An adjoint construction for topological models of intuitionistic modal logic Extended abstract

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1 An adjoint construction for topological models of intuitionistic modal logic Extended abstract M.J. Collinson, B.P. Hilken, D.E. Rydeheard April 2003 The purpose of this paper is to investigate topological aspects of the relationship of modal logics to their Kripke models. In doing so, we extend the traditional treatment of models to a broad range of non-classical (intuitionistic) modal logics. The key to this is a construction of topological spaces which is canonical in the strong sense that the relationship of models to modal algebras is an adjunction, and an adjunction that yields full categorical duality results for interesting classes of models. This is an extended abstract. We give only a basic mathematical outline. A full paper is in preparation. Those interested in this approach to models of intuitionistic modal logic may consult related publications at uk/~david. 1 Introduction A traditional account of Kripke-style semantics of classical propositional modal logic goes as follows. Let R X X be a binary relation (sometimes called the accessibility relation ) on a set X (whose elements are called worlds or states ). Propositions are interpreted as the set of worlds in which they are true. For U X, the modal operators are interpreted as follows: U = {x X y X. xry = y U} U = {x X y U. xry} This interpretation extends to a map from binary relations on X to modal algebras: (X, R) (P (X),, ) where P (X) is the powerset of X and the modal operators are defined above. By a modal algebra we mean (for the moment) a Boolean algebra with modal operators satisfying the classical axioms. david@cs.man.ac.uk 1

2 A map between modal algebras is a morphism of Boolean algebras that commutes with the modal operators (strictly). The maps between relations which correspond to maps of modal algebras are known as p-morphisms. The correspondence sends a p-morphism f to its inverse image map f. Altogether, this defines a functor from the category of relations (with p-morphisms) to that of modal algebras. In the opposite direction, for any modal algebra (A,, ), consider the set pf (A) of prime filters of A. A relation R A pf (A) pf (A) can be defined by pr A q {a A a p} q for all prime filters p, q of A. A completeness result for classical propositional modal logic is based on this construction. From a logic we build a model using the maximally consistent sets of formulae. These constitute the prime filters of a Boolean algebra. Looking over the above development, there are a few comments that we wish to make. Firstly, as we have formulated it, the construction above does not form an adjunction. In addition, there is an implicit topological structure to the above which we have not exploited. Consider a modal algebra (A,, ) and a A. The sets of prime filters O a = {p pf (A) a p} for a A act form a base for a topology on pf (A). This is the so-called Stone topology. It is the purpose of this paper to explore this topological setting. To do so, we put topological spaces in the place of sets, and relations on spaces in the place of relations on sets. It turns out that we can do this in a very general way, without any link between the topology and the relation. This is not an arbitrary generalisation. It combines the Kripke semantics of modal logics with the standard topological semantics of intuitionistic logics in a natural way to provide a framework for the semantics of intuitionistic modal logics. The key to this is the construction of an adjunction from which duality and completeness results follow. These can be regarded as modal extensions of standard results from locale theory [9]. There is a range of interesting models which combine a topology and a relation and which arise naturally in logic and in computation. These include equilogical spaces, the real numbers, and domain models using the Scott topology. 2 Modal frames and relational spaces We begin this brief outline by defining the relevant modal algebras and relational structures. Definition 2.1. A frame is a partial order with finite meets and all joins, with binary meets distributing over arbitrary joins. A modal frame (A,, ) consists 2

3 of a frame A together with a pair, : A A of monotone unary operators which satisfy for all a, b A. (1) a b (a b) (2) a b (a b) (3) (4) There has been considerable discussion in the literature as to what axioms are appropriate for intuitionistic modal logic, in particular whether the final axiom above is required. In the topological approach developed here, these are the exact axioms which ensure a canonical construction of models as an adjunction. Definition 2.2. A modal frame morphism f : (A,, ) (B,, ) is a morphism of frames f : A B, such that f f (5) f f (6) where is the pointwise extension of the order on frames to an order on frame morphisms. We say that the morphism laxly commutes with the modal operators. The morphism is said to be strict if the two inequalities 5, 6 above are equalities. Modal frames and modal frame morphisms form a category MFrm. Definition 2.3. A relational space (X, O, R) consists of a topological space (X, O) together with a binary relation R X X. Let the category RelS consist of relational spaces and continuous, relation-preserving maps between them. Definition 2.4. Let f : (X, O, R) (Y, P, S) be a continuous relation-preserving map between relational spaces. If f satisfies f(x)sy U x X. xrx & f(x ) U (7) for all x X, y Y and U P then we say that f is a topological p-morphism. Relational spaces and topological p-morphisms together form a category RelSp. If f is a topological p-morphism which also satisfies f(x)sy C x X. xrx & f(x ) C (8) for all x X, y Y and closed subsets C then we say that f is a topological pq-morphism. Relational spaces and topological pq-morphisms form a category RelSpq. 3

4 3 An adjoint construction The open set functor extends to a functor from RelSp to MFrm: Definition 3.1. The modal open set functor Ω : RelSp MFrm is defined as follows. Let X = (X, O, R) be a relational space. Then ΩX = (O,, ) where O is the frame of opens and for each U O we define: U = {x X R(x) U} U = {x X R(x) U }. Where () is the interior operation. The functor takes a continuous p-morphism to its inverse image map. We can restrict the modal open set functor to act on the category RelSpq. In the opposite direction, we construct a relational space from a modal frame to provide a contravariant adjunction between MFrm and RelSpq. The construction is fairly intricate but is the core of this paper. We present it in four stages: firstly defining a set of pre-points, then a relation, then restrict the set of pre-points to the set of points, and then we impose a topology on these points. Let A be a modal frame. Definition 3.2. A pre-point of A is a triple (p, a, F ), where 2 is the two-point frame, p : A 2 is a frame morphism, a is an element of A and F is a filter of the frame A, such that p( a) = 0 {c A p( c) = 1} F. The next stage is the construction of a relation between pre-points. Definition 3.3. Two pre-points are related, (p, a, F )R A (q, b, G) if q(a) = 0 F {c A q(c) = 1}. We restrict the set of pre-points to the set of points. The set P A of modal frame points of A is defined to be the largest set of pre-points satisfying the two conditions below. Definition 3.4. The set P A of points of A is the largest set P of pre-points satisfying the conditions: (p, a, F ) P & c a (q, b, G) P. (p, a, F )R A (q, b, G) & q(c) = 1 (p, a, F ) P & c / F (q, b, G) P. (p, a, F )R A (q, b, G) & q(c) = 0. 4

5 The significance of these conditions is discussed below. The topology on the set of points is the standard one from from locale theory, defined as follows: There is a modal frame morphism φ A : A P(P A ) defined by φ A (a) = {(p, a, F ) P A p(a) = 1} for all a A. The topology O A on P A is given by the image of the map φ A. For every modal frame A the modal point construction produces a relational space (P A, O A, R A ). In the above construction, the component a of a point (p, a, F ) determines the interior of the set points unrelated to it. The component F determines the neighbourhood filter of the set of points related to (p, a, F ). It can be shown that φ A is the universal arrow from A to Ω. We then have the following theorem. Theorem 3.5. The assignment A (P A, O A, R A ) is the object part of a contravariant functor, adjoint to Ω : RelSpq MFrm. The unit on the modal frame side is φ. Let the adjoint to the modal open set functor be F : MFrm RelSpq. Let the unit of the adjunction on the topological side be ψ. The construction of points given here is related to that found in [17]. There is an alternative presentation of points in terms of classes of trees. This is somewhat more explicit, at least for calculating with points. See [6] for a version of this alternative construction. 4 Lenses, relational spaces and duality Duality results and completeness for the models may be derived from the above construction of an adjoint as follows. Definition 4.1. A lens of a space (X, O) is the intersection of a closed subset with a saturated subset. That is, every lens L is of the form L = C S where C is closed and S is saturated. Let a lens relation be a binary relation on a space, for which the image of every point along the relation is a lens. That is, R X X is a lens relation if and only if R(x) is a lens of X for every x X. The significance of lenses to modal duality theory comes from the observation that the relation R A in the space (P A, O A, R A ), constructed from a modal frame A is a lens relation. Lemma 4.2. The relation R A is a lens relation. There is a duality between the images of the functors appearing in the above adjunction. A similar result holds for topological p-morphisms, see [7]. Definition 4.3. A modal frame A is modally spatial if φ A is an isomorphism. A space X is modally sober if ψ X, the component of the topological unit at X, is an isomorphism. 5

6 Proposition 4.4. If X is a relational space then Ω(X) is modally spatial. Proposition 4.5. If A is a modal frame then F(A) is modally sober. We summarise these results in the following theorem. Theorem 4.6. There is a duality between the full subcategories of MFrm and of RelSpq whose objects lie in the images of the functors Ω : RelSpq MFrm and F : MFrm RelSpq, respectively. There is an associated spectrality result: Define a modal Heyting algebra to be a Heyting algebra whose elements satisfy the axioms 1, 2, 3, 4 above. The ideal completion functor then takes modal Heyting algebras to modal frames. The modal frames which arise from the ideal completion of a modal Heyting algebra lie within the scope of the above duality. A completeness result follows directly. Fuller details of this result and of the overall mathematical development will be available in the forthcoming paper. 5 Related work The methods and results presented in this paper are developed directly from those of Hilken [7]. The key difference here is the definition of morphisms of relational spaces and the consequent construction of points. This new approach clarifies the way properties of the relational spaces arise from those of modal frames. The interpretation of intuitionistic modal logic in relational spaces has also been considered by Wijesekera [17] who used the same definition of the topological modal operators. In fact, he considers the same definition of modal Heyting algebra and a similar construction of a space from a modal frame. This was an independent discovery which here we generalise and use to define an adjunction between categories. Our approach is also connected, although less directly, with topological approaches to classical modal logic, in particular the work of Halmos [5], Esakia [3], Goldblatt [4], Lemmon [11] and Sambin and Vaccaro [15], all of whom consider restrictions on the notion of relational space to give semantics for various classical modal logics and dualities for modal Boolean algebras. References [1] A. Bauer, L. Birkedal, and D.S. Scott. Equilogical spaces. Theoretical Computer Science, to appear. [2] M.J. Collinson. Modality, Topology and Computation. PhD thesis, The University of Manchester, U.K.,

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