Statistical Reasoning – A Foundation for Semantic Web Reasoning Shashi Kant • Evangelos Mamas Massachusetts Institute of Technology Cambridge, MA 02139 skant@sloan.mit.edu • emamas@sloan.mit.edu Abstract tion, and Robotics etc. The Semantic Web would do well to re-use some of these efforts in building this underlying There has been considerable debate as to the merits framework. and the applicability of probabilistic or statistical reasoning to Semantic Web. Much of this debate seems to have centered on the applicability of sta- 2 Ontologies and Probabilistic Models tistical methods in a supposedly deterministic set- We introduce the notion that Probabilistic Graph Models ting. In this paper, we argue that statistical reason- (PGM) or Bayesian Networks can be viewed as fuzzy On- ing (“reasoning with uncertainty”) need not be a tologies; conversely an Ontology can be viewed as a crisper substitute for traditional Description Logic (DL) / Bayesian Networks. In our proposed architecture, there may First-Order Logic (FOL) reasoning, instead statisti- not be a clear dividing line between them. A good way of cal methods can serve as a complement to logic- visualizing this relation would be to view Ontologies and based reasoning systems in two ways: (i) Offer a Bayesian Networks as ships floating in a sea of statistical meta-reasoning (or audit) mechanism to validate “metadata”. We use this metaphor to describe the notion logical reasoning, and (ii) Act as a “filler” where that the sea of statistical metadata fills-in the gaps between Ontological information either does not exist, or is the islands of Ontologies. Lately there have been some ef- insufficient to reason conclusively. forts to develop Probabilistic Ontologies by annotating OWL or RDF Ontologies with probabilistic information e.g. 1 Introduction BayesOWL [Ding,Peng 2004]. We argue against this ap- proach, and suggest that probabilistic and logic-based rea- Much of the Semantic Web effort has focused on the design soning approaches should be viewed as orthogonal to each and development of Ontologies and related technologies. other. It makes most sense to keep the Ontological informa- This approach presupposes that a critical mass of Ontologies tion separate from the statistical data, along the lines of how will exist that can sufficiently and accurately respond to the WWW operates - in which an HTML page links to a reasoning queries. As Sir Tim Berners-Lee puts it [Berners- “FTP” site or a “mailto” to an email hyperlink and the nec- Lee, 1998]: "The choice of classical logic for the Semantic essary protocols invoked only when clicked. web is not an arbitrary choice among equals. Classical logic is the only way that inference can scale across the Figure 1 illustrates a hierarchical mechanism of aligned On- web." tologies and Bayesian Networks. At the very top are the top- level Ontologies on which there is general agreement and However, a pure logic-based approach looks increasingly acceptance, at the bottom are the fuzzier, grayer-scale implausible given the paucity of Ontologies and the diffi- Bayesian Networks which represent relations between re- culty in constructing and maintaining Ontologies. Just like sources using probabilistic mechanisms. the World Wide Web (WWW) had a ready and mature plat- form to run on i.e. the Internet - which had been in existence BayesianAbstract Ontological Details Fuzzy Crisp for a long time prior to the emergence of the WWW, we feel that the Semantic Web needs an underlying platform, upon which Ontologies can function and interoperate. We argue that this platform should be a web of statistical “metadata” – which expresses semantic relations in prob- abilistic terms. Such systems (e.g. Bayesian Networks, Ontological Abstract Crisp Probabilistic Relational Models) have also been in existence Bayesian Details Fuzzy for a while and are used in various Machine Learning and Query AI applications such as Machine Vision, Speech Recogni- Figure 1: Ontologies vis-à-vis Bayesian Networks We suggest, that probabilistic (or statistical) information be 4 Conclusion encoded using any of the widely accepted Bayesian Inter- change Formats such as XML-BIF [Cozman, 1998], or Mi- “Reasoning with Uncertainty” is probably a misnomer to crosoft Research’s XBN [Microsoft, 1998] or Hugin.net describe the efforts required in this area - a more appropriate [Jensen, 2004] format. We propose that the Ontological phraseology would be “reasoning without certainty”. While model encapsulate what it is designed for - expressing logi- the difference may seem pedantic, the underlying notion is cal relations between resources, and the probabilistic model that “uncertainty” is not a state unto itself, but merely the express the statistical relation between them. We do not see absence of certainty. In a Semantic Web sense, it is a state a need to mix-and-match as they offer very different views where Ontological information is non-existent, incomplete on the same information-set and are perceptually orthogo- or inconclusive. Statistical reasoning could therefore be the nal. bedrock upon which DL/FOL based querying and reasoning can be performed. 3 A Hybrid Reasoning Model This means that the semantic web can operate in areas cur- Reasoning using Ontologies is based on predicate logic and rently out-of-bounds because of a lack of Ontological in- belongs in the classical tradition of monotonic deductive formation. We therefore hypothesize that statistical “meta- reasoning i.e. propositions are either true or false. But this data” could be the building-block of the Semantic Web lead- proposed framework provides a mechanism for handling ing to better and more accurate reasoning mechanisms. fuzzier, incomplete and inaccurate inputs. In this model, reasoning can be performed using a “bottom-up” approach 5 Acknowledgments where a query unanswered by a pure Ontological match is extended further up the hierarchy (Fig 1.) until all required We would like to acknowledge the gracious help and sup- information is found. An adjunct application might be to port of the members of the World Wide Web Consortium validate traditional reasoning with a mathematical confi- (W3C) and faculty of MIT-CSAIL for their help and sup- dence level (meta-reasoning). port. We would especially like to thank Ralph Swick and Sir Tim Berners-Lee for their critique and feedback. Some examples of the reasoning activities possible using this system are: 6 References [Berners-Lee, 1998] Tim Berners-Lee. Axioms of Web Ar- 1. Deductive Reasoning: Deductive reasoning allows a chitecture: n, 1998, Available at: http://www.w3.org/ system to deduce information given a set of (possibly DesignIssues/Rules.html, Accessed on June 20, 2005. incomplete and erroneous) information. For example, it can deduce that the best course to learn “Machine Vi- [Cozman, 1998] Fabio Gagliardi Cozman, The Interchange sion”, “Genomics” and “Political Science” at MIT is Format for Bayesian Networks, 1998, Available at: http://www-2.cs.cmu.edu/afs/cs/user/fgcozman/www/ most probably “6.804J Computational Cognitive Sci- Research/InterchangeFormat/, Accessed on June 23, ence” even though the course does not directly teach 2005. Political Science. It is making a best-guess fit for the requirements [OCW, 2005]. [Microsoft, 1998] Microsoft Research, XML Belief Net- work File Format: Main Page, 1998, Available at: 2. Abductive Reasoning: Abductive reasoning allows a http://research.microsoft.com/dtas/bnformat/ Accessed system to infer the possible causes for a certain effect. on June 23, 2005. For example, the possible courses for learning Artificial [Jensen, 2004] Finn V. Jensen, A Brief Overview of the Intelligence at MIT are 6.803, 6.825 etc. This is the Three Main Paradigms of Expert Systems, Available at: equivalent of diagnostic reasoning in Bayesian Net- http://developer.hugin.com/Getting_Started/Paradigms/, works [OCW, 2005]. Accessed on June 23, 2005. 3. Monotonic reasoning, non-monotonic reasoning and default values: Traditional DL-based Ontologies can [OCW, 2005] MIT Open Courseware, Massachusetts Insti- tute of Technology, Available at: http://ocw.mit.edu, represent information for monotonic reasoning. For ex- Accessed on June 23, 2005. ample, one might declare that Universities in the US have a GPA scale of 4.0, but MIT uses a 5.0 GPA scale [Ding, Peng, 2004] Zhongli Ding and Yun Peng. A Prob- – so the system monotonically cannot reason with that abilistic Extension to Ontology Language OWL, in Pro- information unless it has been explicitly encoded. ceedings of the 37th Hawaii International Conference on System Sciences - 2004 This kind of non-monotonic reasoning is possible with the proposed approach.