On The Logic Of Category Definitions

In their paper on category structures, Gazdar et al. (1988) define a constraint language L c for categories and a logic A c of admissible category structures. ~ The intuitive idea is that for a constraint $ expressed in L c, ~b is a nontrivial constraint if and only if A c 14 th; and it is a satisfiable constraint if and only if Ac 14 -nth. From a practical point of view it is therefore important to know whether A c is decidable and even better that the decision can be given in a time bounded by a recursive function on the length of ~b. However , the remarks made in their paper only suffice to show that the modal fragment of Ac 2 contains S4.Grz = K(l-lp --> p, C]p --* DDp, [~(D(p --* Dp) --* p) --~ p), which does not show that this fragment is decidable. In this note, I will establish both that the modal fragment of A c and A c itself are decidable, and I will prove it in that order. As a result, I will also axiomatize A c. Thus I show first that the modal reduct of Ac, which I call AM, is decidable. This paper will be rather hardgoing for anyone not acquain ted with modal logic. We advise the reader to have Gazdar et al. (1988) at hand while reading this paper, or better still, to read it once through beforehand. For the modal logics we refer the reader to Boolos (1979), Harel (1984), and Segerberg (1971), but in principle any introduction to modal logic will provide enough background to be able to understand the gist of the arguments. Without going into too many details of the construction, I will show that there is an easy way to give a proof that in fact shows that A M = S4.Grz using the structure of the models those logics admit. Intuitively, categories correspond to Kripke models. For let a be a category. Then a defines a set of categories W, which is obtained by successively applying type 13 features to a. An accessibility relation <1 is defined via ot <1/3 i f f f la) = /3 for some type 1 feature f . This accessibility relation is irreflexive, intransitive, finite, and defines a tree-structure on W. Most importantly, it is cycle free. Thus, if we look at the reflexive and transitive closure <]+ of <1, it is again finite and has no non-trivial cycles. It therefore is an S4.Grz structure (see, e.g., Boolos 1979). Conversely, an S4.Grz structure which is a tree can be represented as a category. If we then take a model based on that frame, where val: X-~ 2 w maps a finite set of propositional variables into 2 w, we can code this model by adding a type 0 featurefp for each p ~ X that takes values Y or ±. Thus the resulting category t~ not only codes the successor function by means of type I features, but also the valuation val. W is in one-to-one correspondence ~b with the set F of categories generated by a. We then make the following definitions: Le t /3 be in F: i . /3 ~ fp : T i f f ~/3) E val(p) iff ~/3) ~ p ii. [3 ~ fp : ± iff qb(/3) ~ val(p) iff ~/3) ~ p It is easy to see that for any modal formula P with variables in X, the corresponding translation ~" induced by p *-~ fp : T satisfies/3 ~ "t(P) iff q~(/3) ~ P. The logic of therefore coincides with the logic of all categories that differ from ot only in the assignment of type 0 features. To conclude, the logic of categories as defined in Gazdar et al. (1988) coincides with the logic of all finite, reflexive, transitive trees. It is easily seen that the finite, reflexive, transitive trees generate the class of finite models for S4.Grz. Thus the logic of categories is the logic of the finite models of S4.Grz which, since S4.Grz has the finite model property, is identical to S4.Grz (end ofproojO. A few remarks are in order: I. I used a purely semantical argument, which in this case is the most direct way, because it is fairly easy to see why we get just the models we get, though there is some footwork to be done. 2. Alternatively I could have built a canonical model out of a category structure X, whose worlds are the categories that X admits and whose accessibility relation is as defined above for categories. The proof is essentially the same. 3. The idea of encoding frames and valuations into a single structure has also been explored in Fagin (1985). 4. In Rautenberg (1983) a simple tableau calculus for S4.Grz is given which shows that S4.Grz consistency is effectively decidable, and that the decision procedure is primitive recursive. Furthermore, the size of a tableau is bounded by a function of the number o~P) of subformulas of P, or, more precisely, the theoremhood of P can be decided with a tableau of length -< 27+60(P). Given the proof, the same holds for A M , since the translation procedure reduces the size of a formula. So we have the same bound for A c. 5. In Gazdar et al. (1988) another logic is mentioned which arises from restricting the number of type 1 features to 1. The resulting logic is equal to S4.3.Grz = S4.Grz(<>p/X ~q . ~ .~ (p /X Oq) X/ O(q/X O p) X/ O(p /X q)), the logic of all finite linear orders, as can be seen in the same way. Since finitely generated S4.3.Grz models are finite, this logic is decidable as well. I will now proceed to the full case. Before I embark on the proof, let me remark on a few things. First, although each particular category structure contains only a finite number of features and values, L c contains infinitely many of them. As regards the type 0 features, this causes no problem, since we t r e a t f : a as a proposition and we allow ourselves infinitely many of those. However, type 1 features will create some problems that are not very serious but have to be dealt with carefully. Second, as we defined a translation of L c into modal logic, we will now define a translation of L c into elementary propositional dynamic logic (EPDL) so that every type 1 feature has a program associated with it