## A complete classification of the complexity of propositional abduction (2006)

Venue: | SIAM Journal on Computing |

Citations: | 19 - 4 self |

### BibTeX

@ARTICLE{Creignou06acomplete,

author = {Nadia Creignou and Bruno Zanuttini},

title = {A complete classification of the complexity of propositional abduction},

journal = {SIAM Journal on Computing},

year = {2006},

volume = {36},

pages = {207--229}

}

### OpenURL

### Abstract

Abstract. Abduction is the process of explaining a given query with respect to some background knowledge. For instance, p is an explanation for the query q given the knowledge p → q. This problem is well-known to have many applications, in particular in Artificial Intelligence, and has been widely studied from both an AI and a complexity-theoretic point of view. In this paper we completely classify the complexity of propositional abduction in Schaefer’s famous framework. We consider the case where knowledge bases are taken from a class of formulas in generalized conjunctive normal form. This means that the propositional formulas considered are conjunctions of constraints taken from a fixed finite language. We show that according to the properties of this language, deciding whether at least one explanation exists is either polynomial, NP-complete or ΣP 2-complete. Our results are stated for a query consisting of a single, positive literal and for assumption-based solutions, i.e., the solutions must be formed upon a distinguished subset of the variables that is part of the input. We however show that our results can be interpreted “dually ” for negative queries, and thus also for unrestricted (positive or negative) queries.

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Citation Context ...on the syntactic form of the knowledge base have been identified that make the problem easier, even polynomial for some of them: 2CNF and monotone knowledge bases (see, e.g., [20]), definite Horn CNF =-=[26, 11]-=- and affine [28] bases, CNF bases with unit-refutable pseudo-completion [14] or with bounded kernel width [10], Horn bases given by their characteristic models [19] and some classes of DNF bases [28].... |

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Citation Context ... e.g., [20]), definite Horn CNF [26, 11] and affine [28] bases, CNF bases with unit-refutable pseudo-completion [14] or with bounded kernel width [10], Horn bases given by their characteristic models =-=[19]-=- and some classes of DNF bases [28]. Among those restrictions some impose local properties to the knowledge bases, such as being in definite Horn CNF, and some impose global properties, such as being ... |

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Citation Context ...roblem easier, even polynomial for some of them: 2CNF and monotone knowledge bases (see, e.g., [20]), definite Horn CNF [26, 11] and affine [28] bases, CNF bases with unit-refutable pseudo-completion =-=[14]-=- or with bounded kernel width [10], Horn bases given by their characteristic models [19] and some classes of DNF bases [28]. Among those restrictions some impose local properties to the knowledge base... |

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Citation Context ...isfiability is NP-complete. From this result we derive two additional trichotomy results by considering the variants of abduction in which the query to be explained is a negative literal, or an 1 See =-=[18]-=- for the definition of IHS-B− or IHS-B+ relations.sCOMPLEXITY OF PROPOSITIONAL ABDUCTION 3 unrestricted (positive or negative) literal. In order to obtain these results, on one hand we exhibit new tra... |

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Citation Context ...s between diseases and symptoms, and finally, the abducible variables are those which model diseases (since we do not want to explain a symptom by another symptom). In the same vein, system diagnosis =-=[6, 27]-=-, where one wants to discover the faulty components of a system that does not behave as desired (e.g., an electronic device), can be modelled as an ∗ LIF, UMR CNRS 6166, Université de la Méditerranée,... |

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Citation Context ...CNRS 6072, Université de Caen, Boulevard du Maréchal Juin, 14032 Caen Cedex, France (zanutti@info.unicaen.fr). 1s2 N. CREIGNOU AND B. ZANUTTINI abduction problem. Another application is configuration =-=[1]-=-, and still another one is text interpretation [16]. Abduction has also been studied with temporal knowledge bases [4]. Closer to the reasoning processes themselves, abduction is a fundamental part of... |

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Citation Context ...no such relation is known for relevance. Other interesting issues concern the complexity of counting or enumerating the solutions. For this latter point we refer the reader to Eiter and Makino’s work =-=[12, 13]-=-. We believe that for such studies one would first need to determine the properties that are preserved by implementations (e.g., the variables involved in a solution, the number of solutions etc...), ... |

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Citation Context ...0 and every xi ′ to 1 except for xi ′ 0 ). We will also often use the well-known fact that all the prime implicates of a given formula ϕ in CNF can be generated by repeatedly applying resolution to ϕ =-=[23]-=-. Recall that resolution is the process of adding the clause C1 ∨ C2 to ϕ if there is a variable x such that both clauses (x ∨ C1) and (¬x ∨ C2) are in ϕ and C1, C2 do not contain two opposite literal... |

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(Show Context)
Citation Context ... some of them: 2CNF and monotone knowledge bases (see, e.g., [20]), definite Horn CNF [26, 11] and affine [28] bases, CNF bases with unit-refutable pseudo-completion [14] or with bounded kernel width =-=[10]-=-, Horn bases given by their characteristic models [19] and some classes of DNF bases [28]. Among those restrictions some impose local properties to the knowledge bases, such as being in definite Horn ... |

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Citation Context ...owever many restrictions on the syntactic form of the knowledge base have been identified that make the problem easier, even polynomial for some of them: 2CNF and monotone knowledge bases (see, e.g., =-=[20]-=-), definite Horn CNF [26, 11] and affine [28] bases, CNF bases with unit-refutable pseudo-completion [14] or with bounded kernel width [10], Horn bases given by their characteristic models [19] and so... |

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Citation Context ...m of the knowledge base have been identified that make the problem easier, even polynomial for some of them: 2CNF and monotone knowledge bases (see, e.g., [20]), definite Horn CNF [26, 11] and affine =-=[28]-=- bases, CNF bases with unit-refutable pseudo-completion [14] or with bounded kernel width [10], Horn bases given by their characteristic models [19] and some classes of DNF bases [28]. Among those res... |

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Citation Context ...d IHS-B+ relations can be recognized efficiently by testing closure under the mappings (m, m ′ , m ′′ ) ↦→ m ∧ (m ′ ∨ m ′′ ) and (m, m ′ , m ′′ ) ↦→ m ∨ (m ′ ∧ m ′′ ), respectively; this is proved in =-=[8]-=- and can also be derived from [3] 2 . 2.4. Complexity classes. This paper concerns the computational complexity of the abduction problem, and more precisely of the associated decision problem. We assu... |

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(Show Context)
Citation Context ...riables; that is, the literals occuring in an explanation are restricted, instead of only the variables upon which they are formed. It turns out that a trichotomy has also been found for this problem =-=[21]-=-. Current work aims at classifying every intermediate model, considering various restrictions on the queries and hypotheses, with the goals18 N. CREIGNOU AND B. ZANUTTINI Query pos. neg. any L affine ... |

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Citation Context ...s between diseases and symptoms, and finally, the abducible variables are those which model diseases (since we do not want to explain a symptom by another symptom). In the same vein, system diagnosis =-=[6, 27]-=-, where one wants to discover the faulty components of a system that does not behave as desired (e.g., an electronic device), can be modelled as an ∗ LIF, UMR CNRS 6166, Université de la Méditerranée,... |

3 |
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Citation Context ...-B−, IHS-B+, complementive, 1-valid or 0-valid. Proof. This is obvious for bijunctive, affine and (dual) Horn languages given the closure properties above; more efficient algorithms are also given in =-=[9, 29, 15]-=-. As for definite Horn languages, simply observe that a relation is definite Horn if and only if it is Horn and contains the vector (1, . . . , 1), which can again be tested efficiently; thes6 N. CREI... |

2 |
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(Show Context)
Citation Context ...CREIGNOU AND B. ZANUTTINI abduction problem. Another application is configuration [1], and still another one is text interpretation [16]. Abduction has also been studied with temporal knowledge bases =-=[4]-=-. Closer to the reasoning processes themselves, abduction is a fundamental part of the CMS/ATMS [24], and it has many relations with default reasoning [26]. From a complexity-theoretic point of view, ... |

1 |
The United Nations Statistical Office has undertaken a number of projects dealing with the use of household surveys for development purposes. In the mid-1980s they commissioned a project in Zimbabwe to see whether a model survey of literacy use and skill
- unknown authors
(Show Context)
Citation Context ...ND B. ZANUTTINI Proof. For sake of completeness we demonstrate or cite the implementations in the appendix, but the five points may be obtained by techniques from universal algebra and Post’s lattice =-=[17, 2]-=-. We now make a case distinction depending on whether L is 0-valid (resp. 1-valid) or not. Proposition 7.3 (neither 0- nor 1-valid). If L is neither Schaefer nor 0-valid nor 1-valid, then the problem ... |