Description logics (DLs) [
The first group consists of automata-based approaches for DLs such as OWL 2
EL [
The second group consists of rewriting-based approaches. Roughly speaking, these
approaches rewrite the ontology and/or the query into another formalism, typically a
union of conjunctive queries or a datalog program; the relevant answers can then be
obtained by evaluating the rewriting over the data. Rewriting-based approaches were
developed for members of the DL-Lite family [
The third group consists of combined approaches, which use a three-step process:
first, they augment the data with certain consequences of the ontology; second, they
evaluate the CQ over the augmented data; and third, they filter the result of the
second phase to eliminate unsound answers. The third step is necessary because, to ensure
termination, the first step is unsound and may introduce facts that do not follow from
the ontology; however, this is done in a way that makes the third step feasible. Such
approaches have been developed for logics in the DL-Lite [
None of the combined approaches proposed thus far, however, handles nominals—
concepts containing precisely one individual. Nominals are included in OWL 2 EL, and
they are often used to state that all instances of a class have a certain property value,
such as ‘the sex of all men is male’, or ‘each German city is located in Germany’. In this
paper we present a combined approach for E LHO?r —the DL that covers all features of
OWL 2 EL apart from transitive roles and complex role inclusions. To the best of our
knowledge, this is the first combined approach that handles nominals. Our extension is
nontrivial because nominals require equality reasoning, which increases the complexity
of the first and the third step of the algorithm. In particular, nominals may introduce
recursive dependencies in the filtering conditions used in the third phase; this is in
contrast to the known combined approach for E L [
Following Kro¨tzsch, Rudolph, and Hitzler [
We have implemented a prototypical system using our algorithm, and we carried out a preliminary empirical evaluation of (i) the blowup in the number of facts introduced by the datalog program, and (ii) the number of unsound answers obtained in the second phase. Our experiments show both of these numbers to be manageable in typical cases, suggesting that our algorithm provides a practical basis for answering CQs in an expressive fragment of OWL 2 EL.
The proofs of our technical results are provided in the technical report [
Logic Programming. We use the standard notions of variables, constants, function
symbols, terms, atoms, formulas, and sentences [
In this paper, we interpret a logic program in a model that can be constructed bottom-up. The Herbrand universe of is the set of all terms built from the constants and the function symbols occurring in . Given an arbitrary set of facts B, let (B) be the smallest superset of B such that, for each clause ' ! 2 and each substitution mapping the variables occurring in the clause to the Herbrand universe of , if (') B, then ( ) (B). Let I0 be the set of all facts occurring in ; for each i 2 N, let Ii+1 = (Ii); and let I = Si2N Ii. Then I is the minimal Herbrand model of , and it is well known that I satisfies 8~x:C for each Horn clause C 2 and ~x the vector of all variables occurring in C.
In this paper we allow a logic program to contain the equality predicate . In
first-order logic, is usually interpreted as the identity over the interpretation domain;
however, can also be explicitly axiomatised [
x1 x1
x2 ^ x2 R(~x) ^ xi
! x x2 ! x2 x3 ! x1 x x1 x3 x0i ! R(x1; : : : ; x0i; : : : ; xn) xi x0i ! f (: : : ; xi; : : :) f (: : : ; x0i; : : :) (1) (2) (3) (4) (5) The minimal Herbrand model of a logic program that contains is the minimal Herbrand model of [ .
Conjunctive Queries. A conjunctive query (CQ) is a formula q = 9~y: (~x; ~y) with a conjunction of function-free atoms over variables ~x [ ~y. Variables ~x are the answer variables of q. Let NT (q) be the set of terms occurring in q.
For a substitution such that rng( ) contains only constants, let (q) = 9~z: ( ), where ~z is obtained from ~y by removing each variable y 2 ~y for which (y) is defined. Note that, according to this definition, non-free variables can also be replaced; for example, given q = 9y1; y2:R(y1; y2) and = fy2 7! ag, we have (q) = 9y1:R(y1; a).
Let be a logic program, let I be the minimal Herbrand model of , and let q = 9~y: (~x; ~y) be a CQ that uses only the predicates occurring in . A substitution is a candidate answer for q in if dom( ) = ~x and rng( ) contains only constants; furthermore, such a is a certain answer to q over , written j= (q), if a substitution exists such that dom( ) = ~x [ ~y, = j~x, and (q) I.
Description Logic. DL E LHO?r is defined w.r.t. a signature consisting of
mutually disjoint and countably infinite sets NC , NR, and NI of atomic concepts (i.e., unary
predicates), roles (i.e., binary predicates), and individuals (i.e., constants), respectively.
Furthermore, for each individual a 2 NI , expression fag denotes a nominal—that is,
a concept containing precisely the individual a. Also, we assume that > and ? are
unary predicates (without any predefined meaning) not occurring in NC . We consider
only normalised knowledge bases, as it is well known [
We interpret K as a logic program. Table 1 shows how to translate a TBox T into a logic program (T ). Moreover, let >(T ) be the set of the following clauses instantiated for each atomic concept A and each role R occurring in T .
A(x) ! >(x)
R(x; y) ! >(x)
R(x; y) ! >(y)
A KB K = hT ; Ai is translated into the logic program (K) = (T ) [ >(T ) [ A.
Then, K is unsatisfiable if (K) j= 9y:?(y). Furthermore, given a conjunctive query q
and a candidate answer for q, we write K j= (q) iff K is unsatisfiable or (K) j= (q).
Given a candidate answer for q, deciding whether (K) j= (q) holds is NP-complete
in combined complexity, and PTIME-complete in data complexity [
Datalog Rewriting of E LHO?r TBoxes For the rest of this section, we fix an arbitrary E LHO?r knowledge base K = hT ; Ai. We next show how to transform K into a datalog program D(K) that can be used to check the satisfiability of K. In the following section, we then show how to use D(K) to answer conjunctive queries.
Due to axioms of type 6 (cf. Table 1), (K) may contain function symbols and
is generally not a datalog program; thus, the evaluation of (K) may not terminate.
To ensure termination, we eliminate function symbols from (K) using the technique
by Kro¨tzsch, Rudolph, and Hitzler [
Definition 1. Datalog program D(T ) is obtained by translating each axiom of type other than 6 in the TBox T of K into a clause as shown in Table 1, and by translating each axiom A1 v 9R:A in T into clauses A1(x) ! R(x; oR;A) and A1(x) ! A(oR;A). Furthermore, the translation of K into datalog is given by D(K) = D(T ) [ >(T ) [ A.
J ai taught krtautaught g t h john ≈ fT,J(kr) advisor I ai taught oT,Padvisor
taught kr taught
advisor advisor oA,P john ≈ oT,J fT,P(ai) advisor fA,P (fT,P(ai)) advisor fT,P(kr) advisor fA,P (fT,P(kr)) advisor fA,P (≈john) advisor fA,P (fT,J(kr)) Then, D(T ) contains the following clauses: KRC v Course range(taught ; Prof ) KRC (x) ! taught (x; oT;J ) Prof (x) ! advisor (x; oA;P ) KRC (kr) KRC (x) ! JProf (oT;J ) Prof (x) ! Prof (oA;P ) KRC (x) ! Course(x) Course(x) ! taught (x; oT;P ) JProf (x) ! x john taught (x; y) ! Prof (y) Course(x) ! Prof (oT;P ) taught (x; y) ! Course(x)
The following result readily follows from the definition of (K) and D(K). Proposition 1. Program D(K) can be computed in time linear in the size of K.
Next, we prove that the datalog program D(K) can be used to decide the satisfiability of K. To this end, we define a function that maps each term w in the Herbrand universe of (K) to the Herbrand universe of D(K) as follows: (w) = (w oR;A if w 2 NI ; if w is of the form w = fR;A(w0): Let I and J be the minimal Herbrand models of (K) and D(K), respectively. Mapping establishes a tight relationship between I and J as illustrated in the following example. Example 2. Let A = fCourse(ai )g, let T be as in Example 1, and let K = hT ; Ai. Figure 1 shows a graphical representation of the minimal Herbrand models I and J of (K) and D(K), respectively. The grey dotted lines show how relates the terms in I to the terms in J. For the sake of clarity, Figure 1 does not show the reflexivity of .
Mapping is a homomorphism from I to J.
Lemma 1. Let I and J be the minimal Herbrand models of (K) and D(K), respectively. Mapping satisfies the following three properties for all terms w0 and w, each B 2 NC [ f>; ?g, and each R 2 NR. 1. B(w) 2 I implies B( (w)) 2 J. 2. R(w0; w) 2 I implies R( (w0); (w)) 2 J. 3. w0 w 2 I implies (w0) (w) 2 J.
For a similar result in the other direction, we need a couple of definitions. Let H be an arbitrary Herbrand model. Then, dom(H) is the set containing each term w that occurs in H in at least one fact with a predicate in NC [ f>; ?g [ NR; note that, by this definition, we have w 62 dom(H) whenever w occurs in H only in assertions involving the predicate. Furthermore, auxH is the set of all terms w 2 dom(H) such that, for each term w0 with w w0 2 H, we have w0 62 NI . We say that the terms in auxH are ‘true’ auxiliary terms—that is, they are not equal to an individual in NI . In Figure 1, bold terms are ‘true’ auxiliary terms in I and J.
Lemma 2. Let I and J be the minimal Herbrand models of (K) and D(K), respectively. Mapping satisfies the following five properties for all terms w1 and w2 in dom(I), each B 2 NC [ f>; ?g, and each R 2 NR. 1. B( (w1)) 2 J implies that B(w1) 2 I. 2. R( (w1); (w2)) 2 J and (w2) 62 auxJ imply that R(w1; w2) 2 I. 3. R( (w1); (w2)) 2 J and (w2) 2 auxJ imply that (w2) is of the form oP;A, that
R(w1; fP;A(w1)) 2 I, and that a term w10 exists such that R(w10; w2) 2 I. 4. (w1) (w2) 2 J and (w2) 62 auxJ imply that w1 w2 2 I. 5. For each term u occurring in J, term w 2 dom(I) exists such that (w) = u.
Lemmas 1 and 2 allow us to decide the satisfiability of K by answering a simple query over D(K), as shown in Proposition 2. The complexity claim is due to the fact that each clause in D(K) contains a bounded number of variables [17].
Proposition 2. For K an arbitrary E LHO?r knowledge base, (K) j= 9y:?(y) if and only if D(K) j= 9y:?(y). Furthermore, the satisfiability of K can be checked in time polynomial in the size of K. 4
In this section, we fix a satisfiable E LHO?r knowledge base K = hT ; Ai and a conjunctive query q = 9~y: (~x; ~y). Furthermore, we fix I and J to be the minimal Herbrand models of (K) and D(K), respectively.
While D(K) can be used to decide the satisfiability of K, the following example shows that D(K) cannot be used directly to compute the answers to q.
Example 3. Let K be as in Example 2, and let q1, q2, and q3 be the following CQs: q1 = taught (x1; x2) q3 = 9y: advisor (y; y) q2 = 9y1; y2; y3: taught (x1; y1) ^ taught (x2; y2) ^ advisor (y1; y3) ^ advisor (y2; y3) Furthermore, let i be the following substitutions: 1 = fx1 7! kr ; x2 7! oT;P g 2 = fx1 7! kr ; x2 7! ai ; y1 7! oT;P ; y2 7! oT;P ; y3 7! oA;P g 3 = fy 7! oA;P g Finally, let each i be the projection of i to the answer variables of qi. Using Figure 1, one can readily check that D(K) j= i(qi), but (K) 6j= i(qi), for each 1 i 3.
This can be explained by observing that J is a homomorphic image of I. Now homomorphisms preserve CQ answers (i.e., (K) j= (q) implies D(K) j= (q)), but they can also introduce unsound answers (i.e., D(K) j= (q) does not necessarily imply (K) j= (q)). This gives rise to the following notion of spurious answers. Definition 2. A substitution with dom( ) = ~x [ ~y and D(K) j= (q) is a spurious answer to q if j~x is not a certain answer to q over (K).
Based on these observations, we answer q over K in two steps: first, we evaluate q over D(K) and thus obtain an overestimation of the certain answers to q over (K); second, for each substitution obtained in the first step, we eliminate spurious answers using a special function isSpur. We next formally introduce this function. We first present all relevant definitions, after which we discuss the intuitions. As we shall see, each query in Example 3 illustrates a distinct source of spuriousness that our function needs to deal with.
Definition 3. Let be a substitution s.t. dom( ) = ~x [ ~y and D(K) j= (q). Relation
NT (q) NT (q) for q, , and D(K) is the smallest reflexive, symmetric, and transitive relation closed under the fork rule, where auxD(K) is the set containing each individual u from D(K) for which no individual c 2 NI exists such that D(K) j= u c. (fork) s0 s t0 t
R(s; s0) and P (t; t0) occur in q, and (s0) 2 auxD(K)
Please note that the definition auxD(K) is actually a reformulation of the definition of auxJ , but based on the consequences of D(K) rather than the facts in J.
Relation is reflexive, symmetric, and transitive, so it is an equivalence relation, which allows us to normalise each term t 2 NT (q) to a representative of its equivalence class using the mapping defined below. We then construct a graph Gaux that checks whether substitution matches ‘true’ auxiliary individuals in a way that cannot be converted to a match over ‘true’ auxiliary terms in I.
Definition 4. Let and be as specified in Definition 3. Function : NT (q) 7! NT (q) maps each term t 2 NT (q) to an arbitrary, but fixed representative (t) of the equivalence class of that contains t. Furthermore, the directed graph Gaux = hVaux; Eauxi is defined as follows.
– Set Vaux contains a vertex (t) for each term t 2 NT (q) such that (t) 2 auxD(K). – Set Eaux contains an edge h (s); (t)i for each atom of the form R(s; t) in q such that f (s); (t)g Vaux.
Query q is aux-cyclic w.r.t. acyclic w.r.t. and D(K).
and D(K) if Gaux contains a cycle; otherwise, q is aux
We are now ready to define our function that checks whether a substitution spurious answer. is a Definition 5. Let and be as specified in Definition 3. Function isSpur(q; D(K); ) returns t if and only if at least one of the following conditions hold. (a) Variable x 2 ~x exists such that (x) 62 NI . (b) Terms s and t occurring in q exist such that s t and D(K) 6j= (s) (t). (c) Query q is aux-cyclic w.r.t. and D(K).
We next discuss the intuition behind our definitions. We ground our discussion in minimal Herbrand models I and J, but our technique does not depend on such models: all conditions are stated as entailments that can be checked using an arbitrary sound and complete technique. Since K is an E LHO?r knowledge base, the shape of model I resembles a forest: roughly speaking, the role assertions in I that involve at least one functional term are of the form R(w1; fR;A(w1)) or R(w1; a) for a 2 NI ; thus, I can be viewed as a family of directed trees whose roots are the individuals in NI and whose edges point from parents to children or to the individuals in NI . This is illustrated in Figure 1, whose lower part shows the the forest-model of the knowledge base from Example 3. Note that assertions of the form R(w1; a) are introduced via equality reasoning and may result in links from functional terms to individuals.
Now let be a substitution such that D(K) j= (q), and let = j~x. If is not a spurious answer, it should be possible to convert into a substitution such that = j~x and (q) I. Using the queries from Example 3, we next identify three reasons why this may not be possible.
First, may map an answer variable of q to an auxiliary individual, so by the definition cannot be a certain answer to q; condition (a) of Definition 5 identifies such cases. Query q1 and substitution 1 from Example 3 illustrate such a situation: 2(x2) = oT ;P and oT ;P is a ‘true’ auxiliary individual, so 1 is not a certain answer to q1.
The remaining two problems arise because model J is not forest-shaped, so might map q into J in a way that cannot be converted into a substitution that maps q into I.
The second problem is best explained using substitution 2 and query q2 from Example 3. Query q2 contains a ‘fork’ advisor (y1; y3) ^ advisor (y2; y3). Now, substitution 2 maps y3 to ‘true’ auxiliary individual oA;P which represents ‘true’ auxiliary terms fA;P (fT;P (ai )), fA;P (fT;P (kr )), and so on. Since I is forest-shaped, a match 2 for q in I obtained from 2 would need to map y3 to one of these terms; let us assume that 2 (y3) is mapped to fA;P (fT;P (ai )). Since I is forest-shaped and fA;P (fT;P (ai )) is a ‘true’ auxiliary term, this means that both y1 and y2 must be mapped to the same term (in both J and I). This is captured by the (fork) rule: since is reflexive, the rule derives y1 y2, and condition (b) of Definition 5 checks whether 2 maps y1 and y2 in a way that satisfies this constraint. Note that, due to role hierarchies, the rule needs to be applied to atoms R(s; s0) and P (t; t0) with R 6= P . Moreover, such constraints must be propagated further up the query. In our example, due to y1 y2 and 2(y1) is a ‘true’ auxiliary individual, atoms taught (x1; y1) ^ taught (x2; y2) in q2 also constitute a ‘fork’, so the rule derives x1 x2; this allows condition (b) of Definition 5 to correctly identify 2 as spurious.
The third problem is best explained using substitution 3 and query q3 from Example 3. Model J contains a ‘loop’ on individual oA;P , which allows 3 to map q3 into J. In contrast, model I is forest-shaped, and so the ‘true’ auxiliary terms that correspond to oA;P do not form loops. Condition (c) of Definition 5 detects such situations using the graph Gaux. The vertices of Gaux correspond to the terms of q that are matched to ‘true’ auxiliary individuals (mapping simply ensures that equal terms are represented as one vertex), and edges of Gaux correspond to the role atoms in q. Hence, if Gaux is cyclic, then the substitution obtained from would need to match the query q over a cycle of ‘true’ auxiliary terms, which is impossible since I is forest-shaped.
Unlike the known combined approaches, our approach does not extend q with conditions that detect spurious answers. Due to nominals, the relevant equality constraints have a recursive nature: they depend on both the substitution and on the previously derived constraints. Given that first-order queries do not allow for expressing recursive properties, extending q with these constraints seems to necessarily result in an exponential blow-up in the size of the query. Consequently, filtering in our approach is realised as postprocessing. The following theorem proves the correctness of our approach. Theorem 1. Let K = hT ; Ai be a satisfiable E LHO?r KB, let q = 9~y: (~x; ~y) be a CQ, and let : ~x 7! NI be a candidate answer for q. Then, (K) j= (q) iff a substitution exists such that dom( ) = ~x [ ~y, j~x = , D(K) j= (q), and isSpur(q; D(K); ) = f.
Furthermore, isSpur(q; D(K); ) can be evaluated in polynomial time, so the main source of complexity in our approach is in deciding whether D(K) j= (q) holds. This gives rise to the following result.
Theorem 2. Deciding whether K j= (q) can be implemented in nondeterministic polynomial time w.r.t. the size of K and q, and in polynomial time w.r.t. the size of A. 5
To gain insight into the practical applicability of our approach, we implemented our technique in a prototypical system. The system uses HermiT, a widely used ontology reasoner, as a datalog engine in order to materialise the consequences of D(K) and evaluate q. The system has been implemented in Java, and we ran our experiments on a MacBook Pro with 4GB of RAM and an Intel Core 2 Duo 2.4 Ghz processor. We used two ontologies in our evaluation, details of which are given below. The ontologies, the queries, and the prototype are available at http://www.cs.ox.ac.uk/isg/tools/KARMA/.
The LSTW benchmark [18] consists of an OWL 2 QL version of the LUBM ontology [19], queries q1l; : : : ; q1l1, and a data generator. The LSTW ontology extends the standard LUBM ontology with several axioms of type 6 (see Table 1). To obtain an E LHO?r ontology, we removed inverse roles and datatypes, added 11 axioms using 9 freshly introduced nominals, and added one axiom of type 4 (see Table 1). These additional axioms resemble the ones in Example 1, and they were designed to test equality reasoning. The resulting signature consists of 132 concepts, 32 roles, and 9 nominals, and the ontology contains 180 axioms. From the 11 LSTW queries, we did not consider queries q4l, q6l, q7l, and q1l1 because their result sets were empty: q4l relies on existential quantification over inverse roles, and the other three are empty already w.r.t. the original LSTW ontology. Query q2l is similar to query q2 from Example 3, and it was designed to produce only spurious answers and thus stress the system. We generated data sets with 5, 10 and 20 universities. For each data set, we denote with L-i the knowledge base consisting of our E LHO?r ontology and the ABox for i universities (see Table 2).
SEMINTEC is an ontology about financial services developed within the SEMINTEC project at the University of Poznan. To obtain an E LHO?r ontology, we removed inverse roles, role functionality axioms, and universal restrictions, added nine axioms of type 6 (see Table 1), and added six axioms using 4 freshly introduced nominals. The resulting ontology signature consists of 60 concepts, 16 roles, and 4 nominals, and the ontology contains 173 axioms. Queries qs–q5s are tree-shaped queries used in 1 the SEMINTEC project, and we developed queries qs–q9s ourselves. Query q6s resembles 6 query q2l from LSTW, and queries q8s and q9s were designed to retrieve a large number of answers containing auxiliary individuals, thus stressing condition (a) of Definition 5. Finally, the SEMINTEC ontology comes with a data set consisting of approximately 65,000 facts concerning 18,000 individuals (see row SEM in Table 2).
The practicality of our approach, we believe, is determined mainly by the following two factors. First, the number of facts involving auxiliary individuals introduced during the materialisation step should not be ‘too large’. Table 2 shows the materialisation results: the first column shows the number of individuals before and after materialisation and the percentage of ‘true’ auxiliary individuals, the second column shows the number of unary facts before and after materialisation and the percentage of facts involving a ‘true’ auxiliary individual, and the third column does the same for binary facts. As one can see, for each data set, the materialisation introduces few ‘true’ auxiliary individuals, and the number of facts at most doubles. The number of unary facts involving a ‘true’ auxiliary individual does not change with the size of the input data set, whereas the number of such binary facts increases by a constant factor. This is because, in clauses of type 6, atoms A(oR;A) do not contain a variable, whereas atoms R(x; oR;A) do.
Second, evaluating q over D(K) should not produce too many spurious answers. Table 3 shows the total number of answers for each query—that is, the number of answers obtained by evaluating the query over D(K); moreover, the table shows what percentage of these answers are spurious. Queries q2l, q1l0, q6s, and q8s retrieve a significant percentage of spurious answers. However, only query q2l has proven to be challenging for our system due to the large number of retrieved answers, with an evaluation time of about 40 minutes over the largest knowledge base (L-20). Surprisingly, q1l also performed rather poorly despite a low number of spurious answers, with an evaluation time of about 20 minutes for L-20. All other queries were evaluated in at most a few seconds, thus suggesting that queries q1l and q2l are problematical mainly because HermiT does not implement query optimisation algorithms typically used in relational databases. 6
We presented the first combined technique for answering conjunctive queries over DL ontologies that include nominals. A preliminary evaluation suggests the following. First, the number of materialised facts over ‘true’ anonymous individuals increases by a constant factor with the size of the data. Second, query evaluation results have shown that, while some cases may be challenging, in most cases the percentage of answers that are spurious is manageable. Hence, our technique provides a practical CQ answering algorithm for a large fragment of OWL 2 EL.
We anticipate several directions for our future work. First, we would like to investigate the use of top-down query evaluation techniques, such as magic sets [20] or SLG resolution [21]. Second, tighter integration of the detection of spurious answers with the query evaluation algorithms should make it possible to eagerly detect spurious answers (i.e., before the query is fully evaluated). Lutz et al. [18] already implemented a filtering condition as a user-defined function in a database, but it is unclear to what extent such an implementation can be used to optimise query evaluation. Finally, we would like to extend our approach to all of OWL 2 EL.
This work was supported by the Royal Society; Alcatel-Lucent; the EU FP7 project OPTIQUE; and the EPSRC projects ExODA, MASI3, and QueRe. 17. Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and expressive power of logic programming. ACM Computing Surveys 33(3) (2001) 374–425 18. Lutz, C., Seylan, I., Toman, D., Wolter, F.: The Combined Approach to OBDA: Taming Role Hierarchies using Filters. In Fokoue, A., Liebig, T., Goodman, E., Weaver, J., Urbani, J., Mizell, D., eds.: Proceedings of the Joint Workshop on Scalable and High-Performance Semantic Web Systems (SSWS+HPCSW 2012). Volume 943 of CEUR Workshop Proceedings., CEUR-WS.org (2012) 16–31 19. Guo, Y., Pan, Z., Heflin, J.: LUBM: A benchmark for OWL knowledge base systems. Journal of Web Semantics 3(2–3) (2005) 158–182 20. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley (1995) 21. Chen, W., Warren, D.S.: Query evaluation under the well-founded semantics. In: Proceedings of the 12th ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems. PODS ’93, New York, NY, USA, ACM (1993) 168–179