## Algorithms, datastructures, and other issues in efficient automated deduction (2001)

Venue: | Automated Reasoning. 1st. International Joint Conference, IJCAR 2001, number 2083 in LNAI |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Voronkov01algorithms,datastructures,,

author = {Andrei Voronkov},

title = {Algorithms, datastructures, and other issues in efficient automated deduction},

booktitle = {Automated Reasoning. 1st. International Joint Conference, IJCAR 2001, number 2083 in LNAI},

year = {2001},

pages = {13--28},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and datastructures, and to efficient theorem proving in general. 1

### Citations

223 | Hope theory
- Snyder, Rand, et al.
- 2002
(Show Context)
Citation Context ...icult for a prover in which AC is implemented efficiently, including term indexing modulo AC. So far research on built-in equational theories was mainly built around equational unification (see e.g., =-=[4]-=- for a recent overview). The special case of ACunification was discussed in a number of papers, dating back to [85,86], but only recently efficient algorithms have been described [1]. The experience w... |

157 | Resolution theorem proving
- Bachmair, Ganzinger
(Show Context)
Citation Context ...progress has been made in the theory of resolution-based systems. This theory is build upon completeness theorems for resolution calculi with notions of redundant inferences and redundant derivations =-=[6,63]-=-. The theory is well-understood, but a good theory alone is not enough to implement an efficient prover. The progress in theory can be characterized by the following observation: a prover based on the... |

65 |
Basic paramodulation and superposition
- Bachmair, Ganzinger, et al.
- 1992
(Show Context)
Citation Context ...ementation.sAlgorithms, Datastructures, and Other Issues 19 As a result, some enhancements of theorem provers well-known in theory have never been implemented. A typical example is the basic strategy =-=[17,7,60]-=-. Although in theory basic superposition saves from performing some redundant inferences, in practice implementation of basic superposition requires considerable changes in all algorithms and datastru... |

51 | Solving symbolic ordering constraints
- Comon
- 1990
(Show Context)
Citation Context ...ons. The most advanced results on symbolic constraints are in the area of solving constraints over simplification orderings. The first algorithm for solving RPO ordering constraints were described in =-=[14,35]-=-, followed by a number of results on solving RPO ordering constraints [55,15,54], but only recently an efficient algorithm was designed [58]. In the case of Knuth-Bendix ordering constraints, the deci... |

42 | The automation of proof by mathematical induction
- Bundy
(Show Context)
Citation Context ...ll mention some of them very briefly. In the future modern first-order theorem prover will be tightly integrated with other systems for automated reasoning, for example inductive theorem provers (see =-=[12]-=- for an overview), and proof assistants such as Isabelle [66] and HOL [28], and maybe model checkers. It is likely that first-order provers integrated in such systems will also partially implement pro... |

37 |
DISCOUNT: A system for distributed equational deduction
- Avenhaus, Denzinger, et al.
- 1995
(Show Context)
Citation Context ...s and increase the weight limit, if the prover unsuccessfully terminates with the current weight limit. The most radical solution to the passive clauses problem was originally implemented in DISCOUNT =-=[2]-=- and is now used in Waldmeister and E (and can be used as an option in Spass and Vampire). These provers use a different main loop, in which passive clauses do not participate in backward simplificati... |

32 | FDPLL - a first-order DavisPutnam-Logeman-Loveland procedure
- Baumgartner
- 2000
(Show Context)
Citation Context ...el elimination [41] implemented in SETHEO [40] often performs very well on problems difficult for resolution-based provers. A recent adaptation of propositional splitting to the full first-order case =-=[8]-=- seems to be promising. However, non-resolution based procedures often have difficulties with equality and other built-in theories, as witnessed by the results overviewed in [21]. They have been sever... |

23 |
Flatterms, Discrimination Nets, and Fast Term rewriting
- Christian
- 1993
(Show Context)
Citation Context ...ory consumption. Therefore, along the last two decades a significant number of results on new indexing techniques for theorem proving have been published and successfully applied in different provers =-=[81,32,33,69,84,64,49,13,30,31,93,76,72,26,57]-=-. For every retrieval condition used in theorem provers, it is not necessary to retrieve exactly all candidates satisfying this retrieval condition. If the retrieval condition is used for inferences (... |

22 |
Associative-commutative discrimination nets
- Bachmair, Chen, et al.
- 1993
(Show Context)
Citation Context ...g. Moreover, there are essentially no publications on term indexing in presence of built-in theories. The only paper we know about discusses a special case of indexing for AC-matching of linear terms =-=[5]-=-. Vampire has commutativity built-in in some term indexes and retrieval algorithms (see e.g., [73]). We conclude that efficient algorithms and datastructures for built-in equational theories should be... |

20 | Ordering constraints on trees
- Comon, Treinen
(Show Context)
Citation Context ...ving constraints over simplification orderings. The first algorithm for solving RPO ordering constraints were described in [14,35], followed by a number of results on solving RPO ordering constraints =-=[55,15,54]-=-, but only recently an efficient algorithm was designed [58]. In the case of Knuth-Bendix ordering constraints, the decidability of constraint solving was proved in 1999 [38,39], but no simple algorit... |

11 | Contejean: Avoiding slack variables in the solving of linear Diophantine equations and inequations. Principles and practice of constraint programming
- Ajili, E
- 1997
(Show Context)
Citation Context ...cation (see e.g., [4] for a recent overview). The special case of ACunification was discussed in a number of papers, dating back to [85,86], but only recently efficient algorithms have been described =-=[1]-=-. The experience with the non-AC theorem proving shows that efficient matching is more important than efficient unification, 3 but there is essentially no literature on efficient ACmatching. Moreover,... |

11 | Extensional higher-order resolution
- Benzmüller, Kohlhase
- 1998
(Show Context)
Citation Context ... proofs about inductively defined types, or even restricted forms of higher-order theorem proving. This will require new lines of research, for example – saturation-based higher-order theorem proving =-=[10,11]-=-; – intelligent work with definitions [67,20]; – built-in data types; – recognition of irrelevant axioms; – propositional reasoning; – built-in theories (not only equational) [83]; – reasoning with no... |

10 | LEO – a higher-order theorem prover
- Benzmüller, Kohlhase
- 1998
(Show Context)
Citation Context ... proofs about inductively defined types, or even restricted forms of higher-order theorem proving. This will require new lines of research, for example – saturation-based higher-order theorem proving =-=[10,11]-=-; – intelligent work with definitions [67,20]; – built-in data types; – recognition of irrelevant axioms; – propositional reasoning; – built-in theories (not only equational) [83]; – reasoning with no... |

9 | Equality elimination for the tableau method
- Degtyarev, Voronkov
- 1996
(Show Context)
Citation Context ...ith equality and other built-in theories, as witnessed by the results overviewed in [21]. They have been several proposals on combination of paramodulation-based reasoning and tableau-based reasoning =-=[18,19,52]-=-, but none of them was implemented. 9.2 Parallel and Agent-Based Reasoning Parallel computing is becoming cheaper. Networks of computers are now readily available. This makes parallel and agent-based ... |

7 | Equality elimination for semantic tableaux
- Degtyarev, Voronkov
- 1994
(Show Context)
Citation Context ...ith equality and other built-in theories, as witnessed by the results overviewed in [21]. They have been several proposals on combination of paramodulation-based reasoning and tableau-based reasoning =-=[18,19,52]-=-, but none of them was implemented. 9.2 Parallel and Agent-Based Reasoning Parallel computing is becoming cheaper. Networks of computers are now readily available. This makes parallel and agent-based ... |

4 |
CCE: Testing ground joinability
- Avenhaus, Lochner
- 2001
(Show Context)
Citation Context ...lification rules. There are several examples of successful implementation of “cheap substitutes” for expensive operations. For example, Waldmeister implements specialized algorithms for AC-completion =-=[3]-=-. In resolution theorem proving it is desirable to quickly check whether one can apply a resolvent of two clauses which subsumes one of these clauses. Such an application simplifies the search space, ... |

3 |
On the forms of inference in calculi with equality and paramodulation
- Degtyarev
- 1982
(Show Context)
Citation Context ...ementation.sAlgorithms, Datastructures, and Other Issues 19 As a result, some enhancements of theorem provers well-known in theory have never been implemented. A typical example is the basic strategy =-=[17,7,60]-=-. Although in theory basic superposition saves from performing some redundant inferences, in practice implementation of basic superposition requires considerable changes in all algorithms and datastru... |

2 |
The NP--completeness of subsumption. unpublished manuscript
- Baxter
- 1977
(Show Context)
Citation Context ...6], since overhead on maintaining datastructures for these algorithms does not pay off in practice, where the exponential behavior of Robinson’s algorithm does not show up. Subsumption is NP-complete =-=[9,27]-=-, but the modern provers often make subsumption-checks of 10 5 clauses against a database of 10 4 clauses in a few seconds. A practical approach to comparing implementation of algorithms used in first... |