## A Survey on Complexity Results for Non-monotonic Logics (1993)

Venue: | Journal of Logic Programming |

Citations: | 84 - 6 self |

### BibTeX

@ARTICLE{Cadoli93asurvey,

author = {Marco Cadoli and Marco Schaerf},

title = {A Survey on Complexity Results for Non-monotonic Logics},

journal = {Journal of Logic Programming},

year = {1993},

volume = {17},

pages = {127--160}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper surveys the main results appeared in the literature on the computational complexity of non-monotonic inference tasks. We not only give results about the tractability/intractability of the individual problems but we also analyze sources of complexity and explain intuitively the nature of easy/hard cases. We focus mainly on non-monotonic formalisms, like default logic, autoepistemic logic, circumscription, closed-world reasoning and abduction, whose relations with logic programming are clear and well studied. Complexity as well as recursion-theoretic results are surveyed. Work partially supported by the ESPRIT Basic Research Action COMPULOG and the Progetto Finalizzato Informatica of the CNR (Italian Research Council). The first author is supported by a CNR scholarship 1 Introduction Non-monotonic logics and negation as failure in logic programming have been defined with the goal of providing formal tools for the representation of default information. One of the ideas und...

### Citations

11605 |
Johnson: Computers and Intractability: A guide to theory of NP hardness, Feeman and
- Garey, S
- 1979
(Show Context)
Citation Context ... this paper we are interested in both kinds of results, privileging the analysis of propositional, decidable languages. We use the jargon of computational complexity and recursion theory, as found in =-=[54]-=- and in [131], respectively. In particular we make use of the notions of polynomial, arithmetical and analytical hierarchies. Using notions of higher-order complexity is necessary, since a general pro... |

1564 | The Stable Model Semantics for Logic Programming
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...ff in logic programming, where not denotes negation as failure. This analogy has been stressed and used by Gelfond in [56] to define the iterative fixedpoint semantics and by Gelfond and Lifschitz in =-=[57]-=- to define the stable model semantics for logic programs with negation. An issue of obvious computational interest concerns the compact representation of stable expansions, which is analogous to the i... |

978 |
Negation as failure
- Clark
- 1977
(Show Context)
Citation Context ...urthermore, Marek and Subrahmanian show in [93] that deciding whether a model is a stable model of a program P is \Pi 0 2 -hard and is in \Pi 0 3 . As far as the predicate completion defined by Clark =-=[35]-=- is concerned, for propositional languages it has been proven by Kolaitis and Papadimitriou in [78] that deciding whether a ground atom belongs to the completion of a program is a co-NP-complete probl... |

889 | Classical negation in logic programs and disjunctive databases
- GELFOND, LIFSCHITZ
- 1991
(Show Context)
Citation Context ...ect model semantics and the (partial) disjunctive stable model semantics is a \Pi p 2 -complete problem. A different extension of the stable model semantics has been given by Gelfond and Lifschitz in =-=[58]-=- through the notion of answer set. While inference has been shown to be a problem complete for the second level of the polynomial hierarchy, see Eiter and Gottlob [44], nevertheless some classes have ... |

631 | Towards a theory of declarative knowledge - APT, BLAIR, et al. - 1988 |

436 |
Logic programming
- Apt
- 1990
(Show Context)
Citation Context ...ors have studied the complexity of closed-world inference. Apt and Blair have proven in [5] that reasoning under the closed world assumption (2) is \Pi 0 1 -complete when T is a logic program. Apt in =-=[3]-=- had previously shown that the task is polynomial for propositional Horn formulae. The same bounds hold for the weak generalized CWA defined in [126]. Weak generalized CWA has also been studied by Cha... |

326 |
Linear-time algorithms for testing the satisfiability of propositional Horn formulae
- Dowling, Gallier
- 1984
(Show Context)
Citation Context ...y of the underlying monotonic logic. As an example, restricting the expressiveness of the language to Horn clauses allows for polynomial inference as far as classical propositional logic is concerned =-=[40]-=-, but the inference task becomes NP-hard when propositional default logic [150] or circumscription [26] are considered. The fact that non-monotonicity adds complexity to reasoning was clear from the f... |

323 |
Logic programming and databases
- Ceri, Gottlob, et al.
- 1990
(Show Context)
Citation Context ..., since this survey focuses on the logic programming semantics more directly related to NMR. For a survey on logic programming and databases we refer the reader to the book by Ceri, Gottlob and Tanca =-=[30]-=-. We survey the perfect model, stable model, default model, well-founded, supported model, positivistic, inflationary semantics and the program completion. The expressive power of the various semantic... |

205 |
Abduction compared with negation as failure
- Eshghi, Kowalski
- 1989
(Show Context)
Citation Context ...edrich, Gottlob and Nejdl in [53]. In this paper we focus on logic-based abduction, since it has been shown to be tightly related to negation in logic programming (see for example Eshghi and Kowalski =-=[49]-=- and Kakas and Mancarella [75]) and default logic (see [144]). Furthermore, we only consider propositional languages, since very little is known about first-order abduction. Logic-based abduction has ... |

191 | On the complexity of propositional knowledge base revision, updates and counterfactual
- Eiter, Gottlob
- 1992
(Show Context)
Citation Context ...heless, we want to give some pointers to the complexity results presented in the literature. In the area of belief revision and update important results can be found in the works by Eiter and Gottlob =-=[45, 43]-=-, Nebel [105, 106] and Grahne and Mendelzon [68]. The complexity of reasoning in non-monotonic inheritance networks has been analyzed by Horty, Thomason and Touretzky [71], Selman and Levesque [143], ... |

182 |
Complexity results for nonmonotonic logics
- Gottlob
- 1992
(Show Context)
Citation Context ...arithmetical or analytical hierarchy [5, 25, 34, 135]. Analogous completeness results have been found for the propositional versions of the same formalisms wrt some levels of the polynomial hierarchy =-=[41, 64]-=-. Part of the increase in the complexity of inference can be explained by noticing that the semantic definitions of most non-monotonic formalisms are either based on fixpoint constructions or on condi... |

176 |
Acyclic programs
- Apt, Bezem
- 1991
(Show Context)
Citation Context ...y Cholak and Blair [32] and Blair, Marek and Schlipf [17]. A further class of programs that has been analyzed is the class of acyclic programs. For this class of programs Apt and Bezem have proven in =-=[4]-=- that deciding membership of ground atoms in the perfect model is a decidable task. The class of acyclic programs is a proper subset of the class of locally stratified ones, and checking acyclicity of... |

171 | The complexity of logic-based abduction
- Eiter, Gottlob
- 1995
(Show Context)
Citation Context ...ctive reasoning). The search problem of finding one (best) explanation has also been studied. The most comprehensive work on the complexity of logic-based abduction has been done by Eiter and Gottlob =-=[47]-=-. In this paper, they analyze the complexity of all the above problems under different preference criteria. In particular they show that when no restrictions are imposed on the syntactic form of the t... |

146 | Propositional Semantics for Disjunctive Logic Programs
- Ben-Eliyahu, Dechter
- 1996
(Show Context)
Citation Context ... a problem complete for the second level of the polynomial hierarchy, see Eiter and Gottlob [44], nevertheless some classes have been shown to be computationally simpler by Ben-Eliyahu and Dechter in =-=[11]-=-, where they show a polynomial mapping from a subclass of disjunctive logic programs into a propositional theory. Another semantics, called disjunctive database rule has been defined by Ross and Topor... |

136 | Embedding defaults into terminological knowledge representation formalisms
- Baader, Hollunder
- 1995
(Show Context)
Citation Context ... on decidable fragments of first-order logic are concerned, decidability of credulous reasoning is guaranteed. When open defaults are considered this is no longer true: Baader and Hollunder showed in =-=[7]-=- that credulous default reasoning is undecidable for default theories built on a decidable fragment of first-order logic and containing a finite number of open defaults. Apt and Blair in [5] performed... |

131 |
A fundamental tradeoff in knowledge representation and reasoning
- Levesque, Brachman
- 1985
(Show Context)
Citation Context ... Schlipf [139]. Many results that have appeared in the literature are concerned with decidable fragments of non-monotonic logics. In the spirit of a well-established trend in knowledge representation =-=[20]-=-, they aim at the characterization of the expressive power of languages having polynomial time reasoning procedures. Other works deal with fully expressive languages and try to characterize the precis... |

118 | The computational complexity of abduction
- Bylander, Allemang, et al.
- 1991
(Show Context)
Citation Context ...s of abduction have been presented in the literature, but they are less related to NMR, and for this reason we will not survey them here. However it is worth mentioning the work of Bylander et al. in =-=[1, 21, 22]-=-, who take into account a very general definition of abduction which subsumes that of [144], proving many interesting polynomial as well as NP-hardness results. 8 Polynomial Reductions between NMR Pro... |

110 |
On stratified autoepistemic theories
- Gelfond
- 1987
(Show Context)
Citation Context ... between the interpretation of :Lff in autoepistemic logic and the meaning of not ff in logic programming, where not denotes negation as failure. This analogy has been stressed and used by Gelfond in =-=[56]-=- to define the iterative fixedpoint semantics and by Gelfond and Lifschitz in [57] to define the stable model semantics for logic programs with negation. An issue of obvious computational interest con... |

110 |
Rank-based systems: A simple approach to belief revision, belief update and reasoning about evidence and actions
- Goldszmidt, Pearl
- 1992
(Show Context)
Citation Context ...nce, restricting the attention to base languages where satisfiability can be checked in polynomial time (e. g. Horn clauses) delivers polynomial deduction tasks (see the works of Goldszmidt and Pearl =-=[62]-=-). We would like to conclude this section by summarizing the results previously presented. Propositional default reasoning is in some formal sense strictly harder than classical propositional reasonin... |

97 | Propositional Circumscription and Extended Closed World Reasoning are P 2 -complete
- Eiter, Gottlob
(Show Context)
Citation Context ...arithmetical or analytical hierarchy [5, 25, 34, 135]. Analogous completeness results have been found for the propositional versions of the same formalisms wrt some levels of the polynomial hierarchy =-=[41, 64]-=-. Part of the increase in the complexity of inference can be explained by noticing that the semantic definitions of most non-monotonic formalisms are either based on fixpoint constructions or on condi... |

94 |
On the relationship between Circumscription and Negation as Failure
- Gelfond, Przymusinska, et al.
- 1989
(Show Context)
Citation Context ...Rajasekar, Lobo and Minker [126] the weak generalizedsCWA, Yahya and Henschen [156] the extended generalized CWA, Gelfond and Przymusinska [59] the careful CWA, Gelfond, Przymusinski and Przymusinska =-=[60]-=- the extended CWA and the iterated CWA. The notion of fixed and varying predicates has been used in the careful, in the extended and in the iterated CWA. Extended CWA is the most general of all the ab... |

93 |
Reasoning with Incomplete Information
- Etherington
- 1988
(Show Context)
Citation Context ... theories have been considered more promising, since they support goal-directed algorithms for credulous reasoning (see [129]), while the algorithms proposed for non-normal theories by Etherington in =-=[50]-=- and Zhang and Marek in [158] use exponential amount of space to avoid non-termination. Skeptical default reasoning in the prerequisite-free normal case has been shown to be \Pi p 2 -complete with a s... |

69 |
A circumscriptive theorem prover
- Ginsberg
- 1989
(Show Context)
Citation Context ...ide the scope of this survey, and we refer to the book by Grahne [67] for a general overview. Several algorithms for inferencing under circumscription have been proposed: Przymusinski [122], Ginsberg =-=[61]-=-, Inoue and Helft [73], Bossu and Siegel [19], Olivetti [110], Bell et al. [8, 107]. Each of them presupposes languages of limited expressiveness. The algorithms in [122, 73, 110] use satisfiability t... |

63 |
General logic databases and programs: default semantics and stratification
- Bidoit, Froidevaux
- 1991
(Show Context)
Citation Context ...y Reiter in [129] and it is one of the more extensively studied non-monotonic formalisms. Interesting relations between default logic and logic programming have been shown by Bidoit and Froidevaux in =-=[13]-=-. They used default logic for defining a semantics for negation in logic programming. In default logic the knowledge about the world is divided into two parts, representing certain knowledge and defea... |

56 |
Negation by default and unstratifiable logic programs
- Bidoit, Froidevaux
- 1991
(Show Context)
Citation Context ...ems have been analyzed in the propositional case: deciding the existence of a stable model has been proven NP-complete by Marek and Truszczy`nski in [96] and independently by Bidoit and Froidevaux in =-=[14]-=-. In [97] Marek and Truszczy`nski prove that deciding whether an atom is a member of one stable model is NP-complete and membership in all stable models is co-NP-complete. The supported model semantic... |

51 | The complexity of model checking for circumscriptive formulae
- Cadoli
- 1992
(Show Context)
Citation Context ...sult holds when T is a universal-existential first-order sentence and Z 6= ;. co-NP is actually the upper bound for the model checking of the circumscription of all first-order formulae T . Cadoli in =-=[23]-=- strengthens this result in several directions, by proving co-NP-hardness of model checking when the formula T to be circumscribed is propositional, satisfiable, and Q = Z = ;. co-NP-hardness in the c... |

49 |
Arithmetic classification of perfect models of stratified programs
- Apt, Blair
- 1988
(Show Context)
Citation Context ...ot r. e. The issue of determining precise lower and upper bounds was addressed later and some formalisms have been proved to be complete for precise levels of the arithmetical or analytical hierarchy =-=[5, 25, 34, 135]-=-. Analogous completeness results have been found for the propositional versions of the same formalisms wrt some levels of the polynomial hierarchy [41, 64]. Part of the increase in the complexity of i... |

33 | Complexity results for disjunctive logic programming and application to nonmonotonic logics
- Eiter, Gottlob
- 1993
(Show Context)
Citation Context ...iven by Gelfond and Lifschitz in [58] through the notion of answer set. While inference has been shown to be a problem complete for the second level of the polynomial hierarchy, see Eiter and Gottlob =-=[44]-=-, nevertheless some classes have been shown to be computationally simpler by Ben-Eliyahu and Dechter in [11], where they show a polynomial mapping from a subclass of disjunctive logic programs into a ... |

32 | The complexity of closed world reasoning and circumscription
- Cadoli, Lenzerini
- 1994
(Show Context)
Citation Context ...orn clauses allows for polynomial inference as far as classical propositional logic is concerned [40], but the inference task becomes NP-hard when propositional default logic [150] or circumscription =-=[26]-=- are considered. The fact that non-monotonicity adds complexity to reasoning was clear from the first studies: in his seminal paper, Reiter [129] showed that inference in default logic is not r. e. Th... |

29 | Hierarchical knowledge bases and efficient disjunctive reasoning
- Borgida, Etherington
- 1989
(Show Context)
Citation Context ...positive ground literal. The main result of [34] is that inference of a ground atom under the generalized CWA is complete for the class \Pi 0 2 of the arithmetic hierarchy. Borgida and Etherington in =-=[18]-=- have shown a very specialized language for representing taxonomies in which some form of generalized closed-world reasoning is polynomial. The complexity of closed-world reasoning in relational datab... |

29 |
Removing redundancy from a clause
- Gottlob, Fermuller
- 1993
(Show Context)
Citation Context ...a search problem. The upper bound of the analogous task for plain propositional logic is F\Delta p 2 , hence a polynomial number of calls to an NP oracle is sufficient. Gottlob and Fermuller prove in =-=[66]-=- that unless P=NP a logarithmic number of calls to an oracle in NP is necessary. Papadimitriou proves that, if minimal satisfying truth assignments are searched, then a polynomial number of calls is s... |

28 |
The generalized completeness of Horn predicate logic as a programming language. Acta Cybernetica 4, 3–10. (This is the published version of a 1975 report entitled “General Completeness of PROLOG
- ANDRÉKA, I
- 1978
(Show Context)
Citation Context ...ip of positive and negative literals in the least Herbrand model of a Horn program has been shown r. e.-complete and co-r. e.-complete, respectively, by Smullyan in [148] and by Andreka and Nemeti in =-=[2]-=-. For a first-order language, when negation is allowed there are two different membership problems depending on which sets of models we are interested in. Traditionally, the semantics of logic program... |

28 | A Possible Worlds Semantics for Disjunctive Databases
- Chan
- 1991
(Show Context)
Citation Context ... logic programs into a propositional theory. Another semantics, called disjunctive database rule has been defined by Ross and Topor in [132] and it has been shown to be polynomially tractable by Chan =-=[31]-=-. Chan has also extended this rule to the possible worlds semantics which correctly handles negative clauses. However, literal inference under this new semantics is co-NP-complete [31, 42]. 6 Circumsc... |

28 | Eliminating the Fixed Predicates from a Circumscription
- Kleer, Konolige
- 1989
(Show Context)
Citation Context ...positional, regardless of the presence of varying and/or fixed predicates. The fact that fixed predicates do not contribute to the complexity of circumscription was proved by de Kleer and Konolige in =-=[37]-=-. On the other hand the fact that varying predicates don't affect the complexity may seem surprising, in view of the fact that the algorithms for circumscription that have been proposed (for example P... |

28 |
Complexity of reasoning with parsimonious and moderately grounded expansions. Fundamenta Informaticae
- Eiter, Gottlob
- 1992
(Show Context)
Citation Context ...can be checked by non-deterministically selecting a T 0 and checking whether the condition holds, but this obviously adds one further level of non-determinism to the computation. Eiter and Gottlob in =-=[46]-=- show that it cannot be eliminated by proving that credulous reasoning wrt moderately grounded expansions is \Sigma p 3 -complete and skeptical reasoning \Pi p 3 -complete. Autoepistemic logic is a re... |

27 |
Negation in rule-based database languages: a survey, in: Selected papers of the workshop on Deductive database theory
- Bidoit
- 1991
(Show Context)
Citation Context ...n in logic programming. This is not intended as an overview of such semantics and their relative merits, a detailed and interesting discussion on this issue can be found in the survey paper by Bidoit =-=[12]-=- and in the paper by Schlipf [138]. We will only marginally refer to the huge body of literature on complexity in rule-based database languages (DATALOG), since this survey focuses on the logic progra... |

26 |
Relating default logic and circumscription
- Etherington
- 1987
(Show Context)
Citation Context ...ze the case in which W is 1CNF and D has only disjunction free ordered defaults, which are strictly more expressive than Horn defaults. Orderedness is a property of defaults defined by Etherington in =-=[51]-=-, analogous to the idea of stratification in logic programming. A disjunction-free default has the form a 1s\Delta \Delta \Deltasa l : b 1s\Delta \Delta \Deltasb msc 1s\Delta \Delta \Deltasc n =b 1s\D... |

25 |
Default logic, propositional logic and constraints
- Ben-Eliyahu, Dechter
- 1991
(Show Context)
Citation Context ...ph, finding an extension can be done in polynomial time. This class generalizes the class of disjunction-free ordered theories, thus improving the previous results in [76]. Ben-Eliyahu and Dechter in =-=[9, 10]-=- propose a different technique for performing default reasoning: translating a default theory hD; W i into a propositional theory P so that there is a one-to-one correspondence between the models of P... |

25 | Complexity aspects of various semantics for disjunctive databases - Eiter, Gottlob - 1993 |

23 |
Negation as Failure: Careful Closure Procedure
- Gelfond, Przymusinska
- 1986
(Show Context)
Citation Context ...everal authors: Minker [102] introduced the generalized CWA, Rajasekar, Lobo and Minker [126] the weak generalizedsCWA, Yahya and Henschen [156] the extended generalized CWA, Gelfond and Przymusinska =-=[59]-=- the careful CWA, Gelfond, Przymusinski and Przymusinska [60] the extended CWA and the iterated CWA. The notion of fixed and varying predicates has been used in the careful, in the extended and in the... |

23 |
The Problem of Incomplete Information
- Grahne
- 1991
(Show Context)
Citation Context ...ld reasoning in relational databases has been studied by many authors, like for example Vardi in [155]. The problems addressed are outside the scope of this survey, and we refer to the book by Grahne =-=[67]-=- for a general overview. Several algorithms for inferencing under circumscription have been proposed: Przymusinski [122], Ginsberg [61], Inoue and Helft [73], Bossu and Siegel [19], Olivetti [110], Be... |

20 |
Hypothesis classification, abductive diagnosis and therapy
- Friedrich, Gottlob, et al.
- 1990
(Show Context)
Citation Context ...hods as defined by Reggia, Nau and Wang in [127]. Although the two methods look formally different, strong connections have been pointed out by Bylander in [21] and by Friedrich, Gottlob and Nejdl in =-=[53]-=-. In this paper we focus on logic-based abduction, since it has been shown to be tightly related to negation in logic programming (see for example Eshghi and Kowalski [49] and Kakas and Mancarella [75... |

17 |
nonmonotonic reasoning and the closed world assumption
- Bossu, Siegel, et al.
- 1995
(Show Context)
Citation Context ... the book by Grahne [67] for a general overview. Several algorithms for inferencing under circumscription have been proposed: Przymusinski [122], Ginsberg [61], Inoue and Helft [73], Bossu and Siegel =-=[19]-=-, Olivetti [110], Bell et al. [8, 107]. Each of them presupposes languages of limited expressiveness. The algorithms in [122, 73, 110] use satisfiability testers as subroutines. 6.2 Satisfiability Sat... |

16 | An efficient method for eliminating varying predicates from a circumscription
- Cadoli, Eiter, et al.
- 1992
(Show Context)
Citation Context ...ot r. e. The issue of determining precise lower and upper bounds was addressed later and some formalisms have been proved to be complete for precise levels of the arithmetical or analytical hierarchy =-=[5, 25, 34, 135]-=-. Analogous completeness results have been found for the propositional versions of the same formalisms wrt some levels of the polynomial hierarchy [41, 64]. Part of the increase in the complexity of i... |

15 |
Computational complexity of hypothesis assembly
- Allemang, Tanner, et al.
- 1987
(Show Context)
Citation Context ...s of abduction have been presented in the literature, but they are less related to NMR, and for this reason we will not survey them here. However it is worth mentioning the work of Bylander et al. in =-=[1, 21, 22]-=-, who take into account a very general definition of abduction which subsumes that of [144], proving many interesting polynomial as well as NP-hardness results. 8 Polynomial Reductions between NMR Pro... |

15 | The Expressiveness of Locally Stratified Programs
- Blair, Marek, et al.
- 1995
(Show Context)
Citation Context ...al class of locally stratified programs turns out to be computationally more complex. In fact, membership is \Delta 1 1 -complete over ! as shown by Cholak and Blair [32] and Blair, Marek and Schlipf =-=[17]-=-. A further class of programs that has been analyzed is the class of acyclic programs. For this class of programs Apt and Bezem have proven in [4] that deciding membership of ground atoms in the perfe... |

15 | The complexity of local stratification
- Cholak, Blair
- 1994
(Show Context)
Citation Context ...he problem is r. e. The more general class of locally stratified programs turns out to be computationally more complex. In fact, membership is \Delta 1 1 -complete over ! as shown by Cholak and Blair =-=[32]-=- and Blair, Marek and Schlipf [17]. A further class of programs that has been analyzed is the class of acyclic programs. For this class of programs Apt and Bezem have proven in [4] that deciding membe... |

15 | The complexity of nested counterfactuals and iterated knowledge base revisions
- Eiter, Gottlob
- 1996
(Show Context)
Citation Context ...heless, we want to give some pointers to the complexity results presented in the literature. In the area of belief revision and update important results can be found in the works by Eiter and Gottlob =-=[45, 43]-=-, Nebel [105, 106] and Grahne and Mendelzon [68]. The complexity of reasoning in non-monotonic inheritance networks has been analyzed by Horty, Thomason and Touretzky [71], Selman and Levesque [143], ... |

11 |
Implementing Deductive Database by Linear Programming
- Bell, Nerode, et al.
- 1992
(Show Context)
Citation Context ...eral overview. Several algorithms for inferencing under circumscription have been proposed: Przymusinski [122], Ginsberg [61], Inoue and Helft [73], Bossu and Siegel [19], Olivetti [110], Bell et al. =-=[8, 107]-=-. Each of them presupposes languages of limited expressiveness. The algorithms in [122, 73, 110] use satisfiability testers as subroutines. 6.2 Satisfiability Satisfiability is another computational t... |

10 |
The recursion-theoretic complexity of the semantics of predicate logic as a programming language
- Blair
- 1982
(Show Context)
Citation Context ... g. Kunen's paper [81]) it is assumed that models are interpretations over a larger universe containing an infinite number of constants as well as function symbols. As already pointed out by Blair in =-=[16]-=- the second alternative leads, in general, to computationally simpler problems. Where not otherwise indicated, we implicitly deal with Herbrand models. We now analyze the complexity of reasoning tasks... |