### Representation, reasoning, and relational structures: a hybrid logic manifesto

P Blackburn
2000 Logic Journal of the IGPL
This paper is about the good side of modal logic, the bad side of modal logic, and how hybrid logic takes the good and fixes the bad. In essence, modal logic is a simple formalism for working with relational structures (or multigraphs). But modal logic has no mechanism for referring to or reasoning about the individual nodes in such structures, and this lessens its effectiveness as a representation formalism. In their simplest form, hybrid logics are upgraded modal logics in which reference to
more » ... ndividual nodes is possible. But hybrid logic is a rather unusual modal upgrade. It pushes one simple idea as far as it will go: represent all information as formulas. This turns out to be the key needed to draw together a surprisingly diverse range of work (for example, feature logic, description logic and labelled deduction). Moreover, it displays a number of knowledge representation issues in a new light, notably the importance of sorting. Modal Logic and Relational Structures To get the ball rolling, let's recall the syntax and semantics of (propositional) multimodal logic. Definition 1.1 (Multimodal languages) Given a set of propositional symbols PROP = {p, q, p , q , . . . }, and a set of modality labels MOD = {π, π , . . . }, the set of well-formed formulas of the multimodal language (over PROP and MOD) is defined as follows: for all p ∈ PROP and π ∈ MOD. As usual, ϕ ↔ ψ := (ϕ → ψ) ∧ (ψ → ϕ). Definition 1.2 ((Kripke) models) Such a language is interpreted on models (often called Kripke models). A model M (for a fixed choice of PROP and MOD) is a triple is a non-empty set (I'll call its elements states, or nodes), and each R π is a binary relations on W . The pair (W, {R π | π ∈ MOD}) is called the frame underlying M, and M is said to be a model based on this frame. V (the valuation) is a function with domain PROP and range Pow (W ); it tells us at which states (if any) each propositional symbol is true. 339 L. A Hybrid Logic Manifesto Definition 1.3 (Satisfaction and validity) Interpretation is carried out using the Kripke satisfaction definition. Let M = (W, {R π | π ∈ MOD}, V ) and w ∈ W . Then: iff ∀w (wR π w ⇒ M, w ϕ). If M, w ϕ we say that ϕ is satisfied in M at w. If ϕ is satisfied at all states in all models based on a frame F , then we say that ϕ is valid on F and write F ϕ. If ϕ is valid on all frames, then we say that it is valid and write ϕ. Now, you've certainly seen these definitions before -but if you want to understand contemporary modal logic you need to think about them in a certain way. Above all, please don't automatically think of models as a collection of "worlds" together with various "accessibility relations between worlds", and don't think of modalities as "nonclassical logical symbols" suitable only for coping with intensional concepts such as necessity, possibility, and belief. Modal logic can be viewed in these terms, but it's a rather limited perspective. Instead, think of models as relational structures, or multigraphs. That is, think of a model as an underlying set together with a collection of binary and unary relations. We use the modalities to talk about the binary relations, and the propositional symbols to talk about the unary relations. Remark 1.4 (Kripke models are relational structures) Let's make this precise. Consider a model M = (W, {R π | π ∈ MOD}, V ). The underlying frame (W, {R π | π ∈ MOD}) is already presented in explicitly relational terms, and it is trivial to present the information in the valuation in same way: in fact M can be presented as the following relational structure M = (W, {R π | π ∈ MOD}, {V (p) | p ∈ PROP}). Why think in terms of relational structures? Two reasons. The first is: relational structures are ubiquitous. Virtually all standard mathematical structures can be viewed as relational structures, as can inheritance hierarchies, transition systems, parse trees, and other structures used in AI, computer science, and computational linguistics. Indeed, anytime you draw a diagram consisting of nodes, arcs, and labels, you have drawn some kind of relational structure. There are no preset limits to the applicability of modal logic: as it is a tool for talking about relational structures, it can be applied just about anywhere. Secondly, relational structures are the models of classical model theory (see, for example, Hodges  ). Thus there is nothing intrinsically "modal" about Kripke models, and we're certainly not forced to talk about them using modal languages. On the contrary, we can talk about models using any classical language we find useful (for example, a first-order, infinitary, fixpoint, or second-order language). Unsurprisingly, this means that modal and classical logic are systematically related. Remark 1.5 (Modal logic is a fragment of classical logic) To talk about a Kripke model in a classical language, all we have to do is view it as a relational structure (as described in the previous example) and then 'read off ' from the signature (that is, MODAL LOGIC AND RELATIONAL STRUCTURES