## Theorem proving with structured theories (full report (2001)

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@TECHREPORT{Mcilraith01theoremproving,

author = {Sheila Mcilraith and Eyal Amir},

title = {Theorem proving with structured theories (full report},

institution = {},

year = {2001}

}

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### Abstract

Motivated by the problem of query answering over multiple structured commonsense theories, we exploit graph-based techniques to improve the efficiency of theorem proving for structured theories. Theories are organized into subtheories that are minimally connected by the literals they share. We present message-passing algorithms that reason over these theories using consequence finding, specializing our algorithms for the case of first-order resolution, and for batch and concurrent theorem proving. We provide an algorithm that restricts the interaction between subtheories by exploiting the polarity of literals. We attempt to minimize the reasoning within each individual partition by exploiting existing algorithms for focused incremental and general consequence finding. Finally, we propose an algorithm that compiles each subtheory into one in a reduced sublanguage. We have proven the soundness and completeness of all of these algorithms. 1

### Citations

948 |
A machine-oriented logic based on the resolution principle
- Robinson
- 1965
(Show Context)
Citation Context ...e finding and show how to exploit them to focus reasoning within a partition. 3 Local Inference Using Resolution In this section, we specialize FMP to resolution-based consequence finders. Resolution =-=[26]-=- is one of the most widely used reasoning methods for automated deduction, and more specifically for consequence finding. It requires the input formula to be in clausal form, i.e., a conjunction of di... |

399 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1979
(Show Context)
Citation Context ...FOL as well as propositional theories. In the area of FOL theorem proving, our work is related to research on parallel theorem proving (see surveys in [6; 15]) and on combining logical systems (e.g., =-=[24; 4]-=-). Most parallel theorem prover implementations are guided by lookahead and subgoals to decompose the search space dynamically or allow messages to be sent between different provers working in paralle... |

145 |
Bayesian updating in recursive graphical models by local computations
- Jensen, Lauritzen, et al.
- 1990
(Show Context)
Citation Context ...aph-based techniques for improving the efficiency of reasoning. Graph-based algorithms are commonly used as a means of exploiting structure to improve the efficiency of reasoning in Bayes Nets (e.g., =-=[18]-=-), Constraint Satisfaction Problems (CSPs) (e.g., [12]) and most recently in logical reasoning (e.g., [11; 3; 25]). In all cases, the basic approach is to # Knowledge Systems Laboratory (KSL) convert ... |

103 |
Linear reasoning: A new form of the Herbrand-Gentzen theorem
- Craig
- 1957
(Show Context)
Citation Context ...olution) does not guarantee completeness for L-generation for the corresponding full FOL language, L. This problem is also reflected in a slightly different statement of Craig's interpolation theorem =-=[10]-=- that applies for resolution [27] . A simple way of addressing this problem is to add all constant and function symbols to the communication language between every connected set of partitions. This ha... |

101 | I.: Directional resolution: the davis-putnam procedure, revisited
- Dechter, Rish
- 1994
(Show Context)
Citation Context ...lution) [14] are two first-order resolution-based L-consequence finders. SFK-resolution is complete for first-order L-consequence finding, reducing to Directional Resolution in the propositional case =-=[13]-=- . In contrast, SOL-resolution is not complete for first-order L-consequence finding, but is complete for first-order incremental L-consequence finding. Given new inputs#, an incremental L-consequence... |

98 |
Induction as consequence finding
- Inoue
- 2004
(Show Context)
Citation Context ...on, for any language L that is closed under subsumption [14] . The notion of a consequence finder restricting consequence generation to consequences in a designated sublanguage was discussed by Inoue =-=[17]-=- , and further developed by del Val [14] and others. Most results on the completeness of consequence finding exploit resolution-based reasoners, where completeness results for L-consequence finding ar... |

89 | Efficient approximation for triangulation of minimum treewidth
- Amir
- 2001
(Show Context)
Citation Context ... reasoning over multiple KBs. Other times, a partitioning is induced automatically to improve the efficiency of reasoning. (Some automated techniques for performing this partitioning are discussed in =-=[3; 2]-=-.) This partitioning can be depicted as a graph in which each node represents a particular partition or subtheory and each arc represents shared vocabulary between subtheories. Theorem proving is perf... |

70 | Steps Towards Formalizing Context
- Akman, Surav
- 1996
(Show Context)
Citation Context ...s and the efficiency of reasoning with consequence finders and theorem provers. We do not have any restrictions on the language besides finiteness. Work on formalizing and reasoning with context (see =-=[1]-=- for a survey) can be related to theorem proving with structured theories by viewing the contextual theories as interacting sets of theories. Unfortunately, to introduce explicit contexts, a language ... |

57 | Partition-based logical reasoning, in
- Amir, McIlraith
- 2000
(Show Context)
Citation Context ... all of these algorithms. 1 Introduction Theorem provers are becoming increasingly prevalent as query-answering machinery for reasoning over single or multiple large commonsense knowledge bases (KBs) =-=[3]-=- . Commonsense KBs, as exemplified by Cycorp's Cyc and the High Performance Knowledge Base (HPKB) systems developed by Stanford's Knowledge Systems Lab and by SRI often comprise tens/hundreds of thous... |

50 | Resolution versus Search: Two Strategies for SAT
- Rish, Dechter
- 2000
(Show Context)
Citation Context ...s a means of exploiting structure to improve the efficiency of reasoning in Bayes Nets (e.g., [18]), Constraint Satisfaction Problems (CSPs) (e.g., [12]) and most recently in logical reasoning (e.g., =-=[11; 3; 25]-=-). In all cases, the basic approach is to # Knowledge Systems Laboratory (KSL) convert a graphical representation of the problem into a treestructured representation, where each node in the tree repre... |

43 |
A Completeness Theorem and a Computer Program for Finding Theorems Derivable from Given Axioms
- Lee
- 1967
(Show Context)
Citation Context ...re is in the communication language l of the adjacent node, and that node is on the path to the node containing the query. Recall, consequence finding (as opposed to proof finding) was defined by Lee =-=[19]-=- to be the problem of finding all nontautological logical consequences of a theory or sentences that subsume them. A prime implicate generator is a popular example of a consequence finder 1 . To deter... |

26 |
Combination of constraint solvers for free and quasi-free structures
- Baader, Schulz
- 1998
(Show Context)
Citation Context ...FOL as well as propositional theories. In the area of FOL theorem proving, our work is related to research on parallel theorem proving (see surveys in [6; 15]) and on combining logical systems (e.g., =-=[24; 4]-=-). Most parallel theorem prover implementations are guided by lookahead and subgoals to decompose the search space dynamically or allow messages to be sent between different provers working in paralle... |

26 |
Locking: ~ Restriction £[ Resolution
- Boyer
- 1971
(Show Context)
Citation Context ...servative extension of A j ). Resolution strategies that can be readily used in RES-MP, while preserving completeness, include linear resolution [23] , directional resolution [25] and lock resolution =-=[7]-=- . Strategies such as set-of-support and semantic resolution can be used with somewhat different treatments. 4 Minimizing Node Coupling Using Polarity FMP and RES-MP use the communication language to ... |

22 | A new method for consequence finding and compilation in restricted languages
- Val, A
- 1999
(Show Context)
Citation Context ...there is a substitution # such that # # #. Observe that every reasoner that is complete for Lconsequence finding is also complete for L-generation, for any language L that is closed under subsumption =-=[14]-=- . The notion of a consequence finder restricting consequence generation to consequences in a designated sublanguage was discussed by Inoue [17] , and further developed by del Val [14] and others. Mos... |

16 |
Parallelization of deduction strategies: an analytical study
- Bonacina, Hsiang
- 1994
(Show Context)
Citation Context ...e passing algorithms and our algorithms apply to FOL as well as propositional theories. In the area of FOL theorem proving, our work is related to research on parallel theorem proving (see surveys in =-=[6; 15]-=-) and on combining logical systems (e.g., [24; 4]). Most parallel theorem prover implementations are guided by lookahead and subgoals to decompose the search space dynamically or allow messages to be ... |

15 |
T.: A complete, nonredundant algorithm for reversed skolemization
- Cox, Pietrzykowski
- 1980
(Show Context)
Citation Context ...receiving partition. The process of conservatively replacing function and constant symbols by existentially quantified variables is called unskolemization or reverse Skolemization and is discussed in =-=[5; 9; 8]-=- . [8] presents an algorithm U that is complete for our purposes and generalizes and simplifies an algorithm of [9] . (Space precludes inclusion of the algorithm.) Theorem 3.1 ([8]) Let V be a vocabul... |

14 |
A note on linear resolution strategies in consequence-finding
- Minicozzi, Reiter
- 1972
(Show Context)
Citation Context ...orm, i.e., a conjunction of disjunctions of unquantified literals. The resolution rule is complete for consequence finding (e.g., [19; 27]) and so is linear resolutionsand some of its variants (e.g., =-=[23]-=-). We present algorithm RESOLUTION-M-P (RES-MP), that uses resolution (or resolution strategies), in Figure 4. The rest of this section is devoted to explaining four different implementations for subr... |

13 |
An interpolation theorem in the predicate calculus
- Lyndon
(Show Context)
Citation Context ...er of messages that are passed between adjacent partitions, and thus a reduction in the search space size of the global reasoning problem. Our results are predicated on Lyndon's Interpolation Theorem =-=[20]-=- , an extension to Craig's Theorem [10] . Theorem 4.1 (Lyndon's Interpolation Theorem) Let #, # be sentences such that # # #. Then there exists a sentence # such that # # # and # # #, and that every r... |

12 |
Tree clustering schemes for constraintprocessing
- Dechter, Pearl
(Show Context)
Citation Context ...easoning. Graph-based algorithms are commonly used as a means of exploiting structure to improve the efficiency of reasoning in Bayes Nets (e.g., [18]), Constraint Satisfaction Problems (CSPs) (e.g., =-=[12]-=-) and most recently in logical reasoning (e.g., [11; 3; 25]). In all cases, the basic approach is to # Knowledge Systems Laboratory (KSL) convert a graphical representation of the problem into a trees... |

11 |
Cooperating theorem provers
- Denzinger, Dahn
- 1998
(Show Context)
Citation Context ...e passing algorithms and our algorithms apply to FOL as well as propositional theories. In the area of FOL theorem proving, our work is related to research on parallel theorem proving (see surveys in =-=[6; 15]-=-) and on combining logical systems (e.g., [24; 4]). Most parallel theorem prover implementations are guided by lookahead and subgoals to decompose the search space dynamically or allow messages to be ... |

10 |
GETFOL Manual - GETFOL version 2.0
- Giunchiglia
- 1994
(Show Context)
Citation Context ...more expressive than FOL is needed. Consequently, a number of researchers have focused on context for propositional logic, while much of the reasoning work has focused on proof checking (e.g., GETFOL =-=[16]-=-). 7 Summary In this paper we exploited graph-based techniques to improve the efficiency of theorem proving for structured theories. Theories were organized into subtheories that were minimally connec... |

8 |
Utilizing knowledge-base semantics in graph-based algorithms
- Darwiche
- 1996
(Show Context)
Citation Context ...s a means of exploiting structure to improve the efficiency of reasoning in Bayes Nets (e.g., [18]), Constraint Satisfaction Problems (CSPs) (e.g., [12]) and most recently in logical reasoning (e.g., =-=[11; 3; 25]-=-). In all cases, the basic approach is to # Knowledge Systems Laboratory (KSL) convert a graphical representation of the problem into a treestructured representation, where each node in the tree repre... |

7 |
Interpolation theorems for resolution in lower predicate calculus
- Slagle
- 1970
(Show Context)
Citation Context ... consequence finding. It requires the input formula to be in clausal form, i.e., a conjunction of disjunctions of unquantified literals. The resolution rule is complete for consequence finding (e.g., =-=[19; 27]-=-) and so is linear resolutionsand some of its variants (e.g., [23]). We present algorithm RESOLUTION-M-P (RES-MP), that uses resolution (or resolution strategies), in Figure 4. The rest of this sectio... |

3 |
Finding logical consequences using unskolemization
- Chadha, Plaisted
- 1993
(Show Context)
Citation Context ...receiving partition. The process of conservatively replacing function and constant symbols by existentially quantified variables is called unskolemization or reverse Skolemization and is discussed in =-=[5; 9; 8]-=- . [8] presents an algorithm U that is complete for our purposes and generalizes and simplifies an algorithm of [9] . (Space precludes inclusion of the algorithm.) Theorem 3.1 ([8]) Let V be a vocabul... |

2 |
Consequence finding algorithms. In Algorithms for Defeasible and Uncertain Reasoning
- Marquis
- 2000
(Show Context)
Citation Context ...eduction steps to those leading to interestingsconsequences, skipping deduction steps that do not. In the propositional case, the most popular algorithms are certain L-(prime) implicate finders. (See =-=[21]-=- for an excellent survey.) SOL-resolution (skipping ordered linear resolution) [17] and SFK-resolution (skip-filtered, kernel resolution) [14] are two first-order resolution-based L-consequence finder... |