## LEO-II — A cooperative automatic theorem prover for higher-order logic (2008)

### Cached

### Download Links

Venue: | In Fourth International Joint Conference on Automated Reasoning (IJCAR’08), volume 5195 of LNAI |

Citations: | 34 - 22 self |

### BibTeX

@INPROCEEDINGS{Benzmüller08leo-ii—,

author = {Christoph Benzmüller and Lawrence C. Paulson and Frank Theiss},

title = {LEO-II — A cooperative automatic theorem prover for higher-order logic},

booktitle = {In Fourth International Joint Conference on Automated Reasoning (IJCAR’08), volume 5195 of LNAI},

year = {2008},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. LEO-II is a standalone, resolution-based higher-order theorem prover designed for effective cooperation with specialist provers for natural fragments of higher-order logic. At present LEO-II can cooperate with the first-order automated theorem provers E, SPASS, and Vampire. The improved performance of LEO-II, especially in comparison to its predecessor LEO, is due to several novel features including the exploitation of term sharing and term indexing techniques, support for primitive equality reasoning, and improved heuristics at the calculus level. LEO-II is implemented in Objective Caml and its problem representation language is the new TPTP THF language. 1

### Citations

719 |
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
- Nipkow, Paulson, et al.
- 2002
(Show Context)
Citation Context ... of HO logic. The idea is to combine the strengths of the different systems. On the other hand, LEO-II itself, as an external reasoner, is designed to support HO proof assistants such as Isabelle/HOL =-=[18]-=-, HOL [13], and Ωmega [22] by efficiently automating subproblems and thereby reducing user effort.1.1 Motivation for LEO-II LEO-II is the successor of LEO [4], which was implemented in Allegro Common... |

303 | N.: Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation.; Indag
- Bruijn
- 1972
(Show Context)
Citation Context ...s are represented by a single instance. Ideas from FO term sharing are adapted to HO logic by (i) keeping indexed terms in βη normal form (i.e., η short and β normal) and (ii) using de Bruijn indices =-=[12]-=- to allow λ-abstracted terms to be shared. The resulting data structure represents terms in a directed acyclic graph (DAG). LEO-II supports the visualization of such term graphs. 16: @ 0: ¬ 1 15: @ 11... |

125 | E – A Brainiac Theorem Prover
- Schulz
(Show Context)
Citation Context ...ve Caml and its problem representation language is the new TPTP THF language. 1 Introduction and Motivation Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire [20], E =-=[21]-=-, and SPASS [25], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order (FO) logic. Highe... |

44 | Spass version 2.0
- Weidenbach, Brahm, et al.
- 2002
(Show Context)
Citation Context ...problem representation language is the new TPTP THF language. 1 Introduction and Motivation Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire [20], E [21], and SPASS =-=[25]-=-, have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order (FO) logic. Higher-order (HO) log... |

43 |
Introduction to HOL: a theorem-proving environment for higher-order logic
- Gordon, Melham
- 1993
(Show Context)
Citation Context ...ic. The idea is to combine the strengths of the different systems. On the other hand, LEO-II itself, as an external reasoner, is designed to support HO proof assistants such as Isabelle/HOL [18], HOL =-=[13]-=-, and Ωmega [22] by efficiently automating subproblems and thereby reducing user effort.1.1 Motivation for LEO-II LEO-II is the successor of LEO [4], which was implemented in Allegro Common Lisp as a... |

43 | Experiments with Discrimination-Tree Indexing and Path Indexing for Term Retrieval
- McCune
- 1992
(Show Context)
Citation Context ...rformance of rewrite operations, such as global unfolding of definitions. Additionally, LEO-II employs a term indexing data structure, which is based on structural indexing methods from the FO domain =-=[16,23]-=-, as well as road sign techniques. Road signs are features of the data structure which guide 2 To further visualize the evolution of the term graph during proof search, we modified LEO-II to output a ... |

28 |
A higher-order theorem prover
- LEO
- 1998
(Show Context)
Citation Context ...O proof assistants such as Isabelle/HOL [18], HOL [13], and Ωmega [22] by efficiently automating subproblems and thereby reducing user effort.1.1 Motivation for LEO-II LEO-II is the successor of LEO =-=[4]-=-, which was implemented in Allegro Common Lisp as a part of the Ωmega system and which unfortunately was not available as a standalone reasoner. LEO shared the basic data structures such as terms and ... |

24 |
TPS: A hybrid automatic-interactive system for developing proofs
- Andrews, Brown
(Show Context)
Citation Context ... approach that shows some promising results in selected problem domains. It thus provides an interesting alternative to reasoning solely in FO logic and it also differs from other HO ATPs such as TPS =-=[1]-=- and OTTERλ [2]. TPS is based on the mating method and does not cooperate with specialist FO ATPs. It is particularly strong in proving theorems which require selective expansion of definitions and go... |

21 | The path-indexing method for indexing terms
- Stickel
- 1989
(Show Context)
Citation Context ...rformance of rewrite operations, such as global unfolding of definitions. Additionally, LEO-II employs a term indexing data structure, which is based on structural indexing methods from the FO domain =-=[16,23]-=-, as well as road sign techniques. Road signs are features of the data structure which guide 2 To further visualize the evolution of the term graph during proof search, we modified LEO-II to output a ... |

16 | Selectively instantiating definitions
- Bishop, Andrews
- 1998
(Show Context)
Citation Context ...articularly strong in proving theorems which require selective expansion of definitions and goal directed instantiation of set variables. Examples of such theorems are presented in Bishop and Andrews =-=[10]-=- and Brown [11]. Many of these examples will require corresponding mechanisms in LEO-II to be proven automatically. The formalism of OTTER-λ is not simple type theory but λ-logic, which is a novel com... |

16 | Higher-order substitution tree indexing
- Pientka
- 2003
(Show Context)
Citation Context ...es, such as shared terms, used for instance in E [21]. HO term indexing techniques are rarely addressed in the literature, which hampers the progress of systems in this field. An exception is Pientka =-=[19]-=-. LEO-II’s implementation at term level is based on a perfectly shared term graph, i.e., syntactically equal terms are represented by a single instance. Ideas from FO term sharing are adapted to HO lo... |

14 |
On the Representation of Mathematical Concepts and their Translation into First-Order Logic
- Kerber
- 1992
(Show Context)
Citation Context ...tion. Roughly speaking, currently all clauses that do not contain any λ-term and embedded predicate or proposition are light-colored, and our default FO transformation function employs Kerber’s ideas =-=[14]-=-: it recursively translates every application (pα→β qα) into a term @ (α→β) α (p, q), where @ (α→β) α is a new function or predicate constant that encodes the types of its two arguments. LEO-II’s exte... |

14 | On the evaluation of indexing techniques for theorem proving
- Nieuwenhuis, Hillenbrand, et al.
- 2001
(Show Context)
Citation Context ...e large numbers of terms and, for a given query term t, support the fast retrieval of terms from the index that satisfy a certain relation with t, such as matching, unifiability or syntactic equality =-=[17]-=-. Performance can be further enhanced by representation of terms in efficient data structures, such as shared terms, used for instance in E [21]. HO term indexing techniques are rarely addressed in th... |

14 |
Computer supported mathematics with Omega
- Siekmann, Benzmüller, et al.
(Show Context)
Citation Context ...to combine the strengths of the different systems. On the other hand, LEO-II itself, as an external reasoner, is designed to support HO proof assistants such as Isabelle/HOL [18], HOL [13], and Ωmega =-=[22]-=- by efficiently automating subproblems and thereby reducing user effort.1.1 Motivation for LEO-II LEO-II is the successor of LEO [4], which was implemented in Allegro Common Lisp as a part of the Ωme... |

13 | Equality and Extensionality in Automated Higher-Order Theorem Proving
- Benzmüller
- 1999
(Show Context)
Citation Context ...grated with other reasoners and its sources are available from the LEO-II web-site at: http://www.ags.uni-sb.de/~leo/. Proof search in LEO-II, which is based on the extensional HO resolution calculus =-=[3]-=- has been further improved, e.g., it now supports efficient (recursive) expansion of definitions and primitive equality reasoning (see Section 3). For cooperation with FO ATPs, LEO-II offers two diffe... |

13 | THF0 — the core TPTP language for classical higher-order logic
- Benzmüller, Rabe, et al.
(Show Context)
Citation Context ...l polymorphism adds many nontrivial choice points to the already challenging search space of LEO-II [6] and is therefore future work. Moreover, LEO-II employs the new TPTP THF representation language =-=[7]-=-. At present, LEO-II cooperates with FO ATPs only in a sequential mode and not via the agent based architecture ΩAnts. Figure 1 shows that LEO-II’s performance has nevertheless strongly improved (in o... |

12 | Experiments with an AgentOriented Reasoning System
- Benzmüller, Jamnik, et al.
- 2001
(Show Context)
Citation Context ...tures were not designed for efficiency and their term indexing support was limited [15]. LEO’s performance strongly improved after coupling it with the FO ATP Otter in the agent based ΩAnts framework =-=[8]-=-. This integration has subsequently been improved and Otter has been replaced by Bliksem and Vampire [9]. This cooperative theorem proving approach outperforms – modulo different problem representatio... |

11 | Combined reasoning by automated cooperation
- Benzmüller, Sorge, et al.
(Show Context)
Citation Context ...ce strongly improved after coupling it with the FO ATP Otter in the agent based ΩAnts framework [8]. This integration has subsequently been improved and Otter has been replaced by Bliksem and Vampire =-=[9]-=-. This cooperative theorem proving approach outperforms – modulo different problem representations – FO ATPs such as Vampire on problems about sets, relations, and functions as given in the TPTP SET d... |

10 | Progress report on LEO-II – an automatic theorem prover for higher-order logic
- Benzmüller, Paulson, et al.
- 2007
(Show Context)
Citation Context ...ring its proof object, term graph and term index. This includes statistical analysis of the term graph (see Section 4). In addition to a fully automatic mode, LEO-II also provides an interactive mode =-=[6]-=-. This mode supports debugging and inspection of the search space,100 Vampire 9.0 LEO + Vampire LEO-II + E Total proving time in seconds 10 1 0.1 0.01 SET657+3 SET646+3 SET086+1 SET764+4 SET640+3 SET... |

6 |
Indexing für Terme höherer Stufe
- Klein
- 1997
(Show Context)
Citation Context ...ne reasoner. LEO shared the basic data structures such as terms and clauses with Ωmega; these shared basic data structures were not designed for efficiency and their term indexing support was limited =-=[15]-=-. LEO’s performance strongly improved after coupling it with the FO ATP Otter in the agent based ΩAnts framework [8]. This integration has subsequently been improved and Otter has been replaced by Bli... |

6 |
The design and implementation of Vampire. AICOM
- Riazanov, Voronkov
- 2002
(Show Context)
Citation Context ... Objective Caml and its problem representation language is the new TPTP THF language. 1 Introduction and Motivation Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire =-=[20]-=-, E [21], and SPASS [25], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order (FO) logi... |

6 |
Term indexing for the LEO-II prover
- Theiss, Benzmüller
- 2006
(Show Context)
Citation Context ...f has been found (right). operations based on graph traversal. They help to cut branches of the subgraph to be processed early and they are employed, e.g., in the construction of partial syntax trees =-=[24]-=- in which all branches with no occurrences of a given symbol or subterm are cut. This enables LEO-II to avoid potentially costly operations, such as occurs checks, and to speed up basic operations on ... |

5 | 2002. Solving for set variables in higherorder theorem proving
- Brown
- 2002
(Show Context)
Citation Context ...ong in proving theorems which require selective expansion of definitions and goal directed instantiation of set variables. Examples of such theorems are presented in Bishop and Andrews [10] and Brown =-=[11]-=-. Many of these examples will require corresponding mechanisms in LEO-II to be proven automatically. The formalism of OTTER-λ is not simple type theory but λ-logic, which is a novel combination of λ-c... |

3 | Mathematical induction in Otter-Lambda
- Beeson
- 2006
(Show Context)
Citation Context ...shows some promising results in selected problem domains. It thus provides an interesting alternative to reasoning solely in FO logic and it also differs from other HO ATPs such as TPS [1] and OTTERλ =-=[2]-=-. TPS is based on the mating method and does not cooperate with specialist FO ATPs. It is particularly strong in proving theorems which require selective expansion of definitions and goal directed ins... |