Efficiently Enforcing Path Consistency on Qualitative Constraint Networks by Use of Abstraction

Transcription

1 Efficiently Enforcg Path Consistency on Qualitative Constrat Networks by Use of Abstraction Michael Sioutis and Jean François Condotta Université d Artois, CRIL-CNRS UMR 8188, Lens, France {sioutis,condotta}@cril.fr Abstract Partial closure under weak composition, or partial -consistency for short, is essential for tacklg fundamental reasong problems associated with qualitative constrat networks, such as the satisfiability checkg problem, and therefore it is crucial to be able to enforce it as fast as possible. To this end, we propose a new algorithm, called PWC α, for efficiently enforcg partial -consistency on qualitative constrat networks, that eploits the notion of abstraction for qualitative constrat networks, utilies certa properties of partial -consistency, and adapts the functionalities of some state-of-theart algorithms to its design. It is worth notg that, as opposed to a related approach the recent literature, algorithm PWC α is complete for arbitrary qualitative constrat networks. The evaluation that we conducted with qualitative constrat networks of the Region Connection Calculus agast a competg state-of-the-art generic algorithm for enforcg partial -consistency, demonstrates the usefulness and efficiency of algorithm PWC α. 1 Introduction Qualitative Spatial and Temporal Reasong (QSTR) is a major field of study Artificial Intelligence, and Knowledge Representation & Reasong particular. This field has received a lot of attention over the past decades, as it etends to a plethora of areas and domas that clude ambient telligence, dynamic GIS, cognitive robotics, and spatiotemporal design [Bhatt et al., 2011]. QSTR abstracts from numerical quantities of space and time by usg qualitative descriptions stead (e.g., precedes, contas, is left of ), thus providg a concise framework that allows for rather epensive reasong about entities located space and time. The problem of representg and reasong about qualitative formation can be modeled as a qualitative constrat network (QCN), i.e., a network of constrats correspondg to qualitative spatial or temporal relations between spatial or temporal variables respectively, usg a qualitative constrat language. A qualitative constrat language volves constrats defed over a fite set of bary relations, called base relations (or atoms) [Ligoat and Ren, 2004]. The fundamental reasong problems associated with a given QCN N are the problems of satisfiability checkg, mimal labelg (or deductive closure), and redundancy (or entailment) [Ren and Nebel, 2007]. In particular, the satisfiability checkg problem is the problem of decidg if there eists a spatial or temporal valuation of the variables of N that satisfies its constrats, such a valuation beg called a solution of N, the mimal labelg problem is the problem of fdg the strongest implied constrats of N, and the redundancy problem is the problem of determg if a given constrat is entailed by N (that constrat beg called redundant, as its removal does not change the solution set of the QCN). In general, for most qualitative constrat languages the satisfiability checkg problem is NP-complete. Further, the redundancy problem, the mimal labelg problem, and the satisfiability checkg problem are equivalent under polynomial Turg reductions [Golumbic and Shamir, 1993]. The vast amount, if not all, of the published works that study the aforementioned reasong problems, use partial -consistency as a means to defe practical algorithms for efficiently tacklg them [Amanedde et al., 2013; Sioutis et al., 2015; 2016b; Li et al., 2015; Ren and Nebel, 2001; Nebel, 1997]. Given a QCN N and a graph G, partial -consistency with respect to G, denoted by G -consistency, entails (weak) consistency for all triples of variables N that correspond to three-verte cycles (triangles) G. We note that if G is complete, G -consistency becomes identical to -consistency [Ren and Ligoat, 2005]. Hence, -consistency is a special case of G-consistency. In fact, earlier works have relied solely on -consistency; it was not until the troduction of chordal (or triangulated) graphs QSTR, due to some generalied theoretical results of [Huang, 2012], that researchers started restrictg -consistency to a triangulation of the constrat graph of an put QCN and benefitg from better compleity properties. Currently, and to the best of our knowledge, the fastest algorithm for enforcg partial -consistency, which is presented [Long et al., 2016], is complete only for certa subsets of the set of allowed relations that can occur a QCN. This fact significantly limits its practicality for arbitrary QCNs. Further, sce those subsets defe subclasses of relations that are general smaller than the classes of relations for which partial -consistency is able to decide satisfiability, the algorithm of [Long et al., 2016] is able to tackle 1262

2 fewer QCNs comparison with a generic algorithm for enforcg partial -consistency. To ameliorate this issue, we eploit the notion of abstraction for QCNs, which is an idea adopted from concepts of abstract terpretation [Cousot and Cousot, 1992]. In particular, a QCN is typically abstracted by relag some of its constrats order to satisfy some property (if possible); our case, we aim to abstract a given QCN is such a way that all of its constrats volve relations for which the algorithm of [Long et al., 2016] is complete. This should allow us to harvest as much of the performance gas of that algorithm as possible for an arbitrary QCN N, as we can use the algorithm to enforce partial -consistency on a proper abstraction of N very fast, and then crementally apply appropriate constrat propagations on its output QCN to achieve partial -consistency pertag to the origal QCN N. Specifically, we make the followg contributions: (i) we provide an efficient and generic algorithm for enforcg partial -consistency that eploits the notion of abstraction for QCNs and makes full use of the algorithm of [Long et al., 2016], (ii) we prove the correctness of our approach by utilig certa properties of partial -consistency, such as idempotence and monotonicity, and (iii) we eperimentally validate the usefulness and efficiency of our method agast the state-of-the-art generic algorithm for enforcg partial -consistency of [Chmeiss and Condotta, 2011]. 2 Prelimaries y {EC} {EC} {EC} A (bary) qualitative spatial or temporal constrat language, is based on a fite set B of jotly ehaustive and pairwise disjot relations defed over an fite doma D, which is called the set of base relations [Ligoat and Ren, 2004]. The base relations of a particular qualitative constrat language can be used to represent the defite knowledge between any two of its entities with respect to the level of granularity provided by the doma D. The set B contas the identity relation Id, and is closed under the converse operation ( 1 ). Indefite knowledge can be specified by a union of possible base relations, and is represented by the set contag them. Hence, 2 B represents the total set of relations. The set 2 B is equipped with the usual set-theoretic operations of union and tersection, the converse operation, and the weak composition operation denoted by the symbol [Ligoat and Ren, 2004]. For all r 2 B, we have that r 1 = {b 1 b r}. The weak composition () of two base relations b, b B is defed as the smallest (i.e., strongest) relation r 2 B that cludes b b, or, formally, b b ={b B b (b b ) }, where b b ={(, y) D D D such that (, ) b (, y) b } is the (true) composition of b and b. For all r, r 2 B, we have that r r = {b b b r, b r }. As an eample, the Region Connection Calculus (RCC) is a first-order theory for representg and reasong about mereotopological formation [Randell et al., 1992]. The doma D of RCC comprises all possible non-empty regular subsets of some topological space. These subsets do not have to be ternally connected and do not have a particular dimension, but are commonly required to be regular closed [Ren, 2002]. Simply put, a subset X of a topological space is regular closed that space, if X equals the cloy Figure 1: A QCN of RCC-8 along with a solution sure of its terior. The base relations of RCC are the followg ones: disconnected (DC), eternally connected (EC), equal (EQ), partially overlappg (P O), tangential proper part (T P P ), tangential proper part verse (T P P i), nontangential proper part (NT P P ), and non-tangential proper part verse (NT P P i). These eight base relations form the RCC-8 constrat language. Relation EQ is the identity relation Id of RCC-8. Other notable and well known qualitative spatial and temporal constrat languages clude Pot Algebra [Vila and Kaut, 1986], Cardal Direction Calculus [Ligoat, 1998; Frank, 1991], Interval Algebra [Allen, 1983], and Block Algebra [Balbiani et al., 2002]. The weak composition operation, the converse operation 1, the union operation, the complement operation, and the total set of relations 2 B along with the identity relation Id of a qualitative constrat language, form an algebraic structure (2 B, Id,, 1,, ) that can correspond to a relation algebra the sense of Tarski [Tarski, 1941]. Proposition 1 ([Dylla et al., 2013]). The languages of Pot Algebra, Cardal Direction Calculus, Interval Algebra, Block Algebra, and RCC-8 are each a relation algebra with the algebraic structure (2 B, Id,, 1,, ). In what follows, for a qualitative constrat language that is a relation algebra with the algebraic structure (2 B, Id,, 1,, ), we will simply use the term relation algebra, as the algebraic structure will always be of the same format. The problem of representg and reasong about qualitative formation can be modeled as a qualitative constrat network (QCN), defed the followg manner: Defition 1. A qualitative constrat network (QCN) is a tuple (V, C) where: V = {v 1,..., v n } is a non-empty fite set of variables, each representg an entity; and C is a mappg C : V V 2 B such that C(v, v) = {Id} for all v V and C(v, v ) = (C(v, v)) 1 for all v, v V. An eample of a QCN of RCC-8 is shown Figure 1. In particular, the QCN comprises the set of variables {, y, } and the constrats C(, y) = C(y, ) = C(, ) = {EC}; for simplicity, converse relations as well as Id loops are not mentioned or shown the figure. Defition 2. Let N = (V, C) be a QCN, then: a solution of N is a mappg σ : V D such that (u, v) V V, b C(u, v) such that (σ(u), σ(v)) b; N is satisfiable iff it admits a solution; a QCN is equivalent to N iff it admits the same set of solutions as N ; a sub-qcn N of N, denoted by N N, is a QCN (V, C ) such that C (u, v) C(u, v) u, v V ; the constrat graph of N is the graph (V, E) where {u, v} E iff C(u, v) B and u v; and N is trivially consistent iff u, v V such that C(u, v) =. 1263

3 y {DC} {P O, NT P P } {EC, T P P } y {DC} {P O, TPP, NT P P } {EC, PO, T P P } Figure 2: A QCN of RCC-8 along with an abstraction We recall the followg defition of G-consistency, which, as noted the troduction, is the basic local consistency used the literature for solvg fundamental reasong problems of QCNs, such as the satisfiability checkg problem. Defition 3. Given a QCN N = (V, C) and a graph G = (V, E), N is G -consistent iff {v i, v j }, {v i, v k }, {v k, v j } E we have that C(v i, v j ) C(v i, v k ) C(v k, v j ). It is route to formally prove the followg properties of G -consistency, which we will use the sequel, as they naturally derive from respective properties of -consistency: Property 1. Let N = (V, C) be a QCN and G = (V, E) a graph. Then, the followg properties hold: G (N ) N (vi., the G-closure of N ) is the largest (w.r.t. ) G-consistent sub-qcn of N (Domance); G (N ) is equivalent to N (Equivalence); G ( G (N )) = G (N ) (Idempotence); if N N then G (N ) G (N ) (Monotonicity). We note aga that if G is complete, G-consistency becomes identical to -consistency [Ren and Ligoat, 2005], and, hence, -consistency is a special case of G -consistency. Defition 4. A subclass of relations is a subset A 2 B that contas the sgleton relations of 2 B and is closed under converse, tersection, and weak composition. Given a relation r of 2 B and a subclass A 2 B contag B, A(r) denotes the smallest relation of A that cludes r. Defition 5. Given a QCN N = (V, C), a graph G = (V, E), and a subclass A 2 B contag B, the abstraction of N w.r.t. to A and G, denoted by A G (N ), is the QCN (V, C ) where C (v, v ) = A(C(v, v )) {v, v } E. As an eample, Figure 2 illustrates the abstraction of a QCN of RCC-8 w.r.t. to the subclass that results from the closure of the base relations of RCC-8 under converse, tersection, and weak composition (which, addition, contas B) and the complete graph on its set of variables. Given three relations r, r, r 2 B, we say that weak composition distributes over tersection if we have that r (r r ) = (r r ) (r r ) and (r r ) r = (r r) (r r). Defition 6. A subclass A is distributive iff weak composition distributes over non-empty tersection r, r, r A. Distributive subclasses of relations contag the universal relation B are defed for all of the qualitative constrat languages mentioned Proposition 1 [Long and Li, 2015]. 3 G -Consistency by Use of Abstraction The ma idea behd our approach, is to benefit as much as possible from the performance characteristics of the algorithm for enforcg partial -consistency of [Long et al., Algorithm 1: PWC α (N, G, α, A) : A QCN N = (V, C), a graph G = (V, E), a bijection α : V {0, 1,..., V 1}, and a subclass A of 2 B. 1 beg 2 N = (V, C ) A G (N ); 3 (decision, N ) DPWC+(N, G, α); 4 if decision = False then 5 return (False, N ); 6 e ; 7 foreach {v, v } E do 8 r C (v, v ) C(v, v ); 9 if r = then return (False, N ); 10 if r C (v, v ) then 11 e e {{v, v }}; 12 C (v, v ) r; 13 C (v, v) r 1 ; 14 return PWC(N, G, e); 2016], when given an arbitrary QCN N. As noted the troduction, that algorithm is complete only for certa subsets of 2 B. Those subsets essentially defe subclasses of relations that are distributive (see Defition 6). Hence, to be able to make full use of the algorithm of [Long et al., 2016], we utilie an abstraction of N with respect to some distributive subclass of relations, feed that abstraction to the algorithm, compare its output QCN N with our itial QCN N and properly update N, and fally employ a generic algorithm for enforcg partial -consistency to crementally apply appropriate constrat propagations on the updated QCN N and achieve partial -consistency pertag to the origal QCN N. An algorithm that follows the aforementioned steps is presented Algorithm 1, called PWC α (which stands for partial closure under weak composition by use of abstraction), where subroute DPWC+ le 3 (which stands for directional partial closure under weak composition plus) corresponds to a generalied version of the algorithm of [Long et al., 2016] and subroute PWC le 14 (which stands for partial closure under weak composition) is the state-of-the-art generic algorithm for enforcg partial -consistency of [Chmeiss and Condotta, 2011]. As opposed to the origal algorithm of [Long et al., 2016], its generalied version, vi., algorithm DPWC+, restricts consistency checks to constrats correspondg to edges of a given put graph G. Algorithm DPWC+, as well as its subroute, algorithm DPWC, are presented Algorithms 2 and 3 respectively. For completeness, algorithm PWC is presented Algorithm 4. With respect to algorithm DPWC+, we can prove the followg result: Proposition 2 (cf. [Long et al., 2016]). Given a not trivially consistent QCN N = (V, C) defed over a distributive subclass of relations of a relation algebra, a chordal graph G = (V, E), and a bijection α : V {0, 1,..., V 1} such that (α 1 ( V 1),α 1 ( V 2),...,α 1 (0)) is a perfect elimation orderg of G, algorithm DPWC+ returns (True, G (N )) if G (N ) is not trivially consistent, and (False, N ) 1264

4 Algorithm 2: DPWC+(N, G, α) : A QCN N = (V, C), a graph G = (V, E), and a bijection α : V {0, 1,..., V 1}. 1 beg 2 (decision, N ) DPWC(N, G, α); 3 if decision = False then 4 return (False, N ); 5 for from 1 to V 1 do 6 v α 1 (); 7 foreach v V {v, v} E α(v ) < α(v) do 8 adj {v V {v, v}, {v, v } E 9 α(v ) < α(v)}; 10 C(v, v ) v adj C(v, v ) C(v, v ); 11 C(v, v) (C(v, v )) 1 ; 12 return (True, N ); Algorithm 3: DPWC(N, G, α) : A QCN N = (V, C), a graph G = (V, E), and a bijection α : V {0, 1,..., V 1}. 1 beg 2 for from V 1 to 1 do 3 v α 1 (); 4 adj {v V {v, v} E α(v ) < α(v)}; 5 foreach v, v adj {v, v } E 6 α(v ) < α(v ) do 7 r C(v, v ) (C(v, v) C(v, v )); 8 if r = then return (False, N ); 9 if r C(v, v ) then 10 C(v, v ) r; 11 C(v, v ) r 1 ; 12 return (True, N ); with N N otherwise. As shown [Long et al., 2016], algorithm DPWC+ (at least its origal and more particular version as it appears that work) ehibits outstandg performance comparison with algorithm PWC for enforcg partial -consistency on QCNs defed over a distributive subclass of relations. Due to space constrats, it is not possible to detail the functionality of that algorithm; however, it should suffice to say that it utilies partial -consistency as its core local consistency [Sioutis et al., 2016a], which is enforced by algorithm DPWC, and eploits unique properties of distributive subclasses of relations to achieve partial -consistency a given QCN defed over such a subclass. In summary, partial -consistency is partial -consistency restricted (directionally) to a variable orderg α of the considered QCN. On the other hand, algorithm PWC is both efficient and complete for enforcg partial -consistency on arbitrary QCNs. With respect to algorithm PWC, we recall the followg result: Proposition 3 ([Chmeiss and Condotta, 2011]). Given a not trivially consistent QCN N = (V, C) of a relation alge- Algorithm 4: PWC(N, G, e ) : A QCN N = (V, C), a graph G = (V, E), and optionally a set e such that e E. 1 beg 2 Q (e if e else E); 3 while Q do 4 {v, v } Q.pop(); 5 foreach v V {v, v }, {v, v } E do 6 r C(v, v ) (C(v, v ) C(v, v )); 7 if r = then return (False, N ); 8 if r C(v, v ) then 9 C(v, v ) r; 10 C(v, v) r 1 ; 11 Q Q {{v, v }}; 12 r C(v, v ) (C(v, v) C(v, v )); 13 if r = then return (False, N ); 14 if r C(v, v ) then 15 C(v, v ) r; 16 C(v, v ) r 1 ; 17 Q Q {{v, v }}; 18 return (True, N ); bra, and a graph G = (V, E), algorithm PWC returns (True, G (N )) if G (N ) is not trivially consistent, and (False, N ) with N N otherwise. Havg presented the structure and the core components of algorithm PWC α, we obta the followg lemma to be used for establishg the correctness of our algorithm the sequel: Lemma 1. Let N = (V, C ) and N = (V, C) be two QCNs such that N N. Then, we have that G (N ) G (N ) N. Proof. As N N, by monotonicity of G-consistency we have that G (N ) G (N ). Moreover, sce it also holds that G (N ) N, it follows that G (N ) G (N ) N. The followg result states that algorithm PWC α is complete for enforcg partial -consistency on arbitrary QCNs: Theorem 1. Given a not trivially consistent QCN N = (V, C) of a relation algebra, a distributive subclass of relations A of that relation algebra, a chordal graph G = (V, E), and a bijection α : V {0, 1,..., V 1} such that (α 1 ( V 1),α 1 ( V 2),...,α 1 (0)) is a perfect elimation orderg of G, algorithm PWC α returns (True, G (N )) if G (N ) is not trivially consistent, and (False, N ) with N N otherwise. Proof. Let N = A G (N ). Clearly, N N. As N is defed over a distributed subclass of relations, by Proposition 2 we have that, le 3 of algorithm PWC α, algorithm DPWC+ returns (False, N ) with N N if G (N ) is trivially consistent, and (True, G (N )) otherwise. By monotonicity of G -consistency we have that G (N ) G (N ). Thus, algorithm PWC α le 5 returns (False, N ) only if G (N ) is trivially consistent. Assumg that G (N ) is not 1265

5 trivially consistent and the algorithm contues its eecution, the operation G (N ) N is performed les 7 13 of algorithm PWC α. By Lemma 1 we have that G (N ) G (N ) N. Hence, if G (N ) N defes a trivially consistent QCN, then G (N ) is trivially consistent. Therefore, algorithm PWC α le 9 returns (False, N ) only if G (N ) is trivially consistent. Let M = G (N ) N, and further assume that M is not trivially consistent and the algorithm contues its eecution. M is a not trivially consistent QCN that results by tighteng some of the constrats of G (N ). Note also that M N ; hence, by monotonicity of G -consistency we have that G (M) G (N ). In addition, as we have already established that G (N ) M, by monotonicity of G -consistency we have that G ( G (N )) G (M), and by idempotence of G -consistency it follows that G (N ) G (M). Consequently, we have that G (N ) = G (M). Now, by utilig the cremental functionality of algorithm PWC, we need only feed the algorithm with the QCN M, the graph G, and the set of edges that correspond to the tightened constrats of G (N ) order to obta G (M) (see [Gerevi, 2005, Section 3]). Thus, by Proposition 3 we have that, le 14 of algorithm PWC α, algorithm PWC returns (True, G (M)) if G (M) is not trivially consistent, and (False, M ) with M M otherwise. Sce G (N ) = G (M), we deduce that algorithm PWC α returns (True, G (N )) if G (N ) is not trivially consistent, and (False, N ) with N N otherwise. Regardg time compleity, given a QCN N = (V, C) and a graph G = (V, E) with a maimum verte degree, algorithm PWC α runs O(ma{ 2 V, E B }) time, as the run-times of DPWC+ and PWC are O( 2 V ) [Long et al., 2016] and O( E B ) [Chmeiss and Condotta, 2011] respectively. In terms of the number of triangles t G, and assumg that V + E O(t), algorithm PWC α runs O(t B ) time. However, dependg on the percentage of distributive relations that are the given QCN N and the quality of the abstraction that is obtaed with respect to a given distributive subclass of relations, algorithm DPWC+ (employed le 3 of algorithm PWC α ) may dimish that run-time practice, as it runs O(t) time [Long et al., 2016], which is dependent of the sie of the set of base relations B. We will eplore the aforementioned implication by means of a thorough eperimental evaluation the followg section. 4 Eperimental Evaluation We evaluated the performance of an implementation of algorithm PWC α, agast an implementation of the state-ofthe-art generic algorithm for enforcg partial -consistency PWC, with a varied dataset of arbitrary QCNs of RCC-8. Technical Specifications. The evaluation was carried out on a computer with an Intel Core i7-2820qm processor (which has a frequency of 2.30 GH per CPU core), 8 GB of RAM, and the Trusty Tahr OS (Ubuntu Lu). All algorithms were coded Python and run usg the standard CPython 2.7 terpreter. Only one CPU core was used. Datasets and Measures. We considered random scale-free RCC-8 networks generated by the BA(n, m) model [Barabasi and Albert, 1999], the use of which qualitative constratbased reasong is well motivated [Sioutis et al., 2016b], and random (unstructured) RCC-8 networks generated by the A(n, d, l) model [Ren and Nebel, 2001], which has been traditionally employed over the past decades the literature. Each of the aforementioned models was enriched with a parameter r d that allowed us to specify the percentage of distributive relations we would like to have our RCC-8 networks. In particular, we used the enriched BA(n, m, r d ) model to create random scale-free RCC-8 constrats graphs of order n, with a preferential attachment value m, and with r d % of distributive relations, and the enriched A(n, d, l, r d ) model to create random RCC-8 constrat graphs of order n, with an average verte degree d, with an average sie of relation per constrat l (which defaults to B /2 for model BA(n, m, r d )), and with r d % of distributive relations. We generated a total of 400 RCC-8 networks; more specifically, we considered 10 satisfiable and 10 unsatisfiable RCC-8 networks for each of the models BA(n = , m = 2, r d ) and A(n = 1 000, d = 10.0, l = 4.0, r d ), and for all values of r d rangg from 0% to 90% with a step of 10%. Regardg distributive relations particular, we considered the distributive subclass D8 64 of RCC-8 [Li et al., 2015] as our selection pool and as an put to algorithm PWC α. Satisfiability or otherwise of our networks was guaranteed by the generic qualitative constrat-based reasoner GQR [Gantner et al., 2008; Westphal et al., 2009]. Fally, the maimum cardality search algorithm [Tarjan and Yannakakis, 1984] was used to obta a variable elimation orderg α of a given QCN N for PWC α, and a triangulation G of the constrat graph of N based on α for both PWC α and PWC. We note that, with respect to our evaluation, any perfect elimation ordergs and, hence, any respective chordal graphs would have been adequate, as they would have affected all volved algorithms proportionally and would not have qualitatively distorted the obtaed results. An overview of the use of chordal graphs the QSTR literature is presented [Sioutis et al., 2016c]. Our evaluation volved two measures, which we describe as follows. The first measure considers the number of constrat checks performed by an algorithm for enforcg partial -consistency. Given a QCN N = (V, C) and three variables v i, v k, v j V, a constrat check occurs when we compute the relation r = C(v i, v j ) (C(v i, v k ) C(v k, v j )) and check if r C(v i, v j ), so that we can propagate its constraedness if that condition is satisfied. The second measure concerns the CPU time and is strongly correlated with the first one, as the run-time of any proper implementation of an algorithm for enforcg partial -consistency should rely maly on the number of constrat checks performed. Results. The eperimental results for random RCC-8 networks of models BA(n, m, r d ) and A(n, d, l, r d ) are presented Tables 1a and 1b respectively, where a fraction y denotes that an approach required seconds of CPU time and performed y constrat checks on average per dataset of networks durg its operation. Regardg satisfiable networks, despite a rather sluggish performance of PWC α when 0% of distributive relations were considered, terms of beg around 20% slower on average than PWC, it caught up fast, at around 20% of distributive relations, and constantly outperformed PWC from that pot on, beg nearly 80% faster on average than PWC when 90% of distributive relations were 1266

6 Table 1: Evaluation of the performance of algorithms PWC and PWC α (a) Evaluation with random scale-free RCC-8 networks of model BA(n = , m = 2, r d ) m average µ ma standard deviation σ SAT UNSAT SAT UNSAT SAT UNSAT SAT UNSAT r d PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α 0% 9.6s s s s s 6 2.6s 1 5.8s 3 1.8s 1.8s 10% s s s s s 50k 2 7.2s 38k 20% 13.4s 28k 13.4s s 48k 16.2s 5 10k s s 9 9.8s s 1 2.8s 1 30% 17.6s 38k 15.7s 48k s k 2.6s 34.6s s 3 7.5s 4 4.7s 30k 2 1.8s % 18.4s s s s s s 5 4.1s 2 3.6s 2 4.4s 1 3.6s 50% 15.9s s s s s s 28k 1 60% 22.4s 70k 14.4s s s s s s 8 58k 1.5s % 25.6s s s s s s 4 8.7s 7 4.8s % 26.5s s s s s 128k 2 28k 1.5s % s s s 8 (b) Evaluation with random (unstructured) RCC-8 networks of model A(n = 1 000, d = 10.0, l = 4.0, r d ) s 11 10k 5.7s 60k 700 m average µ ma standard deviation σ SAT UNSAT SAT UNSAT SAT UNSAT SAT UNSAT r d PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α PWC PWC α 0% 3.2s 4.0s s 3.4s 1 1 8k 1 1 8k % 3.8s 1 4.0s s s % 3.9s 4.1s s 2 4.4s 2 4.6s % 4.7s 2 4.3s s 2 5.4s 3 4.9s 3 1.7s 1 40% 4.9s 30k s 3 4.6s 3 6.1s 4 38k 1 50% 5.9s 4 4.3s s 5 4.7s 3 7.4s 6 5.6s % 6.7s s s 7 5.3s 5 10k 70% 7.6s 6 3.8s s s 8 1.5s k 80% 8.0s 78k 3.4s k s s 80k % 8.3s 7 2.3s k 14.8s 250k 110k 1 1.9s considered. The same trend held true, more or less, for unsatisfiable networks, with the addition that the mimum time for refutg unsatisfiable networks for PWC α was often significantly less than that of PWC, which is eplaed by the fact that PWC α may generally detect certa consistencies at an earlier stage (see les 5 and 9 Algorithm 1). 5 Conclusion and Future Work Partial -consistency is an essential local consistency for tacklg fundamental reasong problems associated with qualitative constrat networks (QCNs), such as the problems of satisfiability checkg, mimal labelg, and redundancy. We proposed a new algorithm for efficiently applyg partial -consistency on arbitrary QCNs, that eploits the notion of abstraction for QCNs, utilies certa properties of partial -consistency, and adapts the functionalities of some state-of-the-art algorithms to its design. The evaluation that we conducted with QCNs of the Region Connection Calculus showed the usefulness and efficiency of our method. For future work, we would like to obta a variation of our algorithm for efficiently enforcg sgleton partial -consistency (i.e., partial -consistency [Amanedde et al., 2013]) on arbitrary QCNs, which is an important local consistency for solvg the mimal labelg problem, particular, of QCNs. 1267

7 References [Allen, 1983] James F. Allen. Matag Knowledge about Temporal Intervals. Commun. ACM, 26: , [Amanedde et al., 2013] Nouhad Amanedde, Jean- François Condotta, and Michael Sioutis. Efficient Approach to Solve the Mimal Labelg Problem of Temporal and Spatial Qualitative Constrats. In IJCAI, [Balbiani et al., 2002] Philippe Balbiani, Jean-François Condotta, and Luis Fariñas del Cerro. Tractability Results the Block Algebra. J. Log. Comput., 12: , [Barabasi and Albert, 1999] A.-L. Barabasi and R. Albert. Emergence of scalg random networks. Science, 286: , [Bhatt et al., 2011] Mehul Bhatt, Hans W. Guesgen, Stefan Wölfl, and Shyamanta Moni Haarika. Qualitative Spatial and Temporal Reasong: Emergg Applications, Trends, and Directions. Spatial Cognition & Computation, 11:1 14, [Chmeiss and Condotta, 2011] Assef Chmeiss and Jean- Francois Condotta. Consistency of Triangulated Temporal Qualitative Constrat Networks. In ICTAI, [Cousot and Cousot, 1992] Patrick Cousot and Radhia Cousot. Abstract Interpretation Frameworks. J. Log. Comput., 2: , [Dylla et al., 2013] Frank Dylla, Till Mossakowski, Thomas Schneider, and Diedrich Wolter. Algebraic Properties of Qualitative Spatio-Temporal Calculi. In COSIT, [Frank, 1991] Andrew U. Frank. Qualitative Spatial Reasong with Cardal Directions. In ÖGAI, [Gantner et al., 2008] Zeno Gantner, Matthias Westphal, and Stefan Wölfl. GQR-A Fast Reasoner for Bary Qualitative Constrat Calculi. In AAAI Workshop on Spatial and Temporal Reasong, [Gerevi, 2005] Alfonso Gerevi. Incremental qualitative temporal reasong: Algorithms for the pot algebra and the ord-horn class. Artif. Intell., 166(1-2):37 80, [Golumbic and Shamir, 1993] Mart Charles Golumbic and Ron Shamir. Compleity and Algorithms for Reasong about Time: A Graph-Theoretic Approach. J. ACM, 40: , [Huang, 2012] Jbo Huang. Compactness and its implications for qualitative spatial and temporal reasong. In KR, [Li et al., 2015] Sanjiang Li, Zhiguo Long, Weimg Liu, Matt Duckham, and Alan Both. On redundant topological constrats. Artif. Intell., 225:51 76, [Ligoat and Ren, 2004] Gérard Ligoat and Jochen Ren. What Is a Qualitative Calculus? A General Framework. In PRICAI, [Ligoat, 1998] Gerard Ligoat. Reasong about cardal directions. J. Vis. Lang. Comput., 9:23 44, [Long and Li, 2015] Zhiguo Long and Sanjiang Li. On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi. In COSIT, [Long et al., 2016] Zhiguo Long, Michael Sioutis, and Sanjiang Li. Efficient Path Consistency Algorithm for Large Qualitative Constrat Networks. In IJCAI, [Nebel, 1997] Bernhard Nebel. Solvg Hard Qualitative Temporal Reasong Problems: Evaluatg the Efficiency of Usg the ORD-Horn Class. Constrats, 1: , [Randell et al., 1992] David A. Randell, Zhan Cui, and Anthony Cohn. A Spatial Logic Based on Regions & Connection. In KR, [Ren and Ligoat, 2005] Jochen Ren and Gérard Ligoat. Weak Composition for Qualitative Spatial and Temporal Reasong. In CP, [Ren and Nebel, 2001] Jochen Ren and Bernhard Nebel. Efficient Methods for Qualitative Spatial Reasong. J. Artif. Intell. Res., 15: , [Ren and Nebel, 2007] Jochen Ren and Bernhard Nebel. Qualitative Spatial Reasong Usg Constrat Calculi. In Handbook of Spatial Logics, pages [Ren, 2002] Jochen Ren. A Canonical Model of the Region Connection Calculus. J. Appl. Non-Classical Logics, 12: , [Sioutis et al., 2015] Michael Sioutis, Sanjiang Li, and Jean- François Condotta. Efficiently Characterig Non- Redundant Constrats Large Real World Qualitative Spatial Networks. In IJCAI, [Sioutis et al., 2016a] M. Sioutis, Z. Long, and S. Li. Efficiently Reasong about Qualitative Constrats through Variable Elimation. In SETN, [Sioutis et al., 2016b] Michael Sioutis, Jean-François Condotta, and Manolis Koubarakis. An Efficient Approach for Tacklg Large Real World Qualitative Spatial Networks. Int. J. Artif. Intell. Tools, 25:1 33, [Sioutis et al., 2016c] Michael Sioutis, Yakoub Salhi, and Jean-François Condotta. Studyg the use and effect of graph decomposition qualitative spatial and temporal reasong. Knowl. Eng. Rev., 32:e4, [Tarjan and Yannakakis, 1984] R. E. Tarjan and M. Yannakakis. Simple Lear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs. SIAM J. Comput., 13: , [Tarski, 1941] Alfred Tarski. On the calculus of relations. J. Symb. Log., 6:73 89, [Vila and Kaut, 1986] Marc B. Vila and Henry A. Kaut. Constrat Propagation Algorithms for Temporal Reasong. In AAAI, [Westphal et al., 2009] Matthias Westphal, Stefan Wölfl, and Zeno Gantner. GQR: A Fast Solver for Bary Qualitative Constrat Networks. In AAAI Sprg Symposium on Benchmarkg of QSTR Systems,