## Propositional Circumscription and Extended Closed World Reasoning are $\Pi^P_2$-complete (1993)

Venue: | Theoretical Computer Science |

Citations: | 99 - 21 self |

### BibTeX

@ARTICLE{Eiter93propositionalcircumscription,

author = {Thomas Eiter and Georg Gottlob},

title = {Propositional Circumscription and Extended Closed World Reasoning are $\Pi^P_2$-complete},

journal = {Theoretical Computer Science},

year = {1993},

volume = {114},

pages = {231--245}

}

### Years of Citing Articles

### OpenURL

### Abstract

Circumscription and the closed world assumption with its variants are well-known nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$-complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$-complete, and we show how this result applies to other variants of closed world reasoning.

### Citations

11412 |
Computers and Intractability. A Guide to the Theory of NP-Completeness. edited by W.H
- Garey, Johnson
- 1997
(Show Context)
Citation Context ...e size of T if only the variables of T are considered. For the other CWR-rules, EGCWA and ECWA, exponential space is required in the worst case. We refer to the standard notation in complexity theory =-=[10, 15]-=-. Recall that P A (NP A ) corresponds to the class of decision problems that are solved by deterministic (nondeterministic) Turing machines with an oracle for A in polynomial time. Problem B is polyno... |

854 |
Circumscription, a form of non-monotonic reasoning
- McCarthy
- 1980
(Show Context)
Citation Context ...tlobg@vexpert.dbai.tuwien.ac.at 1 and Przymusinska [13], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12]. Circumscription was introduced by McCarthy in =-=[22-=-]. It is known that circumscription as dened in [36] coincides with the ECWA in the case of propositional logic [12]. While much work has been devoted to the study of the logical properties of such fo... |

734 | The semantics of predicate logic as a programming language
- Emden, Kowalski
- 1976
(Show Context)
Citation Context ...al, checking the consistency of the closure with CWA is an additional { though not unrelated problem { to consider. A semantical characterization of CWA in terms of minimal Herbrand models appears in =-=[3-=-2], which states that CWA(T ) is consistent i the intersection of all Herbrand models of T is a model of T . Hence clearly, Lemma 3.5 CWA(T ) is consistent i T has a unique minimal model M and CWA(T )... |

435 |
Introduction to logic programming
- Apt
- 1990
(Show Context)
Citation Context ...er variants of closed world reasoning. 1 Introduction The nonmonotonic inference techniques of closed world reasoning are widely used in Articial Intelligence, Database Theory, and Logic Programming [=-=2]-=-. Starting with the (naive) Closed World Assumption (CWA) introduced by Reiter [26], several formalizations of closed world reasoning have been developed. In this paper, besides the CWA, we consider t... |

328 | Computing circumscription
- Lifschitz
- 1985
(Show Context)
Citation Context ...13], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12]. Circumscription was introduced by McCarthy in [22]. It is known that circumscription as dened in [=-=36]-=- coincides with the ECWA in the case of propositional logic [12]. While much work has been devoted to the study of the logical properties of such forms of closed world reasoning and of their applicabi... |

320 |
Word problems requiring exponential time: Preliminary report
- Stockmeyer, Meyer
- 1973
(Show Context)
Citation Context ... 8x n 9y 1 9ymE(x 1 ; : : : x n ; y 1 ; : : : ; y m ); where E(x 1 ; : : : ; x n ; y 1 ; : : : ; y m ) is a Boolean expression in variables x 1 ; : : : ; x n ; y 1 ; : : : ; y m , is true [30]. The transformation is as follows. Let z 1 ; : : : ; z n and u be new variables. We dene the following theory T : T = [ ^ 1in (x i 6 z i ) ] ^ [ (u ^ y 1 ^ y 2 ^ ^ y m ) _ E(x 1 ; : : : ; x... |

194 | The NP-completeness column: An ongoing guide
- JOHNSON
- 1986
(Show Context)
Citation Context .... Hence clearly, Lemma 3.5 CWA(T ) is consistent i T has a unique minimal model M and CWA(T ) is logically equivalent to M . CWA consistency checking turns out to be the unique solution variant (cf. [=-=14]-=- for uniqueness questions) of the following problem MINSAT: Has T a minimal model ? Note that the latter is simply the NP-complete SATISFIABILITY (SAT) problem, since every consistent theory has a min... |

190 |
The complexity of optimization problems
- Krentel
- 1988
(Show Context)
Citation Context ... CCWA(T ; P ; Q;Z) j= F 2 P -hard P 2 P [O(log n)] ECWA(T ; P ; Q;Z) j= F , CIRC(T ; Q;P ; Z) j= F 2 P -complete Table 1: Complexity Results for Propositional Closed World Deduction problem, cf. [19=-=, 34-=-]. This may be measured by the number of necessary calls to a P k oracle [19, 17, 33]. Closure computation with O(n) oracle calls is straightforward for CWA, GCWA, and CCWA. For CWA, it is not dicult... |

133 |
Bounded Query Classes
- Wagner
- 1990
(Show Context)
Citation Context ...uggests that the deduction problem under GCWA and under CCWA is not p m -complete for P 3 , since it seems rather unlikely that a problem in P P 2 [O(log n)] is p m - complete for P 3 , cf. [16, =-=34-=-]. 4 Conclusion Our main results are summarized in Table 1. Answering the question in [5], we have shown that the deduction problem with ECWA and with circumscription is P 2 -complete for proposition... |

92 |
On the Relationship Between Circumscription and Negation as Failure
- Gelfond, Przymusinska, et al.
- 1989
(Show Context)
Citation Context ...6, A-1040 Wien, Austria. Internet e-mail: feiter,gottlobg@vexpert.dbai.tuwien.ac.at 1 and Przymusinska [13], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski =-=[12-=-]. Circumscription was introduced by McCarthy in [22]. It is known that circumscription as dened in [36] coincides with the ECWA in the case of propositional logic [12]. While much work has been devot... |

92 |
The complexity of facets (and some facets of complexity
- Papadimitriou, Yannakakis
- 1984
(Show Context)
Citation Context ...es the maximum number of a set of clauses is unique [16]. As for USAT and UOASAT, UMINSAT is easily proved co-NP-hard, but it is not clear how to reduce SAT to it. USAT is complete for the class D P [=-=24-=-] under the randomized reduction vv m of Valiant-Vazirani [31] and, as recently proved, USAT is not in co-D P , which contains NP[co-NP, unless the polynomial hierarchy collapses [8, 7]. We now show ... |

88 | How to progress a database
- Lin, Reiter
- 1997
(Show Context)
Citation Context ...echniques of closed world reasoning are widely used in Articial Intelligence, Database Theory, and Logic Programming [2]. Starting with the (naive) Closed World Assumption (CWA) introduced by Reiter [=-=26]-=-, several formalizations of closed world reasoning have been developed. In this paper, besides the CWA, we consider the following well-known approaches to closed world reasoning: the Generalized Close... |

74 |
More complicated questions about maxima and minima, and some closures of NP
- Wagner
- 1987
(Show Context)
Citation Context ...RC(T ; Q;P ; Z) j= F 2 P -complete Table 1: Complexity Results for Propositional Closed World Deduction problem, cf. [19, 34]. This may be measured by the number of necessary calls to a P k oracle [1=-=9, 17, 33]-=-. Closure computation with O(n) oracle calls is straightforward for CWA, GCWA, and CCWA. For CWA, it is not dicult to show that closure computation is (under suitable polynomial transformability) equi... |

73 |
On Indefinite Data Bases and the Closed World Assumption
- Minker
- 1982
(Show Context)
Citation Context ...orld reasoning have been developed. In this paper, besides the CWA, we consider the following well-known approaches to closed world reasoning: the Generalized Closed World Assumption (GCWA) by Minker =-=[23-=-], the Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen [35], the Careful Closed World Assumption (CCWA) by Gelfond Mailing address: TU Wien, Paniglgasse 16, A-1040 Wien, Au... |

71 |
The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses
- Kadin
- 1988
(Show Context)
Citation Context ... if a Boolean expression E has a unique satisfying assignment [4]. Another similar problem is UOASAT, which asks if the truth assignment that satises the maximum number of a set of clauses is unique [=-=16-=-]. As for USAT and UOASAT, UMINSAT is easily proved co-NP-hard, but it is not clear how to reduce SAT to it. USAT is complete for the class D P [24] under the randomized reduction vv m of Valiant-Vaz... |

48 |
Inferring negative information from disjunctive databases
- Ross, Topor
- 1988
(Show Context)
Citation Context ...s well as CCWA, and both EGCWA and CCWA are restrictions of ECWA. Recently, a weakened form of GCWA has been introduced [25] for syntactically restricted theories, called disjunctive database rule in =-=[27]-=-; this CWR-rule has a polynomial-time algorithm for deduction [5]. An alternative characterization of the CWR-rules is possible in terms of minimal Herbrand models. Recall that in the propositional ca... |

44 |
Deduction in non-Horn databases
- Yahya, Henschen
- 1985
(Show Context)
Citation Context ...ollowing well-known approaches to closed world reasoning: the Generalized Closed World Assumption (GCWA) by Minker [23], the Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen =-=[35-=-], the Careful Closed World Assumption (CCWA) by Gelfond Mailing address: TU Wien, Paniglgasse 16, A-1040 Wien, Austria. Internet e-mail: feiter,gottlobg@vexpert.dbai.tuwien.ac.at 1 and Przymusinska ... |

38 |
Weak generalized closed world assumption
- Rajasekar, Lobo, et al.
- 1989
(Show Context)
Citation Context ...however, preserve consistency. Note that GCWA is a restricted version of EGCWA as well as CCWA, and both EGCWA and CCWA are restrictions of ECWA. Recently, a weakened form of GCWA has been introduced =-=[25]-=- for syntactically restricted theories, called disjunctive database rule in [27]; this CWR-rule has a polynomial-time algorithm for deduction [5]. An alternative characterization of the CWR-rules is p... |

32 |
The Complexity of Closed World Reasoning and Circumscription
- Cadoli, Lenzerini
- 1990
(Show Context)
Citation Context ...to circumscription are 1 2 -complete and 1 2 -complete [28]. Other complexity results concerning model checking are provided by Kolaitis and Papadimitriou in [18]. Cadoli and Lenzerini present in [6=-=, 5]-=- a very careful analysis of the computational complexity of the above-mentioned forms of closed world reasoning for various fragments of propositional logic. They mainly consider syntactically restric... |

29 |
Polynomial terse sets
- Amir, Gasarch
- 1988
(Show Context)
Citation Context ...ven Boolean expressions E 1 ; : : : ; Em , compute b 1 ; : : : ; b m where b i = 1 if E i is satisable and b i = 0 if E i is unsatisable. This problem requires at most m oracle calls; by results in [1=-=]-=-, suciency of fewer calls is unlikely. It is unknown, however, whether QUERY is complete for FP NP , the functions computable in deterministic polynomial time with unrestricted NP oracle access. Simil... |

27 |
NP is as easy as detecting unique solutions, Theoretical Computer Science 47
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...or USAT and UOASAT, UMINSAT is easily proved co-NP-hard, but it is not clear how to reduce SAT to it. USAT is complete for the class D P [24] under the randomized reduction vv m of Valiant-Vazirani [=-=31]-=- and, as recently proved, USAT is not in co-D P , which contains NP[co-NP, unless the polynomial hierarchy collapses [8, 7]. We now show that UMINSAT, and thus CWA consistency checking, is at least as... |

26 |
Some computational aspects of circumscription
- Kolaitis, Papadimitriou
- 1990
(Show Context)
Citation Context ...proven that relevant problems related to circumscription are 1 2 -complete and 1 2 -complete [28]. Other complexity results concerning model checking are provided by Kolaitis and Papadimitriou in [1=-=8]-=-. Cadoli and Lenzerini present in [6, 5] a very careful analysis of the computational complexity of the above-mentioned forms of closed world reasoning for various fragments of propositional logic. Th... |

23 |
PNP[logí‘›] and sparse Turing-complete sets for NP
- Kadin
- 1989
(Show Context)
Citation Context ...RC(T ; Q;P ; Z) j= F 2 P -complete Table 1: Complexity Results for Propositional Closed World Deduction problem, cf. [19, 34]. This may be measured by the number of necessary calls to a P k oracle [1=-=9, 17, 33]-=-. Closure computation with O(n) oracle calls is straightforward for CWA, GCWA, and CCWA. For CWA, it is not dicult to show that closure computation is (under suitable polynomial transformability) equi... |

22 |
and Halina Przymusinska. Negation as failure: careful closure procedure
- Gelfond
- 1986
(Show Context)
Citation Context ..., the Careful Closed World Assumption (CCWA) by Gelfond Mailing address: TU Wien, Paniglgasse 16, A-1040 Wien, Austria. Internet e-mail: feiter,gottlobg@vexpert.dbai.tuwien.ac.at 1 and Przymusinska [=-=13-=-], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12]. Circumscription was introduced by McCarthy in [22]. It is known that circumscription as dened in [36... |

9 |
A Catalog of Complexity Classes, volume A of Handbook of Theoretical
- Johnson
- 1990
(Show Context)
Citation Context ...e size of T if only the variables of T are considered. For the other CWR-rules, EGCWA and ECWA, exponential space is required in the worst case. We refer to the standard notation in complexity theory =-=[10, 15]-=-. Recall that P A (NP A ) corresponds to the class of decision problems that are solved by deterministic (nondeterministic) Turing machines with an oracle for A in polynomial time. Problem B is polyno... |

8 |
Generalized Closed World Assumption is
- Chomicki, Subrahmanian
- 1990
(Show Context)
Citation Context ...ld inference rules have degrees of unsolvability at dierent levels of the arithmetical hierarchy. In particular, Apt and Blair [3] show that the CWA is complete for 0 1 ; Chomicki and Subrahmanian [9] show that GCWA is 0 2 -complete; and Schlipf has proven that relevant problems related to circumscription are 1 2 -complete and 1 2 -complete [28]. Other complexity results concerning model che... |

7 |
Arithmetic classi of perfect models of strati programs
- Apt, Blair
- 1990
(Show Context)
Citation Context ... 0 2 . For thesrst order case, it was shown that a number of closed world inference rules have degrees of unsolvability at dierent levels of the arithmetical hierarchy. In particular, Apt and Blair [3] show that the CWA is complete for 0 1 ; Chomicki and Subrahmanian [9] show that GCWA is 0 2 -complete; and Schlipf has proven that relevant problems related to circumscription are 1 2 -complete... |

7 |
The Complexity of Optimization Functions
- Gasarch
- 1986
(Show Context)
Citation Context ... calls is straightforward for CWA, GCWA, and CCWA. For CWA, it is not dicult to show that closure computation is (under suitable polynomial transformability) equivalent to the following problem QUERY =-=[1-=-1]: Given Boolean expressions E 1 ; : : : ; Em , compute b 1 ; : : : ; b m where b i = 1 if E i is satisable and b i = 0 if E i is unsatisable. This problem requires at most m oracle calls; by results... |

6 |
When is Closed World Reasoning Tractable
- Schlipf
- 1988
(Show Context)
Citation Context ...f the logical properties of such forms of closed world reasoning and of their applicability in dierent contexts, the interest in a complexity analysis of these methods has emerged only more recently [=-=2, 20, 21, -=-29]. Papalaskari and Weinstein [37] show that inference from innite propositional theories under minimal consequence (i.e. circumscription) is 0 2 and not 0 2 . For thesrst order case, it was shown ... |

4 |
On the unique satis problem
- Blass, Gurevich
- 1982
(Show Context)
Citation Context ...nt UMINSAT is to decide if T has exactly one minimal model. Note that this problem is similar to the well-studied USAT problem, which asks if a Boolean expression E has a unique satisfying assignment =-=[4-=-]. Another similar problem is UOASAT, which asks if the truth assignment that satises the maximum number of a set of clauses is unique [16]. As for USAT and UOASAT, UMINSAT is easily proved co-NP-hard... |

2 |
On Unique Satis and Randomized Reductions
- Chang, Rohatgi
- 1990
(Show Context)
Citation Context ...r the class D P [24] under the randomized reduction vv m of Valiant-Vazirani [31] and, as recently proved, USAT is not in co-D P , which contains NP[co-NP, unless the polynomial hierarchy collapses [=-=8, 7-=-]. We now show that UMINSAT, and thus CWA consistency checking, is at least as hard as USAT. Lemma 3.6 USAT p m UMINSAT. Proof: Let E(x 1 : : : ; x n ) be a Boolean expression in variables x 1 ; : : ... |

1 |
On the Structure of Uniquely Satis Formulas
- Chang, Kadin
- 1990
(Show Context)
Citation Context ...r the class D P [24] under the randomized reduction vv m of Valiant-Vazirani [31] and, as recently proved, USAT is not in co-D P , which contains NP[co-NP, unless the polynomial hierarchy collapses [=-=8, 7-=-]. We now show that UMINSAT, and thus CWA consistency checking, is at least as hard as USAT. Lemma 3.6 USAT p m UMINSAT. Proof: Let E(x 1 : : : ; x n ) be a Boolean expression in variables x 1 ; : : ... |

1 |
Covering and Disjointness Relationships in Hierarchy Networks
- Lenzerini
- 1988
(Show Context)
Citation Context ...f the logical properties of such forms of closed world reasoning and of their applicability in dierent contexts, the interest in a complexity analysis of these methods has emerged only more recently [=-=2, 20, 21, -=-29]. Papalaskari and Weinstein [37] show that inference from innite propositional theories under minimal consequence (i.e. circumscription) is 0 2 and not 0 2 . For thesrst order case, it was shown ... |

1 |
Reasoning About Inheritance Networks
- Lenzerini
- 1988
(Show Context)
Citation Context ...f the logical properties of such forms of closed world reasoning and of their applicability in dierent contexts, the interest in a complexity analysis of these methods has emerged only more recently [=-=2, 20, 21, -=-29]. Papalaskari and Weinstein [37] show that inference from innite propositional theories under minimal consequence (i.e. circumscription) is 0 2 and not 0 2 . For thesrst order case, it was shown ... |

1 |
Decidability and De with Circumscription
- Schlipf
- 1987
(Show Context)
Citation Context ... complete for 0 1 ; Chomicki and Subrahmanian [9] show that GCWA is 0 2 -complete; and Schlipf has proven that relevant problems related to circumscription are 1 2 -complete and 1 2 -complete [28]=-=-=-. Other complexity results concerning model checking are provided by Kolaitis and Papadimitriou in [18]. Cadoli and Lenzerini present in [6, 5] a very careful analysis of the computational complexity ... |