A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics

: Logic-based knowledge representation is one of the main building blocks of (logic-1 based) artiﬁcial intelligence. While most successful knowledge representation languages are based 2 on classical logic, realistic intelligent applications need to handle uncertainty in an adequate 3 manner. Throughout the years, many different languages for representing uncertain knowledge— 4 often extensions of classical knowledge representation languages—have been proposed. We 5 brieﬂy present some of the deﬁning properties of these languages as they pertain to the family 6 of probabilistic description logics. This limited view is intended as a way to help the interested 7 researcher ﬁnd the most adequate language for their needs, and potentially identify the gaps 8 remaining. 9


Introduction 11
Logic-based knowledge representation [1] is one of the fundamental building blocks 12 for (logic-based) artificial intelligence.In fact, any intelligent application has, as an un-13 avoidable requirement, the need to represent and handle the knowledge about the 14 domain that it works in [2].This need has led to a plethora of knowledge represen-have been proposed; most notably, possibility theory [5] and evidence theory [7,8]. 2 For 91 the scope of this paper, as in most of the literature in uncertain knowledge representation, 92 we give more weight to the advantages of the easiness of presentation over the likelihood 93 of misunderstandings.Thus, we consider probability theory as the basis for representing 94 and managing uncertainty in the context of knowledge representation.However, we 95 still need to take into account the different interpretations of probability as uncertainty.

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Despite its unified name, and the use of probabilities for handling it, not all un-97 certainty is equal.Halpern [15] already hinted at it when describing its different types

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Hence, if we randomly take one of these tests, it has a 95% chance of being a correct one.

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Note that one can only be so specific about the probability if the whole population is  Indeed, there is no-one capable of discerning a probability of 95% from one of 95.5% 148 nor, for that matter, 60% from 70%.Importantly, even subtle differences may cause 149 huge mismatches over a derivation process; they could even lead to inconsistency in the 150 collected knowledge.In these cases, it is perhaps more useful to represent comparative 151 statements, of the form X is more likely than Y.However, this requires the development   propositional logic tends to be too inexpressive, and even the elements which can be 175 expressed sometimes require a complex and difficult to grasp construction to handle 176 correctly.On the other spectrum, in full predicate logic it is known that verifying the 177 satisfiability of a formula (which in terms of knowledge representation translates to 178 deciding whether a knowledge base is consistent) is an undecidable problem; that is, 179 there is no algorithm which can provide a correct answer in finite time for any possible 180 formula.

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For this paper, we focus on a family of formalisms which lies mainly within these 182 two formalisms.More specifically, most languages within this family-the family 183 of Description Logics (DLs) [17]-are more expressive than propositional logic (thus, 184 able to formalise more complex knowledge in a simpler manner) and at the same 185 time less expressive than predicate logic guaranteeing decidable reasoning tasks (with 186 consistency among them).There are a few exceptions to this statement, which only help 187 in increasing the relevance of the family as knowledge representation formalisms.The 188 very inexpressive DLs EL [18] and DL-Lite [19], which are specially targeted for tractable  We will see this in detail later, but in a nutshell and using Halpern's classification, 199 statistical probabilities are handled by adding uncertainty over the elements of the 200 domain (i.e., the population) while subjective probabilities are dealt with through several 201 potential interpretations (possible worlds).As mentioned before, the differences may be 202 important.

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For all these reasons, we consider description logics as a basic formalism for rep-204 resenting uncertain knowledge.This is meant mainly as a prototypical representation: 205 most of the ideas that we describe apply similarly to other formalisms without major 206 modifications.We emphasise, however, that the classical family of description logics 207 has some limitations which we will not consider further.Most notably, it cannot handle 208 non-monotonic [21], nor temporal knowledge [22,23] natively.Importantly, combin-209 ing uncertainty with non-monotonicity and with temporal constructors is known to 210 be specially problematic [24], both in terms of conceptual understanding and in the 211 computational complexity of reasoning.

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of logics of probability.Broadly speaking, Halpern's classification considers two kinds 99 of views on uncertainty: a statistical one referring to a proportion of the population 100 satisfying a property of interest, and a subjective one dealing with beliefs about possible 101 worlds.The difference lies in how the uncertainty is used within a derivation or reason-102 ing process, but mirrors existing differences from real-life use of probabilities.However, 103 it is important to note that Halpern's classification is orthogonal to the usual distinction 104 between frequentist and Bayesian probabilities, about which we refrain from mentioning 105 anything further in this text.106 Statistical probabilities come into play when speaking about proportionality, and 107 a random selection of elements.Hence, when we say that a medical test has a 95% 108 diagnostic specificity-in lay terms, that if the test is positive, then there is a 95% chance 109 that the individual is in fact positive for the disorder under scrutiny-what we are saying 110 is that 95 out of every 100 positive tests are correct (and the remaining 5 are wrong).

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In contrast, subjective probabilities consider unique instances which are charac-115 terised by different possibilities.The prototypical example in this direction is the weather 116 forecast.When a meteorological model predicts a 40% chance of rain tomorrow, it cannot 117 be read as a statistical statement saying that in 40 out of 100 tomorrows rain will be present.118 Instead, it studies different scenarios based on possible parameters like wind speed and 119 direction, temperature, humidity, and others, to verify in which of those scenarios rain is 120 present.Halpern's classification, however, is not fully satisfying in the context of knowledge 122 representation, and in particular in the context of incomplete domain knowledge and 123 expert knowledge.We use these two cases to exemplify the limitations of each of the 124 two types of probabilities.125 As mentioned already, statistical probabilities are derived as proportional obser-126 vations of an event within a given population.The term statistical hence refers to a 127 very basic analysis of data.The name is unfortunate, as it also evokes the use of more 128 advanced statistical analyses, which are not foreseen in these logics.The most basic 129 example is the presence of incomplete knowledge.While it is pretty straightforward 130 to find out the exact proportion of students in a given classroom who are left-handed, 131 the same cannot be said about e.g., COVID-19 patients who have pulmonary scars.To 132 know this latter proportion, it is necessary to identify precisely who has been infected 133 with the disease, and make a pulmonary plaque on all those subjects.Both of these tasks 134 induce high economic, social, and human costs which one might not be willing to cover.135 Instead, it is possible to approximate this knowledge using a statistical analysis on the 136 available data of publicly known infected individuals, and results from hospital analyses 137 from people suspect of having lung issues arising from it.Alternatively, one can also 138 sample the population to estimate these proportions.Both ideas are intended to fill the 139 gap left by the incomplete knowledge of exactly how many people fall into each of the categories of interest.The cost, however, is that there are (uncertain) margins of error 141 that one needs to deal with.142 Let us consider now subjective probabilities, which aim to represent beliefs about the 143 likelihood of specific events.A common use of subjective probabilities is for modelling 144 expert knowledge, where a (human) expert may-perhaps based on past observations-145 assign a probability to an event.In these cases, the numerical values underlying proba-146 bilities (and their algebraic manipulations) become more a hindrance than an advantage. 147

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152of new reasoning techniques, specially in the presence of mixed statements.Moreover, 153 it comes at the price of losing precision.On the other hand, these statements are more 154 easily understandable by the lay person, and describable by the experts.As it can be seen from this section, representing and managing uncertain knowledge 156 is far from trivial, even from the point of view of choosing the measure of uncertainty.157Thelandscape of probabilistic interpretations is vast, and different applications have 158 diverse needs for expressivity.If we attach other practical considerations like complexity 159 of reasoning, availability of resources, or historic knowledge to use, the panorama 160 gets even more diverse.It is important to keep this diversity in mind when studying 161 uncertain knowledge representation languages to avoid getting lost among the variants 162 that they induce.This is, in fact, one of the biggest obstacles faced by researchers trying 163 to get started in the area: not knowing the differences in the probabilistic interpretations, 164 exploring the state of the art seems a Sisyphean task.165 The following section is an attempt to draw a map of the uncertain knowledge 166 representation landscape and highlight active work and potential gaps.

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Representing Uncertain Knowledge 168 Representing uncertain knowledge has a prerequisite representing knowledge, full 169 stop.Knowledge representation, by itself, has a very long history, during which a 170 plethora of variations, limitations, and features have been considered.A natural first 171 step is to consider a known logic for representing knowledge; hence, one cannot avoid 172 mentioning propositional and (first-order) predicate logic as the foundations of logic-173 based knowledge representation languages.However, from a practical point of view, 174 189 reasoning, do not contain the full power of propositional logic although they allow for additional constructors.At the other end of the spectrum, expressive description 191 logics like SROIQ [20] include constructors (like transitive closure) which cannot be 192 directly expressed in first-order logic.These are handled in a manner that prevents 193 undecidability of reasoning.194 The semantics of description logics, which is based on interpretations akin to 195 first-order logic-that is, with a domain representing all the relevant objects, and an 196 interpretation function which expresses the properties of those individuals in relation to 197 each other-is specially useful for dealing with the various interpretations of uncertainty. 198

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Without going into too many details, the basic building blocks in a description logic 213 are concepts (that is, sets of individuals) and roles, which represent relationships between 214 individuals; slightly more formally, concepts are unary predicates, and roles are binary 215 predicates of first-order logic.Hence, Student is a concept that refers to all the students 216 in the world of interest, while supervises expresses the relationship between a supervisor 217 and their student.These symbols receive an interpretation by setting a (potentially 218 infinite) domain, which contains all the objects of interest, and an interpretation function 219 expressing which objects belong to which concepts, and which pairs are related via 220 roles.What differentiates one description logic from another is the class of constructors 221 used to build more complex concepts-e.g., conjunction, negation, number constraints, 222 etc.-and how they are interpreted.223 The goal of description logics is not only to express different kinds of concepts, but 224 to actually represent the knowledge of a domain.This is achieved through a knowledge 225 base which is a finite set of axioms that serve as constraints for the interpretations.That 226 is, each axiom excludes some potential interpretations as not representing the domain 227 knowledge.For example, an axiom could express that "every student must have at least 228 one supervisor."In this case, any interpretation including a supervisor-free student will 229 be excluded as a violation of the constraint.In general, given a knowledge base, there 230 are still many different (actually, infinitely many) interpretations which satisfy all the 231 constraints imposed.These so-called models are the only interpretations of interest in the 232 context of the knowledge base.233 When we use the term reasoning, we refer to the task of extracting consequences 234 which logically follow from the knowledge expressed in the knowledge base.Recall-235 ing that the axioms within the knowledge base are simply constraints in the possible 236 interpretations, reasoning then refers to finding other pieces of knowledge which are 237 guaranteed by these constraints.In other words, the logical consequences of a knowl-238 edge base are those which follow in all possible models of this set of axioms.We usually 239 say that reasoning is the task of making knowledge which is implicitly encoded by the 240 knowledge base explicit.The motivation behind using several models for reasoning is 241 that we consider that a knowledge base is always (necessarily) incomplete.That is, we 242 believe that a knowledge base will always exclude some information, either because it 243 is irrelevant, or because it is not yet known.In those cases, we want to leave open the or false.When dealing with uncertain axioms, we will have some interpretations where 298 the axiom explicitly holds, and some-due to the open world assumption-where it may