## Paritition-based logical reasoning (2000)

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Venue: | In Proc. KR ’2000 |

Citations: | 57 - 15 self |

### BibTeX

@INPROCEEDINGS{Amir00paritition-basedlogical,

author = {Eyal Amir and Sheila Mcilraith},

title = {Paritition-based logical reasoning},

booktitle = {In Proc. KR ’2000},

year = {2000},

pages = {389--400},

publisher = {Morgan Kaufmann}

}

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### Abstract

We investigate the problem of reasoning with partitions of related logical axioms. Our motivation is two-fold. First, we are concerned with how to reason effectively with multiple knowledge bases that have overlap in content. Second, and more fundamentally, we are concerned with how to exploit structure inherent in a set of logical axioms to induce a partitioning of the axioms that will lead to an improvement in the efficiency of reasoning. To this end, we provide algorithms for reasoning with partitions of axioms in propositional and first-order logic. Craig’s interpolation theorem serves as a key to proving completeness of these algorithms. We analyze the computational benefit of our algorithms and detect those parameters of a partitioning that influence the efficiency of computation. These parameters are the number of symbols shared by a pair of partitions, the size of each partition, and the topology of the partitioning. Finally, we provide a greedy algorithm that automatically decomposes a given theory into partitions, exploiting the parameters that influence the efficiency of computation. 1

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Citation Context ... separator between a and b in G. Algorithms for finding maximal flow are abundant in the graph algorithms literature. Prominent algorithms for maxflow include the Simplex method, Ford and Fulkerson's =-=[31]-=-, the push-relabel method of Goldberg and Tarjan [27] (time bound of O (I V I * l EIsla ) and several implementations [11]), and Dinitz's algorithm [21]. When Dinitz's algorithm is used to solve the n... |

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Citation Context ...ngineered physical systems, many of the domain axiomatizations have these structural properties. Indeed, design of engineering artifacts encourages modularization, with minimal interconnectivity (see =-=[2, 34, 12]-=-). More generally, we believe axiomatizers of large corpora of real-world knowledge tend to try to provide structured representations following some of these principles. 4 Decomposing a Logical Theory... |

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Citation Context ...ng maximal flow are abundant in the graph algorithms literature. Prominent algorithms for maxflow include the Simplex method, Ford and Fulkerson's [31], the push-relabel method of Goldberg and Tarjan =-=[27]-=- (time bound of O (I V I * l EIsla ) and several implementations [11]), and Dinitz's algorithm [21]. When Dinitz's algorithm is used to solve the network problem, algorithm MIN-V-SEP-A-B is of time co... |

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Citation Context ...ee Appendix A.2. BREAK-CYCLES is a greedy algorithm that has a worstcase complexity of O(IEI 2 * m) (where m is the number of symbols in E(A)). An algorithm for finding a minimum spanning tree (e.g., =-=[22]-=-) can be used to compute the optimal subgraph for our purpose (minimizing E(i,j)CE II(i,J)l) in time O(ns+ msn), if we assume that the labels are mutually disjoint. It is not clear whether the algorit... |

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Citation Context ...mated reasoning; [6] presents one example. Many AI researchers have exploited some type of structure to improve the efficiency of reasoning (e.g., Bayes Nets [40], Markov decision processes [8], CSPs =-=[19, 18]-=-, and model-based diagnosis [17]). There is also a vast literature in both clustering and decomposition techniques. Most relevant to our work, are CSP decomposition techniques that look for a separati... |

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Citation Context ...s followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g., =-=[38, 45, 3]-=-). Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition. Decomposition for propositional SAT has f... |

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Citation Context ...efficiency of reasoning, and automatic decomposition. Decomposition for propositional SAT has followed different tracks. Some work focused on heuristics for clause weighting or symbol ordering (e.g., =-=[43, 20]-=-). [39] suggested a decomposition procedure that represents the theory as a hypergraph of clauses and divides the propositional theory into two partitions (heuristically minimizing the number of hyper... |

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Citation Context ... The resolution rule is complete for consequence finding (e.g., [33, 46]) and the same is true for semantic resolution [47] (and set-of-support resolution [24]), and linear resolution variants (e.g., =-=[30]-=-). Such consequence finders are used for prime implicate generation in applications such as diagnosis. Inoue [30] provides an algorithm for selectively generating consequences or characteristic clause... |

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Citation Context ... we propose, and little work on automatically partitioning logical theories for this purpose. Nevertheless, there are many areas of related work. Work on formalizing and reasoning with context (e.g., =-=[37, 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 ...d and complete consequence-finding algorithm. The resolution rule is complete for consequence finding (e.g., [33, 46]) and the same is true for semantic resolution [47] (and set-of-support resolution =-=[24]-=-), and linear resolution variants (e.g., [30]). Such consequence finders are used for prime implicate generation in applications such as diagnosis. Inoue [30] provides an algorithm for selectively gen... |

64 |
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Citation Context ...ocedure that represents the theory as a hypergraph of clauses and divides the propositional theory into two partitions (heuristically minimizing the number of hyperedges) modifying ideas described in =-=[25]-=-. [14] developed an algorithm that partitions a propositional theory into connected components. Both [14, 39] performed experiments that demonstrated a decrease in the time required to prove test sets... |

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Citation Context ...ample. Many AI researchers have exploited some type of structure to improve the efficiency of reasoning (e.g., Bayes Nets [40], Markov decision processes [8], CSPs [19, 18], and model-based diagnosis =-=[17]-=-). There is also a vast literature in both clustering and decomposition techniques. Most relevant to our work, are CSP decomposition techniques that look for a separation vertex [ 18]. Also related is... |

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Citation Context ...orld, teamed 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) =-=[23]-=- and by SRI (e.g., [12]). 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, * ... |

46 |
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Citation Context ...med fields (checking S.A = /LA), and discarding those columns that are now duplicated (e.g.,/LA will be discarded). The proposed algorithm shares some intuition with prime implicate generation (e.g., =-=[36, 30]-=-). Briefly, we first compute all the models of each of the partitions (akin to computing the implicates of each partition). We then usesto combine the partition models into models for ,A. The algorith... |

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Citation Context ...y of resolution-based inference, and identify some of the parameters of influence. Current measures for comparing automated deduction strategies are insufficient for our purposes. rroof length (e.g., =-=[28]-=-) is only marginally relevant. More relevant is comparing the sizes of search spaces of different strategies (e.g., [41]). These measures do not precisely address our needs, but we use them here, leav... |

43 |
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Citation Context ...turn YES. Figure 3: A forward message-passing algorithm. This algorithm exploits consequence finding (step 2) to perform reasoning in the individual partitions. Consequence finding was defined by Lee =-=[33]-=- to be the problem of finding all the logical consequences of a theory or sentences that subsume them. In MP, we can use any sound and complete consequence-finding algorithm. The resolution rule is co... |

22 |
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Citation Context ...oblem of defining guidelines and parameters for good decompositions of sets of axioms for the purpose of logical reasoning. Decomposition has not been exploited in theorem proving until recently (see =-=[4, 5]-=-). We believe that part of the reason for this lack of interest has been that theorem proving has focused on mathematical domains that do not necessarily have structure that supports decomposition. Wo... |

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Citation Context ...o become smaller. 3. n - the number of partitions. Also, a simple analysis shows that given fixed values for l, d in Corollary 3.3, the maximal n that maintains l, d such that also nsa * m (a = 0.582 =-=[42, 13]-=-) yields an optimal bound for LINEAR-PART-SAT In Section 2.2 we saw Figure 8: Decomposing .4's symbols graph. Figure 8 (top) illustrates the symbols graph of theory .4 from Figure 1 and the connected ... |

14 |
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Citation Context ...nclude the Simplex method, Ford and Fulkerson's [31], the push-relabel method of Goldberg and Tarjan [27] (time bound of O (I V I * l EIsla ) and several implementations [11]), and Dinitz's algorithm =-=[21]-=-. When Dinitz's algorithm is used to solve the network problem, algorithm MIN-V-SEP-A-B is of time complexity o(IvIsILl) [22]. Finally, to compute the vertex connectivity of a graph and a minimum sepa... |

13 |
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Citation Context ...e sets of sentences [46], and use resolution theorem proving [46, 29], with or without equality [15, 16] (all after proper reformulation of the theorem). Lyndon's version of the interpolation theorem =-=[35]-=- adds sensitivity to the polarity of relation symbols (? includes P positively (negatively) only if P shows positively (negatively) in both a and/3). PROCEDURE BREAK-CYCLES(G = (V, E, 1)) 1. Find a mi... |

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Citation Context ...mated reasoning; [6] presents one example. Many AI researchers have exploited some type of structure to improve the efficiency of reasoning (e.g., Bayes Nets [40], Markov decision processes [8], CSPs =-=[19, 18]-=-, and model-based diagnosis [17]). There is also a vast literature in both clustering and decomposition techniques. Most relevant to our work, are CSP decomposition techniques that look for a separati... |

8 | A multicontext architecture for formalizing complex reasoning
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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 =-=[26]-=-), There have been few reported successes with automated reasoning; [6] presents one example. Many AI researchers have exploited some type of structure to improve the efficiency of reasoning (e.g., Ba... |

7 |
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Citation Context ... and t3, such that ( [- ? and ? Craig's interpolation theorem is true even if we take c,/3 to be infinite sets of sentences [46], and use resolution theorem proving [46, 29], with or without equality =-=[15, 16]-=- (all after proper reformulation of the theorem). Lyndon's version of the interpolation theorem [35] adds sensitivity to the polarity of relation symbols (? includes P positively (negatively) only if ... |

6 |
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Citation Context ...lving only symbols common to both ( and t3, such that ( [- ? and ? Craig's interpolation theorem is true even if we take c,/3 to be infinite sets of sentences [46], and use resolution theorem proving =-=[46, 29]-=-, with or without equality [15, 16] (all after proper reformulation of the theorem). Lyndon's version of the interpolation theorem [35] adds sensitivity to the polarity of relation symbols (? includes... |

5 |
Formalizing Context (Expanded
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Citation Context ... we propose, and little work on automatically partitioning logical theories for this purpose. Nevertheless, there are many areas of related work. Work on formalizing and reasoning with context (e.g., =-=[37, 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 ...uence in complexity theory, namely that NP rq coNP C_ P/poll/[7]. Nevertheless, there is a good upper bound on the length of the interpolation formula as a function of the length of the minimal proof =-=[32]-=-sIf a,/ share l symbols, and the resolution proof of a -/ is of length k, then there is an interpolant ? of length min(kl 0) , 2t). Thus, we can guarantee a small interpolant, if we make sure the comm... |

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Citation Context ...ngineered physical systems, many of the domain axiomatizations have these structural properties. Indeed, design of engineering artifacts encourages modularization, with minimal interconnectivity (see =-=[2, 34, 12]-=-). More generally, we believe axiomatizers of large corpora of real-world knowledge tend to try to provide structured representations following some of these principles. 4 Decomposing a Logical Theory... |

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