled application of the transitivity and the substitutivity axioms. Further redundancies can be eliminated by applying the `basic' refinement to lazy paramodulation =-=[14]-=-. The resulting calculus, lazy basic paramodulasystem -abstract.tex; 10/03/1999; 13:35; no v.; p.6 SETHEO and E-SETHEO 7 tion, is complete, independently of the selection function, an important proper... |

3 |
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Citation Context ...t of clauses without equality. This corresponding set of clauses without equality is the result of transforming the original set of clauses with equality by means of a variant of the STE-modification =-=[16]-=-. Originally, STE-modification was introduced in [4] in order to show that paramodulation without functional reflexive axioms is complete. The STE-modification we use is a variant of [4] with some fur... |

1 |
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Citation Context ...is can be realized by saturation procedures based on Knuth-Bendix completion [10]. In order to provide an efficient equality handling that overcomes the disadvantages of lazy basic paramodulation, in =-=[17, 15]-=- a new calculus for clause logic with equality was developed. This calculus combines goal-directed tableau-based equational reasoning with restricted bottom-up inferences, and thus allows for the orde... |

1 |
The CADE-13 ATP System Competition Results
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(Show Context)
Citation Context ...ries. Further user information is provided in [20]. 4. Performance Details on the performance of SETHEO and E-SETHEO on the 50 selected problems of the CADE-13 ATP system competition can be viewed in =-=[24]-=-. In total, eight systems participated in the category `General Hardware -- Monolithic Systems -- Mixed Problems'. E-SETHEO was the winner with 36 problems solved, SETHEO finished in third place with ... |