1 Higher Order Uncertainty and Evidential Ontologies Justin Brody Department of Mathematics and Computer Science Franklin and Marshall College Lancaster, PA 17604 justin.brody@fandm.edu Abstract—The uncertainties implicit in intelligence gathering assertion. For example, the credibility of a particular source is are not only about the state of the world, but also about the ways not likely to be a static property of that source, but rather in which varying contexts should affect the degree to which a a parameter which changes based on circumstances ( for proposition is believed. We call this latter form of uncertainty higher order uncertainty, and argue that the introduction of a example, if a source is the subject of physical or financial logical operator to K. Laskey’s MEBN specification can allow distress ). While the proposal in [3] allows for a complex for learning about such uncertainty to occur. network of reasoning to support the determination of any one particular credibility distribution, and further allows for a source’s credibility to be indexed by the context in which the I. I NTRODUCTION assertion is made, it does not allow for a general mechanism In [1], Laskey et. al proposed the creation of ontologies which would decrease any source’s credibility if that source of evidence as a way of facilitating intelligence gathering. is under duress. Rather, it relies on the knowledge engineer Their proposal envisioned such ontologies as represented by to make a determination of how credibility should be affected Bayesian belief networks. In particular, the proposal of [1] by duress. Furthermore, it does not allow the network to learn and the similar proposal in [3] envisioned a network in which about the complex ways in which several factors such as various assertions about the world were assigned degrees of age, constitution, family status, etc. might influence the effect belief. The relationships between such beliefs and pertinent of duress on credibility. The immense complexity of human facts were encoded in the structure of the network, and the psychology makes such a determination highly non-trivial, and impact of new facts on the degrees of belief were updated via equally as wanting of computational assistance as many of the Bayesian learning. While a great deal of the relational structure assertions such networks are designed for. 1 pertinent to reasoning in a specific scenario can be captured I will argue in this paper that extending MEBN to allow for in the structure of a pure Bayesian network, an extra level of a new operator will mitigate some of these problems. While expressivity will be represented by an implementation done in passing to a full-blown higher order implementation is fraught MEBN [2], Laskey’s fusion of Bayesian networks with first- with difficulties (lack of deductive completeness, for example), order logic. The introduction of logical formalism allows an an implementation in which a single operator is introduced can ontology to represent and learn more general structure than add the requisite expressivity while preserving fundamental would otherwise be allowed. Laskey has shown that for any meta-logical properties. logical consequence of a MEBN theory, the associated learning algorithm will derive that consequence in the limit. It is worth noting that the expressivity of first-order logic II. H IGHER O RDER U NCERTAINTY is likely to be crucial when working with human intelligence Let us consider a situation in which the human source X sources. While a source may make quantifier-free assertions makes assertions 1-3 above. We would want to represent his (such as “bin Laden is Kandahar”), he or she is also likely assertions, but we would also want to include information to make assertions whose content necessarily uses quantifiers. about the credibility of his assertions, as discussed in [1] and For example, a highly credible source might make the follow- [3]. In particular, we would want to have an assessment of X’s ing assertions: credibility in making his assertion, which would include infor- 1) “bin Laden is somewhere in Afghanistan” mation about his deceptiveness, competence and opportunity 2) “the only luxury cars used by al-Qaeda are in Kandahar, to judge the situation [3]. These are all factors which would Parachinar, or Islamabad” be subject to varying contextual conditions, rather than being 3) “bin Laden was recently seen in an al-Qaeda owned static properties of X. As such, they seem to be best coded luxury car” either as joint properties of the source and the assertion, or To conclude from this information that bin Laden is (probably) else single properties of the assertion (the latter being a form in Kandahar requires the deductive power of first-order logic. One feature which will not automatically be present in 1 One question that is immediately raised in this context is that of determin- such a scheme is the ability to robustly code and learn the ing what the most important factors in determining a source’s credibility are. While this is an important question, its answer is best determined by experts ways in which contextual factors affect the manner in which in the appropriate fields (e.g. psychology for human sources, or engineering evidence determines the appropriate confidence for a particular for mechanical ones) 2 of short-hand for the former, since (tokens of) assertions are of a given assertion, and α, β are parameters as described associated with unique sources). above. One advantage of coding with parameters is that they As a particular case, let us suppose that X makes his are subject to machine learning rather than being completely assertions while under some form of duress. It is clear that determined by the judgement of a knowledge engineer. the duress should have some effect on the credibility of X’s It is notable that many of these predicates (for example assertion, but it is not clear what precisely that effect should Asserts(X, a) and Believes(X, b)) take propositions as pa- be (in fact there are cases in which duress would conceivably rameters. As such they do not necessarily lie within the scope increase the credibility of an assertion, while one would expect of first order logic. 2 One is left with the question of how the credibility to decrease under most circumstances). It will to represent such statements in an ontology of evidence. A apparently depend on a number of factors, including the nature natural solution is provided by the work of Neuhaus and of the duress, various facts about X, and the nature of the Andersen in [4]. In their paper on speech acts and ontolo- statement being made. In fact, it doesn’t take very long to gies, these authors have suggested encoding speech acts by realize that the number of factors and potential ways they introducing an AssertiveSpeechAct predicate along with a can interact quickly defies any easy analysis. This complexity PropositionalContent operator; the role of the latter is to creates a new form of uncertainty. Noting that this is about refer to the content of a given speech act. Introducing such the network structure rather than about what the network an operator into MEBN would allow for universal statements represents, we term this uncertainty higher order. about assertions; the semantics could be defined in such a way One might certainly treat each assertion individually, and [3] that any instantiation of a speech act would come attached suggests the possibility of having a complex network behind with a default network structure, which would be modified as the credibility distribution of any source’s assertion of any learning (for example of the parameters α, β) occurred. particular statement. However this misses an opportunity to discover general truths about the structure of the higher order III. N ESTING S PEECH ACTS uncertainty, and a universal mechanism for coding hypotheses Another scenario which is worth considering is one in about that structure would be beneficial. which a source makes an assertion about another assertion. One obvious way of implementing this would be to allow For example, X might say that Y told him that bin Laden was complex parameter based statements to be made about a in Kandahar. An implementation of the AssertiveSpeechAct source’s assertion. For example, a partial rule for determining mechanism described above would allow such structured as- the effect of duress on the credibility of an assertion of a by sertions to be represented quite naturally. X might encode the following: Such nested speech acts are likely to occur, and any on- • If the assertion is inconsequential, then the credibility is tology of evidence must be prepared to deal with them. As a unaltered simple example, one would be inclined to give less credence • Otherwise, monetary duress combined with a belief by X to an assertion heard third-hand than the same assertion given that he will be paid for his information should result in first-hand. the credibility of the assertion being decreased by some parameter α IV. C ERTAIN K NOWLEDGE • A consequential assertion made while under physical duress should result in a decrease by the parameter β A final benefit of this proposal is possibly of greater if X is deemed insufficiently competent to judge a. theoretical than practical value. Following the example of Formally, this might be written as: the axiomatic method in mathematics, one can argue that the findings in any knowledge base should be certain and ∀a∀X uncontroversial (this is analogous to the fact that the axioms of �� �� Source(X) ∧ Asserts(X, a) ∧ UnderDuress(X) → a mathematical theory are normally taken to be self-evident.) � �� Any knowledge which is controversial should be generated Consequential(a) < χ) → ∆(Cred(a)) = 0 rather than assumed. � By allowing speech acts to be encapsulated within higher (Consequential(a) ≥ χ)∧ order structures, we are able to do away with some of the (DuressType(X) = Monetary)∧ ambiguity that might otherwise be inherent in working with (Believes(X, “WillBePaidFor(X, Asserts(X, a)”)) > ξ) ontology of evidence. For example, rather than assigning a �� credibility score to a particular act or actor, we can merely → (∆(Cred(a)) = −α) � record which facts pertain to credibility, and have the actual (Consequential(a) ≥ χ)∧ score calculated by the network. In particular, in the previously (DuressType(X) = Physical)∧ described example there would no controversy in recording (CompetenceToJudge(X, a) < θ) that Y asserted that bin Laden is in Kandahar and that Y was � under physical duress at the time of the assertion. To assign a → (∆(Cred(a)) = −β) 2 There are first order theories in which propositions can be meaningfully In this coding, χ, ξ and θ represent threshold parameters which thought of as parameters for predicates, a natural example being Robinson’s Q determine whether or not a variable is strong enough to have an (via Gödel coding). This is an exceptional theory however, and many theories impact. The function ∆ represents the change in the credibility exist in which no such coding is possible. 3 credibility of 0.67, on the other hand, is very much a matter of judgement and open to question. A scheme which records the former rather than the latter may have an arbitrariness in our choice of network structure and parameters, but at least the latter are subject to being updated. Further, the arbitrariness of structure and parameters arguably corresponds to the arbitrary choice of a non-logical language and way of expressing a set of first order axioms (in that they represent representational rather than semantic choices). V. F UTURE W ORK While the value of the expressivity contemplated is clear, there are still many questions that need to be answered. Amongst these, the question of what the added computational cost would be seems to be foremost. While the case of modal logic offers reasonable hope that this proposal might be implemented in a way which preserves soundness and completeness of the logical formalism, one would like to know that the added complexity isn’t practically prohibitive. A related question that would be interesting to explore is that of what level of quantifier complexity is actually necessary. It is noteworthy that the immediate examples of “higher order” statements that come to mind seem to be universal (with respect to “higher order” predicates). A bound on the required quantifier complexity could significantly ease the computational cost of an implementation. VI. ACKNOWLEDGEMENTS I am very grateful to Fabian Neuhaus for introducing me to the subject of ontologies and discussing various ideas with me. In particular, I have had several very helpful exchanges with him that have greatly clarified the goals and presentation of this paper. R EFERENCES [1] K. Laskey, D. Schum, P. Costa, and T. Janssen, “Ontology of Evidence,” ONTOLOGY FOR THE INTELLIGENCE COMMUNITY, p. 20. [2] K. Laskey, “MEBN: A logic for open-world probabilistic reasoning,” The Volnegau School of Information Technology and Engineering. George Mason University, Fairfax, VA, USA. Available at http://ite. gmu. edu/k̃laskey/index. html. [3] E. Wright and K. Laskey, “Credibility models for multi-source fusion,” in Information Fusion, 2006 9th International Conference on, 2006, pp. 1–7. [4] F. Neuhaus and B. Andersen, “The Bigger Picture: Speech Acts in Interaction with Ontology-based Information Systems.”