## Partition-Based Logical Reasoning for First-Order and Propositional Theories (2000)

Venue: | Artificial Intelligence |

Citations: | 51 - 8 self |

### BibTeX

@ARTICLE{Amir00partition-basedlogical,

author = {Eyal Amir and Sheila Mcilraith},

title = {Partition-Based Logical Reasoning for First-Order and Propositional Theories},

journal = {Artificial Intelligence},

year = {2000},

volume = {162},

pages = {49--88}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward message-passing and using backward message-passing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our message-passing ...

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Citation Context ...d identify some of the parameters of influence. Current measures for comparing automated deduction strategies are insu#cient for our purposes. Proof length (e.g., [110,52,112] (also see the survey in =-=[22]-=-)) is only marginally relevant. More relevant is comparing the sizes of search spaces induced by di#erent strategies (e.g., resolution of propositional Horn clauses [84], and contraction rules for FOL... |

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Citation Context ...soning with consequence finders, theorem provers and SAT procedures. We do not have any restrictions on the language besides finiteness. Finally, work on formalizing and reasoning with context (e.g., =-=[76,1]-=-) can be related to partition-based logical reasoning by viewing the contextual theories as interacting sets of theories. Unfortunately, to introduce explicit contexts, a language that is more express... |

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Citation Context ...rtitions should receive messages from which other partitions. Second, there are no clear criteria for decomposing a theory into sub-problems. Another line of work focuses on combining logical systems =-=[80,99,7,87,109]-=-. Here, the computational focus has been on treating combinations of signaturedisjoint theories (allowing the queries to include symbols from all signatures), e.g. [7]. Recent work introduced sharing ... |

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Citation Context ...nd #, such that # # # and # # #. Craig's interpolation theorem is true even if we take #, # to be infinite sets of sentences [102] and use resolution theorem proving [102,54] with or without equality =-=[24,25]-=- (all after proper reformulation of the theorem). Theorem 2.3 (Soundness and Completeness) Let A = # i#n A i be a partitioned theory with the intersection graph G being a tree (i.e., no cycles). Let k... |

100 | Directional resolution: The davis-putnam procedure, revisited - Dechter, Rish - 1994 |

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Citation Context ...not complete, but they can be used in our algorithm, if we are willing to give up completeness. E#cient complete algorithms also typically exhibit better than worst-case behavior (see the analysis of =-=[96]-=-). All of this suggests that emphasizing link sizes over partition size is more accurate for the satisfiability problem. Thus, given a theory, we wish to find a partitioning that minimizes the formula... |

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Citation Context ...ing algorithm for reasoning within an individual partition. The resolution rule is complete for consequence finding (e.g., [65,102]) and the same is true for several linear resolution variants (e.g., =-=[79,56]-=-). A weaker version of completeness for consequence finding is also true for semantic resolution [103] and set-of-support resolution. Also, there are reasoning methods that focus on a given sub-langua... |

88 | Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic
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Citation Context ...quence in complexity theory, namely that NP # coNP # P/poly [14]. Nevertheless, there is a good upper bound on the length of the interpolation formula as a function of the length of the minimal proof =-=[64]-=- : If #, # share l symbols, and the resolution proof of # # # is of length k, then there is an interpolant # of length min(kl O(1) , 2 l ). For the first-order case there are very few results. The bes... |

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Citation Context ...) time, and step (2) can be implemented to take O(|E| # m) time using dynamic programming. Other algorithms that we may use in this context are variants on the cutset method for reasoning with graphs =-=[9,8]-=-. Darwiche [27] used an algorithm that is similar to BREAK-CYCLES for the problem of creating a join tree. Our algorithm di#ers from Darwiche's in treating an already formed partition and creating the... |

74 | A New Correctness Proof of the Nelson-Oppen Combination Procedure
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Citation Context ...rtitions should receive messages from which other partitions. Second, there are no clear criteria for decomposing a theory into sub-problems. Another line of work focuses on combining logical systems =-=[80,99,7,87,109]-=-. Here, the computational focus has been on treating combinations of signaturedisjoint theories (allowing the queries to include symbols from all signatures), e.g. [7]. Recent work introduced sharing ... |

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Citation Context ...sizes (typically, they are chosen to be no larger than a constant times the size of the original graph). The problem of finding balanced separators is NP-hard, but several approximations exist (e.g., =-=[66,60,43]-=-). Our time bound for the procedure is lower than #(2 #m ) when l # #m-#m i -lgn d (i = argmax j m j ). In particular, if l > m 2 , a standard deterministic SAT procedure will be better (compared to t... |

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Citation Context ...rs to perform inference. Three such systems are Cycorp's Cyc, and the High Performance Knowledge Base (HPKB) systems developed by Stanford's Knowledge Systems Lab (KSL) (e.g., [44]) and by SRI (e.g., =-=[20]-=-). These KBs comprise tens/hundreds of thousands of logical axioms. One approach to dealing with the size and complexity of these KBs is to structure the content in some way, such as into multiple dom... |

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Citation Context ... must be able to generate enough of the consequences that follow from interactions with the original partition and the messages. Generalizations of C-ordered linear resolution [55] and set-of-support =-=[45]-=- indeed have the property of completeness for such incremental generators. Other reasoning procedures that are complete for consequence finding in this incremental sense are [59,57,30,81]. We do not p... |

57 | Partition-based logical reasoning, in
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Citation Context ...ts computational complexity. Section 5 presents an algorithm for decomposing a logical theory. Finally, Section 6 discusses some related work. Some of the results in this paper appeared previously in =-=[3]-=-. 2 Partition-Based Theorem Proving In this section we address the problem of how to reason with an already partitioned propositional or FOL theory using theorem proving. In particular, we propose for... |

57 | A Prolog technology theorem prover: A new exposition and implementation
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Citation Context ...the query more aggressively by reasoning backwards from the query. Such strategies have proven e#ective for a variety of reasoning problems, such as planning. Indeed, many theorem provers (e.g., PTTP =-=[106]-=-) are built as backward reasoners and must have a query or goal in order to run. One way to use MP for an analogous backward message-passing scheme is to assert Q in A k , choose a partition A j that ... |

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Citation Context ...d Propositional Satisfiability In this section we propose an algorithm for partition-based logical reasoning based on propositional satisfiability (SAT) search (e.g., DPLL [28], GSAT [95] and WALKSAT =-=[94]-=-). We show that the complexity of computation is directly related to the size of the labels in the intersection graph. 4.1 A Partition-Based SAT Procedure The algorithm we propose uses a SAT procedure... |

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Citation Context ...on the two sides of the edge (i, j) in G, it is true that l(i, j) # L(B 1 ) # L(B 2 ). This property is similar to the running intersection property required of join trees for inference in Bayes nets =-=[27,10,98]-=- and constraint satisfaction problems (CSPs) [32,51]. The following lemma provides the main argument behind most of the completeness proofs in this paper. Lemma 2.5 Let A = # i#n A i be a partitioned ... |

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Citation Context ...e finding. It requires the input formula to be in clausal form, i.e., a conjunction of disjunctions of unquantified literals. For general first-order formulae, a transformation to clausal form (e.g., =-=[69]-=-) includes Skolemization, which eliminates quantifiers and possibly introduces new constant symbols and new function symbols. The definition of the MP algorithm introduced in Section 2.1 is amenable t... |

50 | Distributed Repositories of Highly Expressive Reusable Ontologies
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Citation Context ...eamed with theorem provers to perform inference. Three such systems are Cycorp's Cyc, and the High Performance Knowledge Base (HPKB) systems developed by Stanford's Knowledge Systems Lab (KSL) (e.g., =-=[44]-=-) and by SRI (e.g., [20]). These KBs comprise tens/hundreds of thousands of logical axioms. One approach to dealing with the size and complexity of these KBs is to structure the content in some way, s... |

50 | Resolution versus Search: Two Strategies for SAT
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Citation Context ...of size s (i.e., it has s propositional symbols in its language, link languages included), then the time for the algorithm is O(n # 2 s ). This is the analysis done for CSPs and Bayes networks (e.g., =-=[32,33,88,10]-=-), where the utilized algorithms do in fact use time #(2 m ) for a problem with m variables. For satisfiability the situation is slightly di#erent. There are known stochastic algorithms (e.g., [95,94]... |

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Citation Context ...ed fields (checking S.A = R.A), and discarding those columns that are now duplicated (e.g., R.A will be discarded). The proposed algorithm shares some intuition with prime implicate generation (e.g., =-=[74,56]-=-). First, the algorithm computes L(i), the set of symbols on all of A i 's links, in step 2. Then, it finds out which truth values of L(i) are satisfiable (akin to computing the implicates of each par... |

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Citation Context ...ion of the search space [21,41,107,23,12,108], or allowed messages to be sent between the di#erent provers working in parallel, using heuristics to decide on what messages are relevant to each prover =-=[36,38,35]-=- (surveys can be found in [11,37]). Both approaches typically look at decompositions into very few sub-problems (typically less than ten). In addition, the first approach typically requires complete i... |

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The intractability of resolution. Theoretical Computer Science, 39:297–305
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Citation Context ...ency of resolutionbased inference, and identify some of the parameters of influence. Current measures for comparing automated deduction strategies are insu#cient for our purposes. Proof length (e.g., =-=[110,52,112]-=- (also see the survey in [22])) is only marginally relevant. More relevant is comparing the sizes of search spaces induced by di#erent strategies (e.g., resolution of propositional Horn clauses [84], ... |