On Selecting a Conjunction Operation in Probabilistic Soft Logic

Probabilistic Soft Logic has been proposed and used in several applications as an efficient way to deal with inconsistency, uncertainty and relational representation. In several applications, this approach has led to an adequate description of the corresponding human reasoning. In this paper, we provide a theoretical explanation for one of the semi-heuristic choices made in this approach: namely, we explain the choice of the corresponding conjunction operations. Our explanation leads to a more general family of operations which may be used in future applications of probabilistic soft logic. Introduction and Motivation With the maturing of various sub-fields of AI we are now ready to combine approaches and techniques from multiple AI sub-fields to address broader issues. For example, we now have large bodies of knowledge that are available. An example such a knowledge base is ConceptNet (Liu and Singh 2004). Even in the same collection some of the knowledge may be manually curated, while parts may be automatically extracted. Sometimes all of the knowledge may be automatically obtained, such as the similarity knowledge based used in (Beltagy, Erk, and Mooney 2010) that is obtained using distributional semantics (Bruni, Tran, and Baroni 2014) of natural language. There may be inconsistencies lingering inside the knowledge base. For some of the knowledge, we may be able to assign weights. Learning part of this knowledge and reasoning with such knowledge requires approaches that can handle inconsistencies, uncertainty, structured information, and most importantly the approaches need to scale. Among the various approaches that have been proposed Probabilistic Soft Logic (PSL) (Bach et al. 2010; Bach et al. 2013; Kimmig et al. 2012) stands out as it can not only handle relational structure, inconsistencies and uncertainty, thus allowing one to express rich probabilistic graphical models (such as Hinge-loss Markov random fields), but it also seems to scale up better than its alternatives such as Markov Logic Networks (Richardson and Domingos 2006). Probabilistic soft logic (PSL) differs from most other probabilistic formalisms in that its ground atoms, instead Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. of having binary truth values, have continuous truth values in the interval [0,1]. In the original PSL (Bach et al. 2010; Bach et al. 2013; Kimmig et al. 2012) the syntactic structure of rules and the characterization of the logical operations have been chosen judiciously so that the space of interpretations with nonzero density forms a convex polytope. This makes inference in PSL a convex optimization problem in continuous space, which in turn allows efficient inference. The particular conjunction operation used in the above mentioned PSL is the Lukasiewicz t-norm (Klir and Yuan 1995). A different conjunction operation is used in (Beltagy, Erk, and Mooney 2010), where the resulting PSL is used for semantic textual similarity. PSL has been used in many different applications such as ones listed in(Bach et al. 2010; Bach et al. 2013; Beltagy, Erk, and Mooney 2010; Huang et al. 2012; Kimmig et al. 2012; Memory et al. 2012). However, none of these works precisely justify the particular selection of conjunction operation they use, beyond listing a few implications of using those operations. What we plan to do. In this paper, we provide a theoretical explanation of the conjunction operations that are used in (Bach et al. 2010; Bach et al. 2013; Beltagy, Erk, and Mooney 2010; Huang et al. 2012; Kimmig et al. 2012; Memory et al. 2012) and present a more general family of operations which may be used in future applications of probabilistic soft logic. Plan of the paper. In this section, we recalled and contextualized Probabilistic Soft Logic and which conjunction operations are usually selected in this logic. In the next sections, we provide our theoretical explanation for this selection. Rules, implications and the conjunction operation in Probabilistic Soft Logic The corresponding real-life problem. In many practical situations, we have rules r of the type a1, . . . , an → b that connect facts ai and b. For each such rule r, we know its “degree of importance” λr: the larger λr, the larger our degree of confidence in this rule. If we simply combine all these rules (and ignore their degrees of importance), then usually, the resulting set of rules becomes inconsistent. For example, sociologists known that in elections, a person tends to vote the same way as his friends. So, if a person B has a friend A1 who voted for an incumbent and a friend A2 who voted for a challenger, then: • for the first friend, the above sociological observation implies that B voted for the incumbent, while • for the second friend, the same sociological observation implies that B voted against the incumbent. In such situations, we cannot satisfy all the rules. So, it is reasonable to look for solutions in which an (appropriately defined) deviation from the ideal situation – when all the rules are satisfied – is the smallest possible. Need for probabilistic answers. If we have two conflicting rules, then we cannot be 100% sure which of them is not applicable in the current situation. If one of the rules is more important than the other one, i.e., if its degrees of importance λr is higher (λr > λr′ ), then most probably the first rule is applicable – but it is also possible that in this particular situation, the second rule is applicable as well. Thus, from the inconsistent knowledge base, we cannot extract the exact conclusion about the corresponding facts. At best, we can estimate the probabilities that different facts are true. How to deal with implication. If a implies b, this means that b holds in all situations in which a holds – and, maybe, in other situations as well. Thus, the probability p(b) that b is true is larger than or equal to the probability p(a) that a is true: p(b) ≥ p(a). From this viewpoint, if we know the probabilities p(a) and p(b) of two statements a and b, and p(a) ≤ p(b) (i.e., the difference p(a)−p(b) is non-positive), then this inequality is consistent with the rule a → b. On the other hand, if p(a) > p(b) (i.e., if the difference p(a)−p(b) is positive), this clearly is inconsistent with the rule a → b. Intuitively, the larger the positive difference p(a) − p(b), the larger the violation. It is therefore reasonable to take max(p(a) − p(b), 0) as the measure of severity of the rule’s violation. We want to minimize the overall loss of adequacy. Depending on the rule’s degree of importance λr, the same degree of rule’s violation may lead to different severity. In general, this severity is a function of the rule’s degree of importance λr and of its degree of violation max(p(a)− p(b), 0): d = s(λr,max(p(a)− p(b), 0)). We want to find the probabilities for which the overall loss of adequacy is the smallest possible. For rules r of the type ar → br, this means minimizing the sum