MIXING MODAL AND SUFFICIENCY OPERATORS

Transcription

1 Bulletin of the Section of Logic Volume 28/2 (1999), pp Ivo Düntsch Ewa Or lowska MIXING MODAL AND SUFFICIENCY OPERATORS Abstract We explore Boolean algebras with sufficiency operators, and investigate a class of mixed algebras which corresponds to frames < U, R > with a modal operator R and a sufficiency operator [[R]]. 1. Introduction It is well known that not every first order property can be expressed as a statement of modal logic, a case in point being irreflexivity. Noting that a relation is irreflexive if and only if its complement is reflexive and reflexivity is modally expressible Humberstone [1] introduced an inaccessibility operator, which was determined by the complement of a frame relation; a similar idea was put forward by Gargov et. al. [2] who used a sufficiency operator. As they point out, having separate modal and sufficiency operators only mirrors the deficiency of one construction in the other, and thus, mixing operators will lead to higher expressivity. In this note, we will present a representation theorem for frames and algebras with a sufficiency operator along the lines of the corresponding theorem for Boolean algebras with operators [3], and exhibit the duality between the category of frames with suitable morphisms and the sufficiency algebras in parallel to the modal case [4]. We will define a class of mixed The authors gratefully acknowledge support by the KBN/British Council Grant No WAR/992/151

2 100 Ivo Düntsch and Ewa Or lowska algebras and give a necessary and sufficient condition for its canonical extension to reflect both modal and sufficiency properties of a frame. Finally, we give an example for a frame whose properties can be captured by a mixed expression, but not by a modal sentence. We will only present the results; the proofs will appear elsewhere. 2. Definitions and notation We assume a working knowledge of Boolean algebras and modal logic, and invite the reader to consult [5] for the first topic and [6] for the second one. A frame is a pair <U, R >, where U is a set, and R a binary relation on U. We will usually write xry for < x, y > R, and R(x) is the set {y U : xry}. Furthermore, R = {<y, x>: xry} is the converse of R. A modal operator on a Boolean algebra < B, +,,, 0, 1 > is a mapping f : B B which satisfies (2.1) f(0) = 0, (normality) (2.2) f(x + y) = f(x) + f(y). (additivity) The pair <B, f > is called a modal algebra, and the class of all modal algebras is denoted by MOA. A modal operator f is called completely additive, if it preserves arbitrary joins whenever they exist. The dual mapping f defined by f (x) = f( x) is called a necessity operator. It is co normal, i.e. g(1) = 1, and multiplicative. If B is a Boolean algebra (BA), then its canonical extension is a complete and atomic Boolean algebra B σ which satisfies (2.3) Every atom of B σ is the meet of elements of B. (2.4) If A B and B σ A = 1, then there is a finite subset A 0 of A whose join is 1. Each BA has a canonical extension which is unique up to isomorphism [3]. If f is a modal operator on B, then its canonical extension f σ to B σ is defined by (2.5) f σ (x) = { {f(b) : b B, y b} : y At(B σ ), y x}, where At(B σ ) is the set of atoms of B σ. Given a frame <U, R >, the operators R and [R] on 2 U are defined by

3 Mixing modal and sufficiency operators 101 (2.6) R (X) = {y U : R (y) X }, (2.7) [R](X) = {y U : R (y) X}. The following result is fundamental: Proposition 2.1. [3, Theorem 3.3.] 1. If K = U, R is a frame, then R is a complete modal operator on 2 U, and [R] its dual necessity operator. 2. If f is a modal operator on 2 U, and g its dual, then there is exactly one binary relation S on U such that S = f, and [S] = g. This relation is defined by (2.8) xsy y f({x}). The algebra < 2 U, R > is called the full complex algebra of K. Proposition 2.2. [3, Theorem 3.10] If < B, f > is a modal algebra, then there is, up to isomorphism, a unique frame <U, R>, such that < 2 U, R > = < B σ, f σ >. <U, R> as above is called the atom structure of <B, f >. 3. Sufficiency algebras In this section we introduce sufficiency algebras and present representation and duality theorems in analogy to those for modal algebras [3], [4]. A sufficiency operator on B is a function g : B B which satisfies (3.1) g(0) = 1, (3.2) g(a + b) = g(a) g(b), for all a, b B. We call g a complete sufficiency operator, if (3.1) holds, and (3.3) If i I b i exists, then i I g(b i) exists, and is equal to g ( i I b i). A sufficiency algebra is a Boolean algebra with an additional sufficiency operator; the class of sufficiency algebras will be denoted by SUA. With some abuse of language we will use MOA and SUA also for the respective algebras.

4 102 Ivo Düntsch and Ewa Or lowska Our first result is an algebraic version of the correspondence theorem for modal and sufficiency logic of Tehlikeli [7], quoted in [2]: Proposition 3.1. Let g be a necessity (sufficiency) operator on a BA B, and define g c : B B by g c (x) = g( x). Then, g c is a sufficiency (necessity) operator on B. Thus, necessity and sufficiency operators are mutually term definable. As a next step we define the canonical extension of a sufficiency operator g by (3.4) g σ (x) = { {g(z) : p z, z B} : p x, p At(B σ )}, The pair < B σ, g σ > is called the canonical extension of <B, g >, and we have Proposition 3.2. g σ is a complete sufficiency operator, and g σ B = g. In analogy to Proposition 2.1 we obtain Proposition 3.3. Suppose that K =< U, R > is a frame, and B = <2 W, g > is a complete and atomic SUA. 1. The mapping [[R]] : 2 U 2 U with (3.5) [[R]](X) = {y U : X R (y)} is a complete sufficiency operator. 2. Set (3.6) R g = {<x, y > U U : y g(x)}. Then, (3.7) R [[R]] = R. (3.8) [[R g ]] = g. Furthermore, if S is a binary relation on U with [[S]] = g, then S = R g. Observe that (3.9) y [R](X) ( x)[xry x X] (x X is necessary for xry) (3.10) y [[R]](X) ( x)[x X xry]. (x X is sufficient for xry) which explains the names of the operators. Furthermore, (3.11) [R](X) = [[ R]]( X), (3.12) [[R]](X) = [ R]( X), (3.13) [[R]]({x}) = R ({x}).

5 Mixing modal and sufficiency operators 103 We now have the following representation theorem for SUAs, corresponding to Proposition 2.2: Proposition 3.4. If < B, g > is a sufficiency algebra, then there is (up to isomorphism) a unique frame < U, R >, such that < 2 U, [[R]] > = < B σ, g σ >. In the rest of the Section, we will present the duality in analogy to [4]. A co bounded morphism from a frame K =<W, S > to a frame L =<U, R> is a mapping h : W U such that for all x, y W, t U, (3.14) h(x)rh(y) xsy (3.15) h(x)( R)t ( w W )[h(w) = t and x( S)w]. The full co complex algebra [[K]] of a frame K =< U, R > is the Boolean powerset algebra of U with the additional sufficiency operator [[R]] defined by (3.5). Proposition 3.5. Let K =<W, S >, L =<U, R> be frames. 1. If h : W U is a co bounded morphism, then the mapping h + : [[L]] [[K]] defined by h + (X) = {y W : h(y) X} is a complete SUA homomorphism. 2. Let p : [[L]] [[K]] be a complete SUA homomorphism, and B = {p(x) : X U}. Then, the mapping p + : V U with p + (w) = u, where u U and p(u) is the atom of B above {w} is a co bounded morphism. Corollary 3.6. Let K 1 be the category of power set SUAs with complete homomorphisms, and K 2 be the category of frames with co bounded morphism. 1. The assignments r, s <2 W, g > r <W, R g >, <U, R> s <2 U, [[R]]>, are mutually inverse covariant functors. p r p +, h s h +

6 104 Ivo Düntsch and Ewa Or lowska 2. If the homomorphism p : < 2 U, f > < 2 W, g > is injective (surjective), then p + : <W, R g > <U, R f > is surjective (injective). 3. If K =< W, S >, L =< U, R >, and the co bounded morphism h : K L is injective (surjective), then h + : [[L]] [[K]] is surjective (injective). 4. Mixed algebras In this Section we put modalities and sufficiency operators together. Our aim is to define a class of algebras which reflects modality and sufficiency of the same frame relation. A mixed modal sufficiency algebra (or just mixed algebra) (MIA) is a BA B with two additional operators f, g such that (4.1) f is a modal operator. (4.2) g is a sufficiency operator. (4.3) f σ (p) = g σ (p) for each atom p of B σ. The next result shows that condition (4.3) is the connecting link between modality and sufficiency: Proposition 4.1. For each MIA <B, f, g > there is (up to isomorphism) a unique frame <U, R> such that <2 U, R, [[R]] = < B σ, f σ, g σ >. The algebra < 2 U, R, [[R]] > is called the mixed complex algebra of < U, R >. Condition (4.3) is not elementary, and the question arises, whether we can do better. The next result shows that this is not the case: Proposition 4.2. The class MIA is not first order axiomatisable. We can express f and g of a MIA by a single binary operator as follows: Define e : B B B by (4.4) e(x, y) = f (x) g(y). Here, f is the dual (necessity) operator of f. Then, e is in the clone generated by the operations of <B, f, g >. Conversely, (4.5) e(x, 0) = f (x) g(0) = f (x), (4.6) e(1, x) = f (1) g(x) = g(x)

7 Mixing modal and sufficiency operators 105 show that f and g are definable from e and the Boolean operations. Furthermore, we define m : B B by (4.7) m(x) = e(x, x). If < B, f, g >=< 2 U, R, [[R]] > arises from the frame < U, R > and X, Y U, then (4.8) z e(x, Y ) Y R (z) X. In particular, (4.9) z e(x, X) R (z) = X. Furthermore, (4.10) m(x) = { U, if X = U,, otherwise. Finally we give an example of a class of frames which are not modally definable, but can be captured by a mixed expression. A contact relation C on a set U satisfies the following properties: (4.11) C is reflexive. (4.12) C is symmetric. (4.13) C(x) = C(y) implies x = y. Contact relations play a prominent role in the context of qualitative geometry and spatial reasoning, and were first considered by Laguna [8]. While conditions (4.11) and (4.12) can be expressed by equations involving modal operators, we have shown in [9] that the extensionality condition (4.13) is not modally expressible. We also have Proposition 4.3. The class of frames with a contact relation is not closed under co bounded morphisms. On the other hand, Proposition 4.4. Let < U, C > be a frame, and < B, f, g > its mixed complex algebra, i.e. B = 2 U, f = C, and g = [[C]]. Then, C is a contact relation if and only if

8 106 Ivo Düntsch and Ewa Or lowska (4.14) [C](X) X, (4.15) X [[C]][[C]](X), (4.16) m( (e(x, X) Y )) m( (e(x, X) Y )) = U. Here, the mappings m and e are as defined by (4.4) and (4.7). References [1] I. L. Humberstone, Inaccessible worlds, Notre Dame Journal of Formal Logic 24 (1983), pp [2] G. Gargov, S. Passy, and T. Tinchev, Modal environment for Boolean speculations, [in:] D. Skordev (ed.) Mathematical Logic and Applications, pages , New York, Plenum Press. [3] Bjarni Jónsson and Alfred Tarski, Boolean algebras with operators I, Amer. J. Math. 73 (1951), pp [4] Robert Goldblatt, Varieties of complex algebras, Annals of Pure and Applied Logic 44 (1989), pp [5] Sabine Koppelberg, General Theory of Boolean Algebras, volume 1 of Handbook on Boolean Algebras, North Holland, [6] Robert Bull and Krister Segerberg, Basic modal logic, [in:] D. M. Gabbay and F. Guenthner, editors, Extensions of classical logic, volume 2 of Handbook of Philosophical Logic, pages Reidel, Dordrecht, [7] S. Tehlikeli, An alternative modal logc, internal semantics and external syntax (A philosophical abstract of a mathematical essay). Manuscript, [8] T. de Laguna, Point, line and surface as sets of solids The Journal of Philosophy 19 (1922), pp [9] Ivo Düntsch and Ewa Or lowska, A proof system for contact relation algebras. Submitted for publication, Faculty of Informatics University of Ulster at Jordanstown Newtownabbey, BT 37 0QB, N.Ireland I.Duentsch@ulst.ac.uk Institute of Telecommunications Szachowa , Warszawa, Poland orlowska@itl.waw.pl