## On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus (1997)

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Venue: | Artificial Intelligence |

Citations: | 108 - 22 self |

### BibTeX

@ARTICLE{Renz97onthe,

author = {Jochen Renz and Bernhard Nebel},

title = {On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus},

journal = {Artificial Intelligence},

year = {1997},

volume = {108},

pages = {69--123}

}

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### Abstract

The computational properties of qualitative spatial reasoning have been investigated to some degree. However, the question for the boundary between polynomial and NP-hard reasoning problems has not been addressed yet. In this paper we explore this boundary in the "Region Connection Calculus" RCC-8. We extend Bennett's encoding of RCC-8 in modal logic. Based on this encoding, we prove that reasoning is NPcomplete in general and identify a maximal tractable subset of the relations in RCC-8 that contains all base relations. Further, we show that for this subset path-consistency is sufficient for deciding consistency. 1 Introduction When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data. In these cases, qualitative reasoning about spatial configurations may be used. One particular approach in this context has been developed by Randell, Cui, and Cohn [20], the so-called Region Connecti...

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408 |
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372 | P.: “Constraint propagation algorithms for temporal reasoning: a revised report
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Citation Context ..., the strongest implied relation for each pair of spatial regions, and RENT, the problem of whether a spatial constraint is entailed by \Theta. As it was shown for the corresponding temporal problems =-=[42,17]-=-, these problems are equivalent to RSAT under polynomial Turing reductions. 3 Encoding of RCC-8 in Modal Logic Another way of solving problems concerning RCC-8 is using the encoding of the relations i... |

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Citation Context ...e inconsistent, the negations of these formulas are tautologies: ffsfi; ffsfl; ffs:A;:fisB; :flsC; fisA; flsA; flsBs:C; fis:BsC: Proof: Using e.g. tableau based proof procedures for multi-modal logic =-=[7]-=-, a contradiction can be inferred from each of the formulas. ffsfi: contradicts the fact that no region is empty. ffsfl: analogous. ffs:A: results in a contradiction when combined with IX ! X. :fisB: ... |

264 |
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Citation Context ...pty relation occurs while performing this operation the CSP is not path-consistent, otherwise it is. Path-consistency can be decided in times O(n 3 ) where n is the total number of nodes in the graph =-=[15]-=-. Other computational problems include RMIN, the problem of finding the minimal relations, i.e. the strongest implied relation for each pair of spatial regions, 3 ENCODING OF RCC-8 IN MODAL LOGIC 5 an... |

253 |
Constraint satisfaction
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Citation Context ...hold. We call this problem RSAT. When only relations of a specific set S are used in \Theta, the corresponding reasoning problem is denoted by RSAT(S). RSAT is a Constraint Satisfaction Problem (CSP) =-=[14]-=-, where variables are nodes and relations are arcs of the constraint graph. So RSAT can be solved using the standard methods developed for CSP's. One approximative method for deciding consistency of a... |

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Citation Context ...design of efficient algorithms for the case of connected regions with a fixed dimensionality [11] appear to be interesting in an application context. A Basics on Modal Logic Propositional modal logic =-=[8, 4]-=- has the same syntax as standard propositional logic except for an additional unary operator 2. One common approach to interpret modal logical formulas is the Kripke semantics, where models M are buil... |

214 |
Reasoning about Binary Topological Relations
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Citation Context ...ce, which is part of the DFG special research effort on "Spatial Cognition." 1 2 QUALITATIVE SPATIAL REASONING WITH RCC 2 describe the topological relationship between two regions (see also =-=Egenhofer [6]-=-). Some of the computational properties of this calculus have been analyzed by Grigni et al. [11] and Nebel [16]. However, no attempt has yet been made to determine the boundary between polynomial and... |

201 |
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167 |
Modal Correspondence Theory
- Benthem
- 1976
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Citation Context ...j` OEs/ iff M;w j` OE or M;w j` / M;w j` OE ! / iff M;w j6` OE or M;w j` / M;w j` 2OE iff for all u with wRu: M; u j` OE Note that the modal operator 2 is related to the accessibility relation R (see =-=[41]-=-). Other normal modal logics are obtained by extending K with axioms that formalize properties of R. Some well-known examples of modal axioms and 54 corresponding constraints on the accessibility rela... |

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Citation Context ...ni et al. [11] and Nebel [16]. However, no attempt has yet been made to determine the boundary between polynomial and NP-hard fragments of RCC-8, as it has been done for Allen's [1] interval calculus =-=[18]-=-. We address this problem and identify a maximal fragment of RCC-8 that is still tractable and contains all base relations. As in the case of qualitative temporal reasoning, this proof relies on a com... |

98 | Spatial Reasoning with Propositional Logic
- Bennett
- 1994
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Citation Context ... topological space. The RCC theory is formulated in first order predicate calculus [20]. In this work we will focus on RCC-8, but most of our results can easily be applied to RCC-5, a subset of RCC-8 =-=[2]-=-. RCC-8 uses a set of eight pairwise disjoint and mutually exhaustive relations, called base relations, denoted as DC, EC, PO, EQ, TPP, NTPP, TPP \Gamma1 , and NTPP \Gamma1 , with the meaning of DisCo... |

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Citation Context ...r when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data. In these cases, qualitative reasoning about spatial configurations may be used =-=[8]-=-. Since space offers a very rich structure, many different aspects of space such as, e.g., distance, direction, shape, or topology can be treated in a qualitative way. It is therefore the general goal... |

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Citation Context ...as a solution in topological spaces of any dimension. This is not the case if regions must be one-piece [19]. See Section 9 for a further discussion of this topic. 5 with infinite domains (see, e.g., =-=[24]-=-). A partial method for determining inconsistency of a CSP is the path-consistency method which enforces path-consistency on a CSP [30,27]. A CSP is pathconsistentsif and only if for any consistent in... |

86 |
Linear Time Algorithms for testing the Satisfiability of Propositional Horn formulae
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Citation Context ...al time. In our case this is done by first transforming a set of b H 8 formulas to a propositional Horn formula. The resulting Horn formula can then be solved in time linear in the number of literals =-=[5]-=-. Because the number of literals is considerably higher than n 3 , this way of solving RSAT does not appear to be very efficient. One popular algorithm for solving CSP's with a running time of O(n 3 )... |

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Citation Context ...lation for each pair of spatial regions, 3 ENCODING OF RCC-8 IN MODAL LOGIC 5 and RENT, the problem whether a spatial formula is entailed by \Theta. These problems can be polynomially reduced to RSAT =-=[10]-=-. 3 Encoding of RCC-8 in Modal Logic Another way of solving problems concerning RCC is using the encoding of the relations in first order predicate logic. Such an encoding does not lead to efficient d... |

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Citation Context ... ! I:X)s2(X ! :I:IX)s:2:X: Note that the formulas 2(IIX ! IX), 2(I:X ! :X) and 2(:I:IX ! X) are entailed by the other formulas of m 2 and can therefore be ignored. As follows from the work by Bennett =-=[3]-=-, \Theta is consistent iff m(\Theta) is satisfiable. m(\Theta) is satisfiable if it is true in a world w of a Kripke model M = hW; fR 1 = W \Theta W;R 2 ` W \Theta Wg;i, where W is a set of worlds, R ... |

77 | Topological inference
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- 1995
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Citation Context ...TIAL REASONING WITH RCC 2 describe the topological relationship between two regions (see also Egenhofer [6]). Some of the computational properties of this calculus have been analyzed by Grigni et al. =-=[11]-=- and Nebel [16]. However, no attempt has yet been made to determine the boundary between polynomial and NP-hard fragments of RCC-8, as it has been done for Allen's [1] interval calculus [18]. We addre... |

59 | Solving hard qualitative temporal reasoning problems: Evaluating the efficiency of using the ORD-Horn class
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Citation Context ...algorithm. Further, if the application requires an expressive power beyond the polynomial fragment, it can be used to speed up backtracking algorithms as in the case of qualitative temporal reasoning =-=[17]-=-. Research on this topic has to be continued, as it is still an open question whether there are other maximal tractable fragments of RCC-8 that also contain all base relations. Among other open proble... |

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Citation Context ...5], spatial configuration tasks, (robot) navigation [26], computer vision, natural language processing [26], document analysis, visual languages [18], and qualitative simulation of physical processes =-=[10,34]-=-. Even when the expressive power of the calculus is too weak for a particular application, with the efficient reasoning mechanisms resulting from the work of this paper it can be used as fast preproce... |

49 |
Computing transitivity tables: a challenge for automated theorem provers. Lecture notes in computer science
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Citation Context ... and union of relations can easily be obtained by performing the corresponding set theoretic operations. Composition of base relations has to be computed using the formal definitions of the relations =-=[19, 2]-=-. The compositions of the eight base relations are shown in Table 1. Every entry in the composition table specifies the relation obtained by composing the base relation of the corresponding row with t... |

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Citation Context ...rch space is computed as b (n 2 \Gamman)=2 where b is the average branching factor, n the number of spatial variables contained in \Theta, and (n 2 \Gamma n)=2 the number of different constraints. In =-=[39]-=- we made an empirical study of reasoning with RCC-8 by randomly generating instances of up to 100 regions and solving them using different strategies. It turned out that those strategies applying c H ... |

46 | A Canonical Model of the Region Connection Calculus
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- 2002
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Citation Context ...domain of the variables are subsets of a topological space. So RSAT can be solved using the standard methods developed for CSP's 5 Here, the dimension of the topological space is not considered. Renz =-=[37]-=-, however, found that whenever a set of constraints over RCC-8 has a solution in a topological space of some dimension, it has a solution in topological spaces of any dimension. This is not the case i... |

40 |
Unit refutation and Horn sets
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- 1974
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Citation Context ...which in every resolution step at least one of the two resolved clauses is a positive unit clause, i.e., a clause containing a single positive literal. As PUR is refutation-complete for Horn formulas =-=[21]-=-, it follows that the path-consistency method decides consistency of RSAT(H 8 ). 7.1 Applying positive unit resolution to the Horn clauses of RCC-8 The only way to derive the empty clause using PUR is... |

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Citation Context ...WITH RCC 2 describe the topological relationship between two regions (see also Egenhofer [6]). Some of the computational properties of this calculus have been analyzed by Grigni et al. [11] and Nebel =-=[16]-=-. However, no attempt has yet been made to determine the boundary between polynomial and NP-hard fragments of RCC-8, as it has been done for Allen's [1] interval calculus [18]. We address this problem... |

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Citation Context ...istency method applied to \Theta also refines the constraint to a constraint in R fl . This is proven by Noetherian induction on chain structures, which is defined on well-founded relations (see e.g. =-=[43]-=-). Definition 42 A relation OE on a set M is well-founded, if and only if every non empty subset of M has a minimal element, i.e. 8S ` M:(S 6= ; ! 9x 2 S:(:9y 2 S:y OE x)). Theorem 43 (Noetherian Indu... |

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Citation Context ...ma1 gY ) =2(Y ! X)s:2(Y ! IX)s:2(X ! Y ); m 1 (XfNTPPgY ) =2(X ! IY )s:2(Y ! X); m 1 (XfNTPP \Gamma1 gY ) =2(Y ! IX)s:2(X ! Y ); m 1 (XfEQgY ) =2(X ! Y )s2(Y ! X): As follows from the work by Bennett =-=[3,4]-=-, \Theta is consistent if and only if m(\Theta) is satisfiable. It is striking that m(\Theta) is composed only of conjunctions and disjunctions of the model and entailment constraints and the regulari... |

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Citation Context ...t has yet been made to determine the boundary between polynomial and NP-hard fragments of RCC-8, as it has been done for Allen's interval calculus [33] and recently for RCC-5, a sub-calculus of RCC-8 =-=[23]-=-. We address this problem and identify a maximal fragment of RCC-8 that is still tractable and contains all base relations. As in the case of qualitative temporal reasoning, this proof relies on a com... |

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