A Case Study Integrating Knowledge Graphs and Intuitionistic Logic

Abstract

In this work, we present a way to model and reason over Knowledge Graphs via an Intuitionistic Description Logic called iALC. We also introduce a Natural Deduction System for iALC to reason over our modelling of the information of the Knowledge Graphs. Furthermore, we apply this modelling to a case study in a context that aims to support the definition of concepts of Trust, Privacy, and Transparency, and the solution of apparent conflicts between them without the need for additional strategies, using only terms of the logic itself.

The authors gratefully acknowledge financial support from CNPq.

ANatural Deduction System for iALC

In this appendix, we present the rules⁸ of the ND calculus for iALC. In this notation, the premises of each rule are present over a bar (which represents the rule itself), and its conclusion lies below. We obtain proofs in ND a tree-like form by connecting these deduction steps.

In ND, rules come in two kinds: introduction and elimination rules. Introduction rules take some premises and produce a formula in conclusion with the related logical operator. For instance, the $\sqcap -intro$ generates a formula with the $\sqcap$ operator (meaning conjunction) in conclusion. Its meaning is as such: if a concept $\alpha$ is valid and a concept $\beta$ is valid, then we can conclude that the concept $\alpha \sqcap \beta$, meaning the conjunction of $\alpha$ and $\beta$, is also valid. On the other hand, elimination rules, as the name suggests, take one of the premises with the related operator (called the main premise - other premises, if present, are secondary or auxiliary premises) and produce a different formula, thus eliminating the operator. For example, the $\sqsubseteq -elim$ rule takes the main premise, containing the $\sqsubseteq$ operator and an auxiliary premise, resulting in a formula without the operator involved in the main premise. An intuitive approach to this rule is that, given that $\alpha$ is valid and that $\alpha$ is a sub-concept of $\beta$, we can conclude that $\beta$ is valid as well (this rule is, in fact, the modus ponens of iALC). Depending on the logic, the ND calculus may have other kinds of rules, called structural. Such is the case for the rules $Gen$ and $Gen-n$.

An important feature present in ND is the mechanism of hypothesis discharge, present in some rules. When one discharges a hypothesis, it means that the deduction step is self-contained, i.e., the discharged hypothesis is not open in the proof. A closed hypothesis transfers the logical consequence up to the conclusion of the rule that discharged it. For instance, in the $\sqsubseteq -intro$ rule, if we somehow have that $\alpha$ is valid somewhere and that we end up at $\beta$ being valid by applying possibly many ND rules, we can then conclude that $\alpha \sqsubseteq \beta$, and discharge the $\alpha$ above $\beta$. With this, there is no need to assume that $\alpha$ is valid every time one wants to show that $\beta$ is valid since it was already proven, and one is allowed to utilise $\alpha \sqsubseteq \beta$ for the rest of the proof. Essentially, to discharge a hypothesis closes the respective branch. The discharge mechanism reduces the number of hypotheses needed. A proof that has open hypotheses draws the truth of its conclusion from the truth of these hypotheses. We can prove the hypotheses using a set of axioms or, as in this article, the base knowledge of the KGs.

Let $\alpha$ and $\beta$ be concepts, x and y nominals, $\delta$ a formula, and R a role. L represents a list of labels (possibly empty). Labels represent either negation of concepts, or universal or existential restrictions on concepts, made implicit. They indicate a sort of context to the concept to which they are attached. $L^{\forall }$ is a list that restricts all labels in itself to $\forall R$ of some kind, and $L^{\exists }$ restricts all to $\exists R$. The $\bot$ concept is not valid in any world. We can see it representing falsehood.

Rules that utilise nominals and involve x and y assume a role connecting them, i.e. xRy. The n in the names of some rules indicate that those are the versions that utilise nominals. The Gen rules function as a lift to concepts, adding a universal restriction to the end of the list.

The ex falso quodlibet rules, $efq$ and $efq-n$, represent the Principle of Explosion: from falsehood, anything follows.

The reductio ad absurdum rules, namely $raa$ and $raa-n$, are not part of the ND calculus for iALC but are there to show that the calculus can be expanded from iALC to its classical counterpart, ALC, by having both of them. With their addition to the ruleset, nominals will have completely different meanings from the VLSs of iALC. Consequently, the models will lose their refinement obtained by the intuitionistic aspect of the logic. However, the deductions themselves will have the same structure, with rules that elevate the calculus to a classical framework.