## Is Intractability of Non-Monotonic Reasoning a Real Drawback? (1996)

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Venue: | Artificial Intelligence |

Citations: | 43 - 8 self |

### BibTeX

@INPROCEEDINGS{Cadoli96isintractability,

author = {Marco Cadoli and Francesco M. Donini and Marco Schaerf},

title = {Is Intractability of Non-Monotonic Reasoning a Real Drawback?},

booktitle = {Artificial Intelligence},

year = {1996},

pages = {946--951}

}

### OpenURL

### Abstract

Several studies about computational complexity of non-monotonic reasoning (NMR) showed that non-monotonic inference is significantly harder than classical, monotonic inference. This contrasts with the general idea that NMR can be used to make knowledge representation and reasoning simpler, not harder. In this paper we show that, to some extent, NMR fulfills the representation goal. In particular, we prove that non-monotonic formalisms such as circumscription and default logic allow for a much more compact and natural representation of propositional knowledge than propositional calculus. Proofs are based on a suitable definition of compilable inference problem, and on non-uniform complexity classes. Some results about intractability of circumscription and default logic can therefore be interpreted as the price one has to pay for having such an extra-compact representation. On the other hand, intractability of inference and compactness of representation are not equivalent notions: we ex...

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A Logic for Default Reasoning
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Citation Context ...ested by an answer to Problem 2 are preferred. The complexity of the above problems has been already studied in the literature. When T is 2-CNF we have the following figures: Problem 1. is polynomial =-=[13]-=-; Problem 2. is polynomial [4]; Problem 3. is polynomial [13]; Problem 4. is co-NP-complete [9]. In the above scenario NMR seems to do exactly the form of reasoning the student needs. In fact a soluti... |

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Citation Context ...ant to know what researchers in Theoretical Computer Science have discovered about such classes. Relations between non-uniform and uniform complexity classes were studied in the literature, cf. e.g., =-=[20, 45]-=-. In particular, it is proved that if co-NP ` P/poly, then the collapse of the so-called polynomial hierarchy happens. More formally, if co-NP ` P/poly then \Sigma p 2 = \Pi p 2 . Collapse of the poly... |

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Citation Context ...e problems may become intractable (e.g., the complexity raises from polynomial to NP-complete [22]), intractable problems may become "more" intractable (e.g., from NP-complete to \Sigma p 2 =-=-complete [17]), decidab-=-le problems may become undecidable [1], and undecidable problems may become "more" undecidable (e.g., from r.e.-complete to \Pi 1 2 -complete [38]). A survey on computational aspects of NMR ... |

151 |
Foundations of a functional approach to knowledge representation
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- 1984
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Citation Context ...lexity raises from polynomial to NP-complete [22]), intractable problems may become "more" intractable (e.g., from NP-complete to \Sigma p 2 -complete [17]), decidable problems may become un=-=decidable [1], and unde-=-cidable problems may become "more" undecidable (e.g., from r.e.-complete to \Pi 1 2 -complete [38]). A survey on computational aspects of NMR appears as [10]. This aspect of NMR is acknowled... |

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Citation Context ...s for saving storage space, at the cost of loosing both soundness and completeness. Selman & Kautz are the first authors to introduce the idea of compilation in the knowledge representation field. In =-=[40, 23]-=- they propose to compile a propositional theory for obtaining two Horn formulae, called respectively Horn greatest lower bound and Horn least upper bound. The main difference with our approach is in t... |

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Citation Context ...e in a more compact fashion. Noticeably, negation through cut is commonly used among prolog programmers for writing more compact and efficient programs [41, Chap.11]. Moreover, closed-world reasoning =-=[35, 15, 16, 34]-=- allows for effective representation of implicit knowledge in relational as well as deductive databases, and has been widely used among database practitioners for many years now. Apparently, there is ... |

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Citation Context ...e problems may become undecidable [1], and undecidable problems may become "more" undecidable (e.g., from r.e.-complete to \Pi 1 2 -complete [38]). A survey on computational aspects of NMR a=-=ppears as [10]. Thi-=-s aspect of NMR is acknowledged in the AI community. Brachman [3, p. 1090] writes: "An irony of work on NMR is that, while the easy adoption and retraction of assumptions is most useful for speed... |

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Citation Context ... in specific cases. Nevertheless, some forms of inference are simpler if a formula is represented via the set of its characteristic models. This idea has been expanded by Khardon & Roth in two papers =-=[24, 25]-=- where they present a new framework for learning and reasoning. In particular, in [25] they analyze the possibility of rewriting a propositional theory in a new form that admits polynomial-time infere... |

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problems for simple default logics
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Citation Context ...rence has a complexity which is comparable to classical inference, the general picture shows that tractable problems may become intractable (e.g., the complexity raises from polynomial to NP-complete =-=[22]), intractable probl-=-ems may become "more" intractable (e.g., from NP-complete to \Sigma p 2 -complete [17]), decidable problems may become undecidable [1], and undecidable problems may become "more" u... |

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56 | Learning to reason
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Citation Context ... in specific cases. Nevertheless, some forms of inference are simpler if a formula is represented via the set of its characteristic models. This idea has been expanded by Khardon & Roth in two papers =-=[24, 25]-=- where they present a new framework for learning and reasoning. In particular, in [25] they analyze the possibility of rewriting a propositional theory in a new form that admits polynomial-time infere... |

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Citation Context ...representation. In order to prove such results, we use the notion of non-uniform complexity class, a topic whose importance for knowledge representation and reasoning has been recently highlighted in =-=[23]. On the other hand,-=- results of kind 2 show that "non-compilability" is not always related to intractability, since there are polynomially-intractable formalisms that are "compilable". Finally, result... |

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Deduction in non-horn databases
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Citation Context ...A(T ) = T [ f:p j T 6j= pg: (1) This rule has been refined by several authors: Minker [30] introduced the generalized CWA, Rajasekar, Lobo and Minker [34] the weak generalized CWA, Yahya and Henschen =-=[44]-=- the extended generalized CWA, Gelfond and Przymusinska [15] the careful CWA, Gelfond, Przymusinski and Przymusinska [16] the extended CWA. The notion of varying atoms has been used in the careful and... |

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Citation Context ...e in a more compact fashion. Noticeably, negation through cut is commonly used among prolog programmers for writing more compact and efficient programs [41, Chap.11]. Moreover, closed-world reasoning =-=[35, 15, 16, 34]-=- allows for effective representation of implicit knowledge in relational as well as deductive databases, and has been widely used among database practitioners for many years now. Apparently, there is ... |

37 | Reasoning With characteristic models
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Citation Context ...proved that not only no efficient basis exists but also that no other compilation is possible. The possibility of rewriting off-line a propositional theory, was addressed by Kautz, Kearns & Selman in =-=[21]-=-. In particular, they investigate the reformulation of a Horn formula into the set of its characteristic models, where characteristic models are independent models that cannot be obtained as intersect... |

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The complexity of propositional closed world reasoning and circumscription. Journal of computer and system sciences
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Citation Context ...ce. As an example, while deciding if a literal follows from a propositional Horn formula can be done in linear time provided we reason in the classical semantics [11], the same task is co-NP-complete =-=[9]-=- if we reason under circumscription. Although there are cases in which non-monotonic inference has a complexity which is comparable to classical inference, the general picture shows that tractable pro... |

34 | Near-optimal plans, tractability and reactivity
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Citation Context ...ion has size polynomial in the number of propositional variables but whose set of characteristic models has exponential size. Selman studied planning problems structured in fixed and varying parts in =-=[39]-=-, analyzing the impact in terms of computational complexity of such a structure. An instance of a planning problem is defined by an initial state, a final (i.e., desired) state, and a set of operators... |

33 | On compact representations of propositional circumscription - Cadoli, Donini, et al. - 1995 |

32 |
Circumscription implies predicate completion (sometimes
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Citation Context ..., is defined as in formula (2), except that now K is a single atom from P . Relationships between circumscription and closed-world reasoning were studied by many researchers -- see for example Reiter =-=[37]-=-, Lifschitz [27], and all work defining closed-world rules. In particular it has been shown that an abstract notion of minimality underlies both of them. For example, CWA(T ) is consistent iff T has a... |

32 | Hard Problems for Simple Default Logics
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Citation Context ...rence has a complexity which is comparable to classical inference, the general picture shows that tractable problems may become intractable (e.g. the complexity raises from polynomial to NP-complete (=-=Kautz & Selman 1991)), intrac-=-table problems may become "more" intractable (e.g. from NP-complete to \Sigma p 2 -complete (Gottlob 1992)), decidable problems may become undecidable (Baader & Hollunder 1992), and undecida... |

31 |
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Citation Context ...sitional Horn formulae can be done in linear time provided we reason in the classical semantics (Dowling & Gallier 1984), inference under circumscription in such formulae is a co-NP-complete problem (=-=Cadoli & Lenzerini 1990-=-). Although there are cases in which non-monotonic inference has a complexity which is comparable to classical inference, the general picture shows that tractable problems may become intractable (e.g.... |

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Citation Context ...s non-uniform complexity classes to prove non-compilability. We want to point out that some semantical aspects of Horn approximations can be explained in terms of non-monotonic reasoning, as shown in =-=[5]-=-. Other papers discuss the possibility of reducing the complexity of query an36 swering through off-line preprocessing, specifically focusing on inference in classical propositional logic. Moses & Ten... |

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- 1989
(Show Context)
Citation Context ...ving representational space, trading off-line computation for on-line efficiency---appeared in the literature. The approach taken in such papers is now briefly compared to ours. Borgida & Etherington =-=[2]-=- propose a system for representing hierarchical knowledge, i.e., subset relations among classes of individual objects. In the system it is possible to say, for example, that cats, snakes and ferrets a... |

27 |
Itâ€™s not my default: The complexity of membership problems in restricted propositional default logics
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- 1990
(Show Context)
Citation Context ...cific subject. We can formalize these preferences via the set of default rules D = f :s 1 s 1 ; : : : ; :s k s k g. Defaults of this kind are called Prerequisite-Free Positive Normal Unary (PFPNU) in =-=[42]-=-. Intuitively, an extension of the default theory hD; W i is a curriculum which maximizes courses in S, while still satisfying constraints in W . In our fellowship example, we have S = fAlgebra; Compu... |

26 |
Relating default logic and circumscription
- Etherington
- 1987
(Show Context)
Citation Context ...in Table 2, where we divide cases between the two forms of reasoning (credulous and skeptical) and between conjunctive and disjunctive queries. We conclude the section with a remark. It is well known =-=[12]-=- that default logic easily simulates circumscription, using a default ::a :a for each atom a to be minimized. Hence, one may wonder the difference between results in this section and those in the prev... |

25 |
Otter 2.0
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(Show Context)
Citation Context ...ion. Given that the problem is computationally intractable, we address the following question: if the student was to reason in a classical fashion -- e.g. using a classical theorem prover like OTTER (=-=McCune 1990-=-) -- how much implicit information would he/she need to represent explicitly in the new KB? In other words, is it feasible to transform the original KB -- dealt with NMR -- into a new one -- dealt wit... |

22 |
and Halina Przymusinska. Negation as failure: careful closure procedure
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- 1986
(Show Context)
Citation Context |

21 |
Off-line reasoning for on-line efficiency
- Moses, Tennenholtz
- 1996
(Show Context)
Citation Context ...apers discuss the possibility of reducing the complexity of query an36 swering through off-line preprocessing, specifically focusing on inference in classical propositional logic. Moses & Tennenholtz =-=[31]-=- analyze the possibility of speeding up the complexity of query answering through a previous off-line analysis of the knowledge base. Their goal can be considered as a special case of our notion of co... |

18 |
Closed-World Databases and Circumscription
- Lifschitz
- 1985
(Show Context)
Citation Context ...in formula (2), except that now K is a single atom from P . Relationships between circumscription and closed-world reasoning were studied by many researchers -- see for example Reiter [37], Lifschitz =-=[27]-=-, and all work defining closed-world rules. In particular it has been shown that an abstract notion of minimality underlies both of them. For example, CWA(T ) is consistent iff T has a unique minimal ... |

14 |
Computing circumscriptive databases. I: Theory and algorithms
- Nerode, Ng, et al.
- 1995
(Show Context)
Citation Context ... the size of the universal plan. As a matter of facts, Selman is able to prove that there exist cases in which such a super-polynomial blow-up must happen, unless NP ` P/poly. Recently, Nerode et al. =-=[32]-=- investigated reasoning in deductive databases where predicates can be minimized with circumscription. Their explicit goal is to compile the database off-line, trading the space needed to store the co... |

12 |
Incomplete deductive databases
- Imielinski
- 1991
(Show Context)
Citation Context ...em 10. Decide whether a course is mandatory. It can be shown that unsatisfiability of 3CNF formulae can be reduced to this problem, hence we can conclude that the problem is co-NP-complete (cf. e.g., =-=[18]). Of cour-=-se, Problem 10 can be answered "no" if one finds an admissible curriculum not containing the given course. Now suppose the student wants to find such a curriculum. Since the (intractable) pr... |

10 |
On the complexity of model finding for nonmonotonic propositional logics
- Cadoli
- 1992
(Show Context)
Citation Context ... are preferred. The complexity of the above problems has been already studied in the literature. When T is 2-CNF we have the following figures: Problem 1. is polynomial [13]; Problem 2. is polynomial =-=[4]-=-; Problem 3. is polynomial [13]; Problem 4. is co-NP-complete [9]. In the above scenario NMR seems to do exactly the form of reasoning the student needs. In fact a solution to Problem 4 gives some ext... |

9 |
Decidability and definability with circumscription
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- 1987
(Show Context)
Citation Context ...., from NP-complete to \Sigma p 2 -complete [17]), decidable problems may become undecidable [1], and undecidable problems may become "more" undecidable (e.g., from r.e.-complete to \Pi 1 2 =-=-complete [38]). A -=-survey on computational aspects of NMR appears as [10]. This aspect of NMR is acknowledged in the AI community. Brachman [3, p. 1090] writes: "An irony of work on NMR is that, while the easy adop... |

4 |
Decidability and Definability with Circumscription, Annals of Pure and Applied Logic, Volume 35, Number 14
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- 1987
(Show Context)
Citation Context ...2 -complete (Gottlob 1992)), decidable problems may become undecidable (Baader & Hollunder 1992), and undecidable problems may become "more" undecidable (e.g. from r.e.-complete to \Pi 1 2 -=-=complete (Schlipf 1987)). A-=-n upto -date survey on computational aspects of NMR appears as (Cadoli & Schaerf 1993). This aspect of NMR is acknowledged in the AI community. Brachman (1990, p. 1090) writes: "An irony of work ... |

2 |
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- 1995
(Show Context)
Citation Context ...pact representation of knowledge. These results can also be extended to circumscription of general propositional formulae and to many formalisms for belief revision and update (as shown in [8] and in =-=[6]-=-, respectively). As a matter of facts, in the present work we do not prove that the space of a purely propositional representation of a nonmonotonic formula must be super-polynomial (this is actually ... |

2 |
On compact representations of circumscription
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- 1995
(Show Context)
Citation Context ...or more compact representation of knowledge. These results can also be extended to circumscription of general propositional formulae and to many formalisms for belief revision and update (as shown in =-=[8]-=- and in [6], respectively). As a matter of facts, in the present work we do not prove that the space of a purely propositional representation of a nonmonotonic formula must be super-polynomial (this i... |