- Published 2014

ion Component. A is a finite set of abstraction statements of the form R → (c1, . . . , cn)[cscore].sql, (21.4) where sql is an SQL statement returning n-ary tuples 〈c1, . . . , cn〉 (n ≤ 2) with score determined by the cscore column. The tuples have to be ranked in decreasing order of score and, as for the fact component, we assume that there cannot be two records 〈c, s1〉 and 〈c, s2〉 in the result set of sql with s1 = s2 (if there are, then we remove the one with the lower score). The score cscore may be omitted and in that case the score 1 is assumed for the tuples. We assume that R occurs in O, while all of the relational tables occurring in the SQL statement occur in F . Finally, we assume that there is at most one abstraction statement for each abstract relational symbol R. Query Language. The query language enables the formulation of conjunctive queries with a scoring function to rank the answers. More precisely, a ranking query is of the form q(x)[s] ← ∃y R1(z1)[s1], . . . , Rl(zl)[sl], OrderBy(s = f(s1, . . . , sl, p1(z1), . . . , ph(z ′ h)) (21.5) where (1) q is an n-ary relation, every Ri is a ni-ary relation (1 ≤ ni ≤ 2). Ri(zi) may also be of the form (z ≤ v), (z < v), (z ≥ v), (z > v), (z = v), L at es t A dv an ce s in I nd uc tiv e L og ic P ro gr am m in g D ow nl oa de d fr om w w w .w or ld sc ie nt if ic .c om by D r. U m be rt o St ra cc ia o n 01 /1 4/ 15 . F or p er so na l u se o nl y. October 10, 2014 14:25 Latest Advances in Inductive Logic Programming 9in x 6in b1798-ch21 page 202 202 Latest Advances in Inductive Logic Programming (z = v), where z is a variable, v is a value of the appropriate concrete domain; (2) x are the distinguished variables ; (3) y are existentially quantified variables called the non-distinguished variables. We omit to write ∃y when y is clear from the context; (4) zi, zj are tuples of constants or variables in x or y; (5) s, s1, . . . , sl are distinct variables and different from those in x and y; (6) pj is an nj-ary fuzzy predicate assigning a score pj(cj) ∈ [0, 1] to each nj-ary tuple cj of constants; (7) f is a scoring function f : ([0, 1]) → [0, 1], which combines the scores of the l relations Ri(ci) and the n fuzzy predicates pj(c ′′ j ) into an overall score s to be assigned to q(c). We call q(x)[s] its head, ∃y.R1(z1)[s1], . . . , Rl(zl)[sl] its body and OrderBy(s = f(s1, . . . , sl, p1(z′1), . . . , ph(zh)) the scoring atom. We also allow the scores [s], [s1], . . . , [sl] and the scoring atom to be omitted. In this case we assume the value 1 for si and s instead. The informal meaning of such a query is: if zi is an instance of Ri to degree at least or equal to si, then x is an instance of q to degree at least or equal to s, where s has been determined by the scoring atom. The answer set ansK (q) over K of a query q is the set of tuples 〈t, s〉 such that K |= q(t)[s] with s > 0 (informally, t satisfies the query to nonzero degree s) and the score s is as high as possible, i.e. if 〈t, s〉 ∈ ansK (q) then (i) K |= q(t)[s′] for any s′ > s; and (ii) there cannot be another 〈t, s′〉 ∈ ansK (q) with s > s′. 21.3 ILP for Learning Fuzzy DL Inclusion Axioms In this section we consider a learning problem where: • the target concept H is a DL-Lite atomic concept; • the background theory K is a DL-Lite-like knowledge base 〈F ,O,A〉 of the form described in Section 21.2; • the training set E is a collection of fuzzy DL-Lite-like facts of the form (21.1) and labeled as either positive or negative examples for H . We assume that F ∩ E = ∅; • the target theory H is a set of inclusion axioms of the form B H (21.6) L at es t A dv an ce s in I nd uc tiv e L og ic P ro gr am m in g D ow nl oa de d fr om w w w .w or ld sc ie nt if ic .c om by D r. U m be rt o St ra cc ia o n 01 /1 4/ 15 . F or p er so na l u se o nl y. October 10, 2014 14:25 Latest Advances in Inductive Logic Programming 9in x 6in b1798-ch21 page 203 Can ILP Deal with Incomplete and Vague Structured Knowledge? 203 where H is an atomic concept, B = C1 . . . Cm, and each concept Ci has syntax C −→ A | ∃R.A | ∃R. . (21.7) We now show how we may learn inclusion axioms of the form (21.6). To this aim, we define for C = H IILP |= C(t) iff K ∪ E |= C(t)[s] and s > 0. (21.8) That is, we write IILP |= C(t) if it can be inferred from K and E that t is an instance of concept C to a non-zero degree. Now, in order to account for multiple fuzzy instantiations of fuzzy predicates occurring in the inclusion axioms of interest to us, we propose the following formula for computing the confidence degree of an inclusion axiom: cf(B H) = ∑ t∈P B(t) ⇒ H(t) |D| (21.9) where • P = {t | IILP |= Ci(t) and H(t)[s] ∈ E+}, i.e. P is the set of instances for which the implication covers a positive example; • D = {t | IILP |= Ci(t) and H(t)[s] ∈ E}, i.e. D is the set of instances for which the implication covers an example (either positive or negative); • B(t) ⇒ H(t) denotes the degree to which the implication holds for the instance t; • B(t) = min(s1, . . . , sn), with K ∪ E |= Ci(t)[si]; • H(t) = s with H(t)[s] ∈ E . Clearly, the more positive instances supporting the inclusion axiom, the higher the confidence degree of the axiom. Note that the confidence score can be determined easily by submitting appropriate queries via the query language described in Section 21.2. More precisely, proving the fuzzy entailment in (21.8) for each Ci is equivalent to answering a unique ranking query whose body is the conjunction of the relations Rl resulting from the transformation of Cis into FOL predicates and whose score s is given by the minimum between sls. HotelTable id rank noRooms h1 3 21 h2 5 123 h3 4 95 RoomTable id price roomType hotel r1 60 single h1 r2 90 double h1 r3 80 single h2 r4 120 double h2 r5 70 single h3 r6 90 double h3 Tower id t1 Park id p1 p2 DistanceTable id from to time d1 h1 t1 10 d2 h2 p1 15 d3 h3 p2 5 L at es t A dv an ce s in I nd uc tiv e L og ic P ro gr am m in g D ow nl oa de d fr om w w w .w or ld sc ie nt if ic .c om by D r. U m be rt o St ra cc ia o n 01 /1 4/ 15 . F or p er so na l u se o nl y. October 10, 2014 14:25 Latest Advances in Inductive Logic Programming 9in x 6in b1798-ch21 page 204 204 Latest Advances in Inductive Logic Programming For illustrative purposes we consider the case involving the classification of hotels as good ones. We assume to have a background theory K with a relational database F storing facts such as an ontology O4 encompassing the following inclusion axioms Park Attraction, T ower Attraction, Attraction Site and a set A of abstraction statements such as: Hotel → (h.id). SELECT h.id FROM HotelTable h cheapPrice → (h.id, r.price)[score]. SELECT h.id, r.price, cheap(r.price) AS score FROM HotelTable h, RoomTable r WHERE h.id = r.hotel ORDER BY score closeTo → (from, to)[score]. SELECT d.from, d.to closedistance(d.time) AS score FROM DistanceTable d ORDER BY score where cheap(p) is a function determining how cheap a hotel room is given its price, modelled as e.g. a so-called left-shoulder function (defined in Fig. 21.1). We set cheap(p) = ls(p; 50, 100), while closedistance(d) = ls(d; 5, 25). Assume now that our target concept H is GoodHotel, and that • E+ = {GoodHotel+(h1)[0.6], GoodHotel+(h2)[0.8]}, while E− = {GoodHotel−(h3)[0.4]}; • GoodHotel GoodHotel and GoodHotel− GoodHotel occur in K . As an illustrative example, we compute the confidence degree of r : Hotel ∃cheapPrice. ∃closeTo.Attraction GoodHotel i.e. a good hotel is one having a cheap price and close proximity to an attraction. Now, it can be verified that for K ′ = K ∪ E Fig. 21.1 Left shoulder function ls(x;a, b). 4http://donghee.info/research/SHSS/ObjectiveConceptsOntology(OCO).html. L at es t A dv an ce s in I nd uc tiv e L og ic P ro gr am m in g D ow nl oa de d fr om w w w .w or ld sc ie nt if ic .c om by D r. U m be rt o St ra cc ia o n 01 /1 4/ 15 . F or p er so na l u se o nl y. October 10, 2014 14:25 Latest Advances in Inductive Logic Programming 9in x 6in b1798-ch21 page 205 Can ILP Deal with Incomplete and Vague Structured Knowledge? 205

@inproceedings{Lisi2014CanID,
title={Can ILP Deal with Incomplete and Vague Structured Knowledge?},
author={Francesca A. Lisi and Umberto Straccia},
year={2014}
}