Bulletin of the Section of Logic Volume 34/4 (2005), pp. 195 201 Mladen Vuković HENNESSY MILNER THEOREM FOR INTERPRETABILITY LOGIC Abstract Interpretability logic is a modal description of the interpretability predicate. The modal system IL is an extension of the provability logic GL (Gödel Löb). We define bisimulations between generalized Veltman models, i.e. IL set-models. Then we consider some operations between models and prove that the operations are a special case of bisimulation. At the end we prove Hennessy Milner theorem for IL set -models. 1. Introduction The provability logic GL (Gödel-Löb) is a simple modal description of provability. The language of the provability logic contains propositional letters p 0, p 1,..., the logical connectives,,, and, and the unary modal operator. The symbol stands for. We use for falsity and for truth. The axioms of system GL are all tautologies, (A B) ( A B), ( A A) A. The inference rules of GL are modus ponens and necessitation A/ A. The system GL completely expresses provability predicate of Peano arithmetic. The interpretability logic IL is an extension of provability logic with a new modal binary operator. This operator stands for interpretability, considered as a relation between extensions of a fixed theory. The new axioms of the system IL are (A B) (A B), ((A B) (B C)) (A C), ((A C) (B C)) ((A B) C), (A B) ( A B) and A A.
196 Mladen Vuković The paper [4] provides the necessary definitions and detailed explanation and gives several examples of interpretations. We are only interested in IL as a system of modal logic. We introduce our notation and some basic facts, following [4]. Now we give the definition of generalized Veltman semantics. An ordered triple (W, R, {S w : w W }) is called an IL set -frame or generalized Veltman frame if we have: a) (W, R) is a GL-frame, i.e. W is a non-empty set, and R is a transitive and reverse well-founded relation on W ; b) Every w W satisfies S w W [w] P(W [w])\{ }, where W [w] denotes the set {u : wru}; c) The relation S w is quasi-reflexive: for every w W, i.e. wru implies us w {u}; d) The relation S w is quasi-transitive: for every w W, i.e. if us w V and ( v V )(vs w Z v ) then us w ( v V Z v ); e) If wrurv then us w {v}; f) If us w V and V Z W [w] then us w Z. An ordered quadruple (W, R, {S w : w W }, ) is called an IL set - model or generalized Veltman model, and denoted by W, if we have: (1) (W, R, {S w : w W }) is an IL set -frame; (2) is a forcing relation between elements of W and formulas of IL, which satisfies the following: (2a) w and w are valid for every w W ; (2b) commutes with the Boolean connectives; (2c) w A if and only if u(wru u A); (2d) w A B if and only if u((wru & u A) V (us w V & ( v V )(v B))). There are other semantics for interpretability logic. The best known one is Velman semantics. In [5] we have proved that in some cases the generalized Veltman models better distinguish principles of interpretability than Veltman models. The IL set -models are used in [2] for determining of characteristic class of some principles of interpretability, too. In the following text we denote by W an IL set -model (W, R, {S w : w W }, ) and by W we denote an IL set -model (W, R, {S w : w W }, ).
Hennessy Milner theorem for interpretability logic 197 Let we note that we use the same symbol for forcing relation in different models. 2. Bisimulations Bisimulation is a basic relation between two models; see [1], Chapter 2, Section 2. We have defined a notion of bisimulation between two generalized Veltman models in [6]. We modify this definition. Let W and W be two IL set -models. If u W, w W and Z W W then we denote by: [u] w,z := {u w R u and uzu }, [[u]] w,z is the set of all functions f : [u] w,z P(W [w ]) such that, for all u [u] w,z we have u S w f(u ). Definition 1. Let W and W be two IL set -models. We say that a relation Z W W is a bisimulation between models W and W if the following conditions are satisfied. (at) if wzw then the nodes w and w satisfy the same propositional letters. (zig) if wru and wzw, then, for all f [[u]] w,z, there is a set V such that us w V and ( v V ) ( u [u] w,z)( v f(u )) vzv. (zag) if w R u and wzw then, for all f [[u ]] w,z, there is a set V such that u S w V and ( v V ) ( u [u ] w,z 1)( v f(u)) vzv, where Z 1 is a converse of the relation Z. If Z W W is a bisimulation between models W and W then we write Z : W W. We will write W, w W, w if there is a bisimulation Z : W W such that wzw.
198 Mladen Vuković Let W and W be two IL set -models. Let w W and w W be two nodes. We say that nodes w and w are modally equivalent and write w w if w ϕ if and only if w ϕ, for all formulas ϕ. Let we have W, w W, w. It is easy to see by induction of the complexity of formula that then we have w w. It is easy to check that the following proposition is true. Proposition 2. Let W, W and W be IL set -models. a) The relation Z = {(w, w) : w W } is a bisimulation. So, W, w W, w. b) If Z is a bisimulation between models W and W then Z 1 = {(w, w) : wzw } is a bisimulation between models W and W. c) If Z : W W and Z : W W then Z Z : W W. d) If {Z i : i I} is a set of bismulations between W and W then the union Z i is a bisimulation. There is a maximum of all bisimulations between W and W. Definition 3. (cf. [1], Chapter 2, Section 1) a) Let {W i : i I} be a set of IL set -models and W i = (W i, R i, {S w (i) : w W i }, ). Suppose that the sets W i are mutually disjoint. It is easy to define W i and check that this is an IL set -model. It is called disjoint union and we write W i. b) Let W be an IL set -model and w W. Let W be the set W [w ] and R = R W. It is easy to check that W = (W, R, {S w : w W }, ) is an IL set -model. The model W is called generated submodel of the model W. c) Let W and W be IL set -models, and let f : W W be a function that satisfies the following conditions. (i) The nodes w and f(w) satisfy the same propositional letters. (ii) If wrv then f(w)r f(v). (iii) If us w V then f(u)s f(w) f(v ). (iv) If f(w)r v then there is a node v W such that wrv and v = f(v).
Hennessy Milner theorem for interpretability logic 199 (v) If f(u)s f(w) V then there is a set V W [w] such that us w V and V = f(v ). We say that the function f is bounded morphism from W to W. If a bounded morphism f is surjective then we write f : W W. Straightforward proof of the following proposition is left to the reader. Proposition 4. a) Let W i be disjoint union of IL set -models. Then we have W i, w W, w, for each i I and w W i. b) Let W be a generated submodel of W. Then we have W, w W, w, for each w W. c) Let f : W W be a surjective bounded morphism between IL set - models W and W. Then we have W, w W, f(w), for each w W. 3. Hennessy Milner theorem In the following theorem we prove that two modally equivalent nodes will be bisimular if models possess some special property. We say that an IL set model W is image finite if the set W [w] is finite for all w W. In [3], the Hennessy Milner theorem is proved for Veltman semantics. Now we prove the theorem for generalized Veltman semantics. Theorem 5. Let W and W be two image finite IL set -models. Than there is a bisimulation between models W and W. Moreover, if w w then W, w W, w. Proof. We prove that the relation of modal equivalence is a bisimulation between models W and W. The condition (at) from the definition of bisimulation is trivially fulfilled. We check that the relation possesses the property (zig). Let w, u W and w W be nodes such that wru and w w. Let we suppose the condition (zig) is not satisfied for the nodes w, u, w, and a function f [[u]] w,z. Let n be a natural number such that W [w ] = {u 1,..., u n}. Let s be a natural number such that s n and [u] w,z = {u 1,..., u s}. Then we have u i S w f(u i ), for all i {1,..., s}. Now we define a formula ϕ. If we have s = n then we define ϕ =. If we have s < n then u u i, for all i {s + 1,..., n}. There is a formula ϕ i
200 Mladen Vuković such that u ϕ i and u i ϕ i, for all i {s + 1,..., n}. Then we have u ϕ s+1... ϕ n and u i ϕ s+1... ϕ n, for all i {s + 1,..., n}. Now we define ϕ = ϕ s+1... ϕ n. So, if s < n we have u ϕ u i ϕ, for all i {1,..., s} ϕ, for all i {s + 1,..., n}. u i By the assumption of the theorem the set P(W [w]) is finite. There are only finitely many sets V W [w] such that us w V. Let V 1,..., V m be all sets V such that us w V. Then we have ( i {1,..., m})( v i V i )( j {1,..., s})( v f(u j)) v i v. For each i {1,..., m} let we choice v i V i such that ( j {1,..., s})( v f(u j)) v i v. For each j {1,..., s} the set f(u j ) is finite. Let us choose a natural number p(j) such that f(u j ) = {u j1,..., u jp(j) }. Then we have ( i {1,... m})( j {1,..., s})( k {1,..., p(j)}) v i u jk. Let ψ ijk be a formula such that v i ψ ijk and u jk ψ ijk. Let we define ψ = m s p(j) i=1 j=1 k=1 ψ ijk. It is easy to see that f(u j ) ψ for all j {1,..., s}, and V i ψ for all i {1,..., m}. If x W is a node such that w R x and x ϕ then there is some j {1,..., s} such that x = u j. But we know u j S w f(u j ) and f(u j ) ψ. So, we have w ϕ ψ. If V W [w] such that us w V then there is some i {1,..., m} such that V = V i. But V i ψ for all i {1,..., m}. So, w ϕ ψ. This is impossible because we have supposed w w.
Hennessy Milner theorem for interpretability logic 201 In the similar way we can check that the relation possesses the property (zag). Remark. We want to mention that all results will be true if we take the definition of the IL set -models as in the paper [2], i.e. instead the conditions d) and f) of our definition one can take the following condition if us w V then for all v V we have that if v Y and vs w Y then us w Y. Acknowledgments. Author would like to express his gratitude to Professor A. Visser for valuable comments. References [1] P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, Cambridge University Press, 2001. [2] E. Goris, J. Joosten, Modal Matters of Interpretability Logic, Logic Group Preprint Series 226, Department of Philosophy, University of Utrecht, 2004, http://preprints.phil.uu.nl/lgps/?lang=en [3] R. de Jonge, IL modellen en bisimulaties, preprint X-2004-06, ILLC, Amsterdam, 2004, http://www.illc.uva.nl/publications/ [4] A. Visser, An overview of interpretability logic, [in:] Kracht, Marcus (ed.) et al., Advances in modal logic. Vol. 1. Selected papers from the 1st international workshop (AiML 96), Berlin, Germany, October 1996, Stanford, CA: CSLI Publications, CSLI Lect. Notes. 87, pp. 307 359 (1998) [5] M. Vuković, The principles of interpretability, Notre Dame Journal of Formal Logic, 40 (1999), pp. 227 235 [6] M. Vuković, Characteristic classes and bisimulations of generalized Veltman models, Grazer Mathematische Berichte, 341 (2000), pp. 7 16. University of Zagreb Croatia