This extended abstract summarizes our previous work on a defeasible extension of Description Logic (DL) for contextual reasoning.1 Here, we considered on the one hand the addition of multiple dimensions of defeasibility, allowing us to express for example that a rule has to be satisfied no matter the geographical context but that the rule can change in the next years. On the other hand, we showed that Answer Set Programming (ASP) especially when enhanced with algebraic measures provide a powerful tool to implement our framework and open up perspectives for the future.
Reasoning with context dependent knowledge is a classical and fundamental theme in AI [
A rich framework among them are Contextualized Knowledge Repositories (CKR) [
D( ⊑ ) branch : ⊓ ⊑ ⊥ ⊑
D( ⊑ ) local : () world_2019 : ⊓ ⊑ ⊥
D( ⊑ ) branch_2019 : ⊓ ⊑ ⊥
D( ⊑ ) local_2019 : () world_2020 : branch_2020 :
D( ⊑ ) D( ⊑ ) local_2020 : branch_2021 : local_2021 :
For modeling and analyzing complex scenarios where global regulations can be refined by
more specific situations, the CKR model was extended in [ 10] to cater for defeasible axioms in
local contexts and knowledge inheritance across hierarchies, based on a coverage contextual
relation [
Example 1. In the left CKR in Figure 1, we model the employees of a company on three diferent contextual levels, the world, a branch and a local site. There are people working in Electronics (), Robotics () or as a Supervisor (). They can work either onsite () or remote (). At the global level, supervisors should (by default) work in electronics. This is overwritten (by default) in the branch, where supervisors should work in robotics. Therefore, the supervisor at the local context satisfies (), () and (), but not ().
This approach, however, is limited to reason only on hierarchies based on this single type of contextual relation. In practice, defeasible inheritance may be necessary under diferent contextual relations. For instance, in our example, we may want to specify that ⊑ is actually defeasible w.r.t. time and D( ⊑ ) actually only holds since 2020.
As the following extension of the example shows, this shortcoming can be approached by introducing multi-relational CKRs.
Example 2 (cont.). Consider the multi-relational CKR given on the right in Figure 1. Here, denotes the coverage relation, and denotes the time relation. Note also that axioms are not generally defeasible anymore, but only with respect to either coverage or time, denoted by D and D, respectively.
Given the adopted model, we can correctly derive that in 2019 at the local context we still have () instead of (). This changes in the years 2020 and 2021, where we have () due to the defeasible axiom D( ⊑ ) at context local_2020. Here, we also model that until the current context changes with respect to time, supervisors need to work remotely using the axiom D( ⊑ ). Thus, we have () instead of () in 2020 and 2021.
A further limitation is that even for a single coverage relation, it is challenging to encode the induced preference relation over CKR interpretations using ASP because the relation may not be a strict partial order. By default, this is as assumed e.g. in the asprin framework [11] and dropping this assumption in asprin leads to an increase in complexity. A specialized implementation for preferential reasoning was introduced [12], which however needs to consider all answer sets of a program to single out a preferred CKR model.
We showed that we can overcome the first limitation by presenting a multi-relational version of the CKR framework. The second limitation was attacked by encoding reasoning with preferences in a recent extension of ASP with algebraic measures.
We made the following contributions: • We generalized single-relational CKRs to multi-relational CKRs (MR-CKR), where axioms are not defeasible in general but merely with regard to individual relations that model coverage along diferent dimensions such as time or location. By a combination of preferences over the distinct individual relations, we obtain an overall preference over the models of a CKR. • We showed how to model multi-relation CKRs in ASP. Specifically, we use to this end ASP with algebraic measures [13], which is a foundation to express many quantitative reasoning problems. Here, weighted logic formulas [14] measure values associated with an interpretation ℐ by performing a computation over a semiring, whose outcome depends on the truth of the propositional variables in ℐ. Such measures can be used for e.g. weighted model counting, probabilistic reasoning and, as in our case, preferential reasoning. • While asprin is a powerful tool for modeling preferences in ASP, it appears to be illsuited for expressing multi-relational CKR. The reason are eval-expressions in CKRs, which propagate predicate extensions from one local context to another. We showed, however, that under a well-behaved use of such expressions (according to a syntactic disconnectedness condition), multi-relational CKRs can be expressed eficiently in asprin. This enables us to use the asprin solver to evaluate preferences for CKRs, which we showcased in a prototype implementation. • Furthermore, ASP with algebraic measures opens up the possibility of reasoning tasks for CKRs beyond asprin’s capability, even in absence of eval-expression. As examples, we consider obtaining preferred CKR models by overall weight queries and epistemic reasoning, which for description logics is specifically needed in aggregate queries [15].
We considered the application of ASP with algebraic measures for expressing preferences of defeasibility in multirelational CKRs. Specifically, we found special cases in which asprin can be used for eficient reasoning and explored advanced reasoning scenarios, where the expressivity and flexibility of algebraic measures ofers an advantage.
We plan to further study the possibilities for epistemic reasoning on DLs enabled by algebraic measures. With respect to contextual reasoning, a possible continuation of this work may consider a refinement of the definitions of preference and knowledge propagation across diferent contextual relations, possibly moving towards non-Horn DLs in contexts [16]. Apart from further theoretical aspects, we plan to consider a motivating real-world application.
This work was partially supported by the European Commission funded projects “Humane AI: Toward AI Systems That Augment and Empower Humans by Understanding Us, our Society and the World Around Us” (grant #820437) and “AI4EU: A European AI on Demand Platform and Ecosystem” (grant #825619), and the Austrian Science Fund (FWF) project W1255-N23. The support is gratefully acknowledged. [10] L. Bozzato, L. Serafini, T. Eiter, Reasoning with justifiable exceptions in contextual hierarchies, in: M. Thielscher, F. Toni, F. Wolter (Eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Sixteenth International Conference, KR 2018, Tempe, Arizona, 30 October - 2 November 2018, AAAI Press, 2018, pp. 329–338. URL: https://aaai.org/ocs/index.php/KR/KR18/paper/view/18032. [11] G. Brewka, J. P. Delgrande, J. Romero, T. Schaub, asprin: Customizing answer set preferences without a headache, in: B. Bonet, S. Koenig (Eds.), Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, January 25-30, 2015, Austin, Texas, USA, AAAI Press, 2015, pp. 1467–1474. URL: http://www.aaai.org/ocs/index.php/AAAI/AAAI15/paper/ view/9535. [12] L. Bozzato, T. Eiter, L. Serafini, Justifiable exceptions in general contextual hierarchies, in: G. Bella, P. Bouquet (Eds.), Modeling and Using Context. CONTEXT 2019, volume 11939 of Lecture Notes in Computer Science, Springer, 2019, pp. 26–39. doi:10.1007/ 978-3-030-34974-5\_3. [13] T. Eiter, R. Kiesel, Weighted LARS for quantitative stream reasoning, in: G. D. Giacomo, A. Catalá, B. Dilkina, M. Milano, S. Barro, A. Bugarín, J. Lang (Eds.), ECAI 2020 - 24th European Conference on Artificial Intelligence, 29 August-8 September 2020, Santiago de Compostela, Spain, August 29 - September 8, 2020 - Including 10th Conference on Prestigious Applications of Artificial Intelligence (PAIS 2020), volume 325 of Frontiers in Artificial Intelligence and Applications , IOS Press, 2020, pp. 729–736. URL: https://doi.org/ 10.3233/FAIA200160. doi:10.3233/FAIA200160. [14] M. Droste, P. Gastin, Weighted automata and weighted logics, in: L. Caires, G. F. Italiano, L. Monteiro, C. Palamidessi, M. Yung (Eds.), Automata, Languages and Programming, 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005, Proceedings, volume 3580 of Lecture Notes in Computer Science, Springer, 2005, pp. 513–525. URL: https://doi.org/10.1007/11523468_42. doi:10.1007/11523468\_42. [15] D. Calvanese, E. Kharlamov, W. Nutt, C. Thorne, Aggregate queries over ontologies, in: Proceedings of the 2nd International Workshop on Ontologies and Information Systems for the Semantic Web (ONISW 2008), Association for Computing Machinery, New York, NY, USA, 2008, pp. 97–104. URL: https://doi.org/10.1145/1458484.1458500. doi:10.1145/ 1458484.1458500. [16] L. Bozzato, T. Eiter, L. Serafini, Reasoning with Justifiable Exceptions in ℰ ℒ⊥ Contextualized Knowledge Repositories, in: Description Logic, Theory Combination, and All That, volume 11560 of LNCS, Springer, 2019, pp. 110–134.