An Evidential Path Logic for Multi-Relational Networks

Multi-relational networks are used extensively to structure knowledge. Perhaps the most popular instance, due to the widespread adoption of the Semantic Web, is the Resource Description Framework (RDF). One of the primary purposes of a knowledge network is to reason; that is, to alter the topology of the network according to an algorithm that uses the existing topological structure as its input. There exist many such reasoning algorithms. With respect to the Semantic Web, the bivalent, monotonic reasoners of the RDF Schema (RDFS) and the Web Ontology Language (OWL) are the most prevalent. However, nothing prevents other forms of reasoning from existing in the Semantic Web. This article presents a non-bivalent, non-monotonic, evidential logic and reasoner that is an algebraic ring over a multirelational network equipped with two binary operations that can be composed to execute various forms of inference. Given its multi-relational grounding, it is possible to use the presented evidential framework as another method for structuring knowledge and reasoning in the Semantic Web. The benefits of this framework are that it works with arbitrary, partial, and contradictory knowledge while, at the same time, it supports a tractable approximate reasoning process. Knowledge structures are used to represent facts about the world. The most common formal data structure to represent knowledge is the network. With respect to symbolic knowledge representation, the multi-relational network (also known as an edge labeled graph or semantic network) is widely used. A multi-relational network is composed of a set of vertices and a family of edge sets, where each edge set has a different nominal, or categorical, label. Formally, a multi-relational network can be represented as M = (V,E), where V is the set of vertices and E = {E0, E1, . . . , Em ⊆ (V ×V )} is the family of directed edge sets. In recent years, perhaps the most popular instance of a multi-relational data structure for knowledge representation is the Resource Description Framework (Dau 2006) of the Semantic Web iniRodriguez, M.A., Geldart, J., “An Evidential Path Logic for Multi-Relational Networks”, Association for the Advancement of Artificial Intelligence (AAAI): Technosocial Predictive Analytics Symposium, AAAI Press, LA-UR-08-06397, Stanford University, March 2009. tiative (Berners-Lee, Hendler, and Lassila 2001)1. An edge in an RDF network is called a statement, or triple, as it is composed of a subject, predicate, and object. For example, suppose the statement (i, k, j). This statement denotes that i is related to j by means of an k-type relationship. Given the previous definition of M , this is equivalent to the directed edge (i, j) ∈ Ek. A particular instance of a statement is (marko, coauthor, joe). This statement denotes that Marko has a coauthorship relationship to Joe. Languages such as the RDF Schema (RDFS) and the Web Ontology Language (OWL) impose a set of constructs that serve to structure knowledge in a particular manner. The particularities of such a structure are used by an RDFS or OWL reasoner to infer new statements that can be added to the RDF network. The statements inferred by such reasoners are bivalent in that they are either true or false and moreover, their truth value is monotonic as it does not change once it has been asserted. While RDFS and OWL are common languages in the Semantic Web, the flexibility of RDF can easily support other knowledge structures and reasoning algorithms. The purpose of this article is to present a non-bivalent, non-monotonic, evidential logic and reasoner for multirelational networks that leverages many of the ideas from Non-Axiomatic Logic (NAL) (Wang 2006) and the NonAxiomatic Reasoning System (NARS) (Wang 1993). The philosophical foundation of an evidential logic is that no statement is inherently true or false and that a statement only maintains levels of evidence to support or negate its claim. The notion of experience-grounded semantics [is where] the truth value of a judgment indicates the degree to which the judgment is supported by the system’s experience. Defined in this way, truth value is system-dependent and time-dependent. Different systems may have conflicting opinions, due to their different experiences. (Wang 1994) The typical metaphor in an evidential logic system is that of an agent that perceives the world, represents its perceptions in an internal knowledge structure, and reasons on that structure to infer new knowledge (Wang 2004a). Moreover, it is assumed that this agent has limited computational resources in terms of both space and time and thus, does not Other formal models of RDF include a bipartite graph (Hayes and Gutierrez 2004) and hypergraph (Morale and Serodio 2006) representation. ar X iv :0 81 0. 14 81 v2 [ cs .L O ] 3 0 D ec 2 00 8 maintain an objective knowledge structure nor does it necessarily have the ability to reason across its entire subjective knowledge structure. In other words, the agent has only so much information that it can store and process at any one time. This notion is known as the Assumption of Insufficient Knowledge and Insufficient Resources (AIKIR). Nonaxiomatic logic is contrasted to axiomatic logic, where truth is bivalent, is defined independent of the time and space requirements necessary to derive it, can be reasoned from a finite set of premises, and where all reasoning produces true, immutable conclusions. The evidential logic presented in this article forms an algebraic ring over a multi-relational network (i.e. the knowledge structure) equipped with two binary operations (i.e. the atoms of the inferencing algorithms). Given the logic’s multi-relational formulation, it is possible to comfortably represent this structure in RDF and thus, on the Semantic Web. The primary contribution of this article is the application of evidential logics to multi-relational networks and the formulation of an algebraic evidential reasoner. Evidence in an Inheritance Network With evidential logics, there does not exist an objective boolean truth value for every question that can be asked of the world as the world is not reasoned on directly (Wang 2004b). What is reasoned on is the agent’s internal knowledge structure. For the agent, knowledge is gained as new evidence from the world is discovered (either through direct perception or through communication) or as knowledge is inferred given the agent’s internal reasoning system. The Non-Axiomatic Reasoning System (NARS) is an example of an evidential reasoning system (Wang 1993). The data structure proposed for NARS version 2.2 is a directed evidence network denoted G = (V,E, λ), where V is a set of vertices, E ⊆ (V × V ) is a set of directed “inheritance” edges, and λ : E → 〈[0, 1], [0, 1]〉 maps each edge to an evidence tuple. For example, an edge is denoted