Axiomatisation of Hybrid Logic

Transcription

1 Imperial College London Department of Computing Axiomatisation of Hybrid Logic by Louis Paternault Submitted in partial fulfilment of the requirements for the MSc Degree in Advanced Computing of Imperial College London September 2008

2 Abstract In this report, we study the hybrid logics with nominals and We recall ten Cate s method to simulate hybrid logic using modalities, in what he calls nice frames, and we show that the hybrid logic of a class of frames is the modal logic of the class of its corresponding nice frames. Then, we expand this idea by considering the closure of the class of nice frames under disjoint unions, bounded morphic images, ultraroots and generated subframes, and we characterise this new class of frames. Using these notions, we axiomatise the hybrid logic of any elementary class of frames. As last, we study stable logics, which are hybrid logics axiomatised by modal axioms together with hybrid axioms common to any hybrid logic. We show that the hybrid logic of any elementary modally definable class of frames, or of any elementary class of frames closed under disjoint unions, bounded morphic images, ultraroots and generated subframes, are stable. Acknowledgements I am very grateful to my supervisor Ian Hodkinson for his help, and for all the time he spent with me during this project, and during a previous project that led to this one. 2

3 Introduction Hybrid Logic In modal or tense logic, one can deal with worlds by the mean of operators such as and, but it is not possible to refer directly to a world. With an analogy between tense and natural languages, one can say it will rain (for example with a formula such as r, where r means rain), but one cannot say on the 6 th of October 1969, it was raining. Arthur Prior was the first, in the 1960s, to express this idea in what would become hybrid logic. He introduced nominals, which are propositional variables that are true in exactly one world, and operators T and. T(i, φ) means that φ holds in the world named by i. It will be later i φ, and we will use this notation. Operator is an universal quantifier over nominals. With these tools, one is now able to express that it rained at a certain date: if i represents the point 6 th of October 1969, and r still represents rain, one can i r to express that it was raining on the 6 th of October Another operator has been introduced later:. It was discovered at the same time by several authors, and allows us to name the current world. Used together this operator will be very useful to come back to a world we already saw. Hence, a lot of work has been carried out about these binders. Hybrid logic is a technical improvement compared to modal logic. It is more expressive, which means that some hybrid formula have no modal equivalent, whereas every modal formula has an hybrid equivalent. Hybrid languages can also express more classes of frames. For example, although there is no modal formula that is valid in a frame if and only if this frame is irreflexive, we can build such an hybrid formula. One application of hybrid logic to computing is the evaluation of queries and constraints on semistructured data. For example, it is shown in [4] that the answer set of a query in such a data corresponds to the truth set of a certain hybrid formula. Therefore, model checking of hybrid formulas can be applied to semistructured data. Axiomatisation In this report, we focus on the axiomatisation of hybrid logic with nominals and That is, given a class of frames, is it possible to find a set of axioms that generates the logic valid in the considered class? Different approaches have been carried out to try to solve this problem. For example, we can find in [2] axiomatisation for the hybrid logic with and, etc. But all these axiomatisations only deal with the class of all frames. In [1], the authors give a way to axiomatise the hybrid logic of a class of frames characterised by a set of pure formulas (i.e. formulas containing no propositional variables, but only nominals). Here we have an improvement compared to the previous result, in the sense that we can axiomatise any class of frames definable by pure formulas. However, the axiomatisation is done from scratch: even though hybrid logic is based on modal logic, this axiomatisation ignores this. An other flaw is that they need extra rules to make their axiomatisation complete. An interesting result can be found in [11]. In this article, a theorem states that any class of frames defined by Sahlqvist formulas can be axiomatised in a smart way, in the sense that the axiomatisation reflects the fact that hybrid logic is based on modal logic. As we can see, there is a gap in these results. We know that we can axiomatise the class of all frames, any class of frames axiomatised by modal Sahlqvist formulas, any class of frames defined by 3

4 pure sentences (if we add extra rules). But we would like, first, to be able to axiomatise wider classes of frames, and then, to try to separate, in an axiomatisation, the hybrid part from the modal part. That is, we would like to know when it is possible to make the axiomatisation reflects the fact that hybrid logic is based on modal logic, and when we cannot. A first step will be presented as stable classes, in chapter 5. Plan In the first chapter of this report, we recall the definitions and some basic properties about the notions we will use, that is syntax and semantics of hybrid and modal logic, and some frame constructions. Then, in chapter 2, we give ten Cate s method to simulate nominals and hybrid operators using modalities. This method is mainly the definition of nice frames. A problem with these frames is that the valuation of nominals is bound to these frames, thus the validity of a formula in such a frame does not consider every possible valuation. We introduce the hybrid extension of a class of frames to solve this problem, and we show that the hybrid logic of a class of frames is the modal logic of its hybrid extension. We also show that this hybrid extension is elementary if and only if the original class was elementary. Another problem with these nice frames is that the class of nice frames is not closed under bounded morphic images, generated subframes, disjoint unions and ultraroots. We study in chapter 3 the closure of these nice frames under these operations, which happens to be the class of frames that validates the modal logic that is true in the class of nice frames. We also give a characterisation of these frames. In chapter 4, using the nice frames we defined before, we axiomatise the hybrid logic of any elementary class of frames. Then, in chapter 5, we study stable logics, i.e. hybrid logics that are axiomatised by modal logic together with hybrid axioms defining hybrid logic. In these logics, hybrid operators and nominals do not bring anything more (regarding to axiomatisation) than their own definition. We also study Mc Kinsey and Hughes s logics from this point of view, and we show that their hybrid logics might be as interesting as their modal logics were. At last, we tried to gather most of the symbols we use in this report in the index, that can be found page 53. 4

5 Contents Abstract Acknowledgements Introduction Syntax and Semantics of Hybrid Logic Modal Logic Hybrid Logic Frame Constructions Nominals as Modalities Nice Frames Hybrid Extension of Modal Classes Fairly Nice Frames Nice Frames and Fundamental Frames Constructions Preservation Under the Four Operations Closure of Nice Extensions Under the Four Operations Axiomatising the Elementarily Generated Hybrid Logics 29 5 Stable Classes An Attempt to Characterise Stable Classes Mc Kinsey and Hughes s Logics Minimal Stable Classes Conclusion 45 Achievements Further Work A Arbitrary Frames 48 A.1 Similarity Type A.2 Syntax A.3 Semantics B Ultraproducts 50 Bibliography 52 Index 53 List of Figures 54 5

6 Chapter 1 Syntax and Semantics of Hybrid Logic Sections 1.1 and 1.2 of this chapter have been copied with a few modifications from my ISO. Moreover, the definitions and explanations about modal logic are mainly inspired from [9] and [1]. Before studying its axiomatisation, let us formally define what hybrid logic is. Since it is built on modal logic, we begin by giving the syntax and semantics of this logic. Then, we will precisely explain what we mean by hybrid logic. At last, we will give the definition of some class constructions that will be used in this report. 1.1 Modal Logic We give here the syntax and semantics of frames having only one relation. But frames can have several relations, with other arities than two. These frames are described in appendix A Syntax Let us give the syntax of modal logic. Given an infinitely countable set V = {p, q, r...} of propositional symbols, a well formed formula is built by induction with the following rules. φ::=p φ ψ φ φ Other operators are treated as abbreviations. For all formulas φ and ψ: Semantics def φ ψ def ( φ ψ) φ def φ φ ψ def φ ψ Definition A Kripke frame is a pair F = W, R, where: W is a non-empty set of worlds; R is a binary relation on W. From now on, we will use the word frames instead of Kripke frames. We will sometimes talk about state rather than world. These two words refer to the same object. Before defining a model, let us define a valuation. 6

7 1.1. MODAL LOGIC Definition A valuation is a map from the set of propositional variables V to the set P(W) of the subsets of W. V : V P(W) The image of a propositional variable by a valuation is interpreted as the set of worlds in which this variable is true. Now we can define a model. Definition A model is a pair M = F, V where F = W, R is a frame, and V is a valuation from V to the worlds of F. We can also write M = W, R, V. Let us define what the truth of a formula means. Notation The validity in modal evaluation will be noted = m. Definition M, w = m p iff w V (p) M, w = m never M, w = m φ iff not M, w = m φ M, w = m φ ψ iff M, w = m φ or M, w = m ψ M, w = m φ iff for all world v W such that Rwv, we have: M, v = m φ Corollary A consequence of the previous definition is the natural meaning of the abbreviations. M, w = m always M, w = m φ ψ iff M, w = m φ and M, w = m ψ M, w = m φ iff there exists a world v W such that Rwv and M, v = m φ M, w = m φ ψ iff if M, w = m φ, then M, w = m ψ Validity and Satisfiability Let us now define the validity and satisfiability of formulas of modal logic. Definition A formula φ is valid if φ is true in any world for any model: M, w M, w = m φ Definition A formula φ is valid in a frame F if this formula is true in every world, for every model of this frame. For all valuation V and world w W F, V, w = m φ Definition A formula φ is valid in a model M = W, R, V if this formula is true in every world of this model. w W M, w = m φ Definition A formula φ is satisfiable if there exists a model M = W, R, V and a world w W such that φ is true in w in M. M, w M, w = m φ Definition Formulas φ and ψ are equivalent if for every model M, for every world w of M, we have: M, w = m φ iff M, w = m ψ We will write: φ ψ 7

8 CHAPTER 1. SYNTAX AND SEMANTICS OF HYBRID LOGIC 1.2 Hybrid Logic Hybrid logic is modal logic with nominals and operators. Nominals are propositional variables which are true in exactly one world. In other words, a nominal names a world. Then, having these new objects, we can add some operators, that we will describe below. There exists several operators for hybrid logic, but when we deal with this logic, we generally only consider a subset of those operators. Even with no operators (only nominals), hybrid logic can express more than modal logic. Then, adding operators can give even more expressivity, but it is not always the case, as some combination of operators are powerful enough to express other ones Operators The main operators that are used in hybrid logic and. However, in this report, we will mainly study the basic hybrid language H(@), i.e. the hybrid language with nominals and only. We will sometimes use and, but is only given here to show that hybrid logic is not limited to the only and. We mainly study this language H(@) because good completeness theorems exist for elementary classes with,, but not for this one. Operators and will be used as a tool, but the main results we prove Down arrow The down arrow allows us to name a world. The formula i φ binds the nominal i to the world where this formula is evaluated. For example, let us evaluate the formula i i in a world w. i binds i to w, so i is the name of w. Hence, i means that the world denoted by i is accessible, i.e. w is accessible from itself. Therefore, the validity of this formula on a frame means that the relation of this frame is reflexive. operator allows us to move to another world. The i φ is true if φ is true in the world denoted by i. For example, the i j means that the world denoted by i has the world denoted by j as a successor. Exists and for all and For all allows us to express that a formula is true for every nominal. For example, suppose the following formula is true in a world w: i i. This formula means that for every nominal i, i is true; i.e. for every nominal i, the world denoted by i is accessible from w. So, if this formula holds in a world w in a model, this implies that every world is accessible from w Syntax We introduce N, which is a countably infinite set of nominals. We will use the letters i, j... to talk about nominals. One can deal with several hybrid languages, depending on the operators one uses. Here are the rules common to every hybrid languages. p denotes a propositional variable (p V), i a nominal (i N), and φ and ψ formulas. φ::= p i φ ψ φ φ Once again, the usual abbreviations (,,, and ) are introduced. Then, if we want to use operators, we can add the following rules. i φ 8

9 1.2. HYBRID LOGIC Exists, for all, φ::= iφ iφ Note that it is also possible to use as the abbreviation of. Definition Let O {@,,, } be any set of operators. (i) We call H(O) the hybrid language using the operators O. (ii) A H(O)-formula is a formula of the language H(O). (iii) A H(O)-logic is a logic of the language H(O). As I already said, in this report, we will mainly use H(@) and H(, ). So, if we do not specify in which language relies an hybrid formula or logic, it will usually be in the language H(@). Definition We recall that a sentence is a formula that does not contain free propositional variables or nominals. Definition A formula that does not contain any propositional variables, but only nominals is called a pure formula Semantics Frames A frame in hybrid logic is defined the same way it was defined for modal logic: it is a set of worlds W together with an accessibility relation R over these worlds. Nominals In hybrid logic, a model has nearly the same definition it had in modal logic. We simply have to say that nominals are true in only one world. We will do this by giving a new definition to the valuation function. We define V as a map from the set of propositional variables and nominals to the set of the subsets of the worlds W. V : V N P(W) Then, we impose a restriction on V, which is that the image of a nominal must be a singleton. i N V (i) = 1 Semantics of operators Let us define what does truth mean for a formula containing the hybrid and. The hybrid evaluation will be noted = h. To define the semantics of and, we need the following definition. Definition Given two valuations V and V, and a nominal i, the following formula (pronounced V is an i-variant of V ) means that V agrees with V on all arguments save possibly i. V i V Definition Let F = W, R be a frame, where W is a set of worlds and R a binary relation over these worlds, and V be a valuation function. F, V, w = i φ iff F, V, v = h φ, where V (i) = {v} F, V, w = h iφ iff V such that V i V, we have F, V, w = h φ F, V, w = h iφ iff V such that V i V, we have F, V, w = h φ 9

10 CHAPTER 1. SYNTAX AND SEMANTICS OF HYBRID LOGIC Validity and Satisfiability The definitions of validity and satisfiability for an hybrid formula are the very same definitions we had for modal logic Examples Let us give some examples of use of hybrid logic formulas. We say that a formula is associated with a frame property if we have: the property holds in the frame iff the formula is valid in the frame. Some frame properties (like reflexivity or transitivity) can be associated with modal formulas. On the other hand, we can find hybrid formulas associated with conditions (like irreflexivity) that are impossible to associate with modal formulas. Here are some examples. On the left hand side, we give some properties for which we can find modal formulas that hold if and only if the frame has the property. On the right hand side, we give some conditions that cannot be associated with any modal formula. For each of these properties, we give an associated hybrid formula. Some properties that can be associated with modal formulas Some properties that cannot be associated with modal formulas reflexivity i i irreflexivity 1 i i symmetry i i antisymmetry 2 i ( i i) transitivity i i universality 3 i density i i exactly one world i determinism i i at most two i ( j j k 1.3 Frame Constructions We introduce here a few frame constructions that are widely used in this report. The frames we consider here have several modalities, that can be either unary or binary. In this section, W, (R i ), (S i ) will refer to the frame composed of worlds W, binary relations (R i ) i r (r N {ω}) and unary relations (S i ) i s (s N {ω}). See appendix A to see how to deal with frames with an arbitrary number of relations. We do not need, in this report, to study the general case of frames, since we will only use basic frames W, R having one binary relation, and non-standard frames W, R, (R i ), (S i ) (this will be defined in section 2.1), having an infinite number of binary relations R and R i (i N), and an infinite number of unary relations S i (i N). However, in the following definitions, we do not make a difference between relation R and relations R i because, regarding to the definition of these frame constructions, this difference is irrelevant. The definitions and properties in this section also apply for basic frames having only one binary relation. In the following definitions, if we do not precise it, F (respectively F ) is a frame with worlds W (resp. W ), a (possibly infinite) set of r relations (R k ) k r (resp. (R k ) k r), and a (possibly infinite) set of s relations (S k ) k s (resp. (S k ) k s). We say that a frame is weakly connected if for any couple of worlds, there exist a path going from one world to the other, using any relation and its converse. 1 x, y, if Rxy then Ryx 2 x, y, if Rxy and Ryx then x = y 3 x, y Rxy 10

11 1.3. FRAME CONSTRUCTIONS Definition Let F = W, (R i ), (S i ) be a frame. We say that F is weakly connected if for all worlds x and y of W, there exists a sequence of m worlds (w i ) 0 i m 1, where w 0 = x, w m 1 = y, and for all 0 i m 1, there exists a k such that R k w i w i+1 or R k w i+1 w i. Remark One can find in literature the definition of connected frames, which is stronger than the previous definition of weakly connected frame. But as we will only use weakly connected frames, we will talk about connected frames to refer to weakly connected frames. Definition Subframe Let F be a frame. A frame F is a subframe of F if: (i) W W (ii) R k = R k W W for all k r (iii) S k = S k W for all k s Definition A connected component of a frame is a maximal (with respect to inclusion) connected subframe. We now define the four fundamental frame constructions, together with a slightly different version of the generated subframe. Definition Disjoint union Let I be a set, and (F i ) i I be a (possibly infinite) set of frames, each frame F i having a set of worlds W i, a (possibly infinite) set of r binary relations (R i,k ) k r, and a (possibly infinite) set of s unary relations (S i,k ) k s. The disjoint union i I F i of these frames is the frame F = W, (R k ) k r, (S k ) k s, where: W = i I W i {i} For all k r and (u, x), (v, y) in W, we have R k (u, x)(v, y) iff x = y and R x,k (u, v) For all k s and (u, x) in W, we have S k (u, x) iff S x,k u Definition Bounded morphism Let F and F be two frames, and f a function from W to W. The function f is said to be a bounded morphism if the following conditions hold, for any k r and l s: For all x, y W, R k xy implies R k f(x)f(y), and S lx implies S l f(x). For all x W, y W, if R k f(x)y then there exists y W such that f(y) = y and R k xy. For all x W if S l f(x) then S lx. Definition Bounded morphic image Let F and F be two frames. F is called a bounded morphic image of F if there exists a bounded morphism f from W onto W. Some definitions about ultrafilters and ultraproducts, necessary to understand the following definition, are given in appendix B. Although the definition of ultraroots is also given in the appendix, we give it here as well. Definition Ultraroot Let F and F be two frames. F is an ultraroot of F if there exist an ultrafilter U build upon the set I such that: 11

12 CHAPTER 1. SYNTAX AND SEMANTICS OF HYBRID LOGIC The set W is the ultrapower (W ) I /U For any k r, for all worlds x = (x n ) n I /, y = (y n ) n I / of W, we have: R k (x, y) iff {n I; R k x ny n } U For any k s, for all world x = (x n ) n I / of W, we have: Definition Generated subframe S k x iff {n I; S k x n} U (i) Let F be a frame. A subframe F of F is a subframe of F generated by the subset M W if: (a) W is defined as follows. M W W for all k r, x W, y W, if R k xy then y W W is the smallest set matching these conditions. (b) R k = R k W W for all k r (c) S k = S k W for all k s (ii) F is a generated subframe of F if it is a subframe of F generated by some subset of W. (iii) We say that F is a connected generated subframe of F if it is a generated subframe of F and it is connected. Remark A point-generated subframe (i.e. a subframe generated by a single world) is a connected generated subframe as well. These definitions of frame construction lead to the following operations over classes of frames. Definition Let K be a class of frames. (i) Ud K is the class of frames that are the disjoint union of frames of K. (ii) H K is the class of bounded morphic images of frames of K. (iii) Ru K is the class of ultraroots of frames of K. (iv) S K is the class of generated subframes of frames of K. (v) Sc K is the class of connected generated subframes of frames of K. Disjoint union, generated subframe, ultraroot and bounded morphic images will be referred as the four fundamental frame constructions, or the four fundamental operations. Lemma The four fundamental frame constructions preserve modal validity. One can found, for example, in theorem 3.14 (page 139) of [1] a proof of this result for disjoint union, bounded morphic image and generated subframe, and proposition 2.71 (page 105) of the same book can be easily extended to prove this result for disjoint union. 12

13 Chapter 2 Nominals as Modalities Ten Cate et al. have proposed, in [11] for example, to treat nominals as modalities. This allows us to apply some theorems of modal logic to hybrid logic. However, as hybrid logic is more powerful than modal logic, this trick has some limitations. The more obvious one is that nominals, being modalities, do not have their value defined by the valuation function but by the relation of the frame. Thus, nominals are now part of the frames, whereas in hybrid logic they belong to the models. In this chapter, we will first define properly how to consider nominals as modalities, which will mainly be the definition of nice frames (section 2.1). Then, in section 2.2, we will show that when considering the nice frames built over a class of frames by treating all the possible valuations of the nominals (which will be called the hybrid extension), we keep some useful properties, such that: the hybrid logic of a class of frames is the modal logic of its elementary extension (theorem 2.2.5), and a class of frame is elementary iff its hybrid extension is elementary (theorem ). 2.1 Nice Frames In hybrid logic, value of nominals is set by the valuation function. The idea of ten Cate et al. was to simulate nominals in modal logic by treating them as modalities. We explain in this section how to perform this simulation. Most of the definitions and lemmas of this section are taken from section 3 of [11]. We recall that N is the set of all nominals. Definition (from paragraph 3 of [11]) A non-standard frame is a frame of the form F = W, R, (R i ) i N, (S i ) i N. The semantics associated to these frames is: M, w = m i iff w S i M, w = i φ iff w (wr i w = M, w = m φ) From now on, a standard frame will refer to frame with a single relation R, whereas a frame may refer to a standard or non-standard frame. Notation The evaluation of a formula in the hybrid semantics is noted = h, whereas its evaluation in the non-standard semantics (which is defined in definition above, where nominals are considered as modalities) is noted = m, as it is a modal evaluation. Notation In order to get notations that are easier to read, we will now write W, R, (R i ), (S i ) instead of W, R, (R i ) i N, (S i ) i N. Since we will not consider any set of nominals different from N, this will not lead to misunderstanding. 13

14 CHAPTER 2. NOMINALS AS MODALITIES In the definition of a non-standard frame, nothing says that the relations (R i ) and (S i ) represent nominals, i.e. that R i leads to the world named by i, and that S i is a singleton containing this world. The following definition introduces such a condition. Definition (from paragraph 3 of [11]) A non-standard frame is said to be nice if for each i N, S i is a singleton and xy(r i xy S i y). The following lemma shows that evaluating a formula in a nice frame is, in a certain sense, equivalent to evaluating it in the hybrid valuation. Lemma (labelled lemma 3.2 in [11]) Let M m = W, R, (R i ) i N, (S i ) i N, V m be a nice model, and M h = W, R, V h (where V h = V m {(i, S i ) i N }) be its corresponding hybrid model. Then, for any world w W and hybrid formula φ, we have: M h, w = h φ iff M m, w = m φ Definition (from paragraph 2 of [11]) We call the following set of axioms, where i and j are any nominals, and p is any propositional variable. i i p i i p (elimination) (@ i p i) p i i i i p Each of these axioms is a Sahlqvist formula or equivalent to a conjunction of Sahlqvist formulas. Thus, each of them has a first order correspondent, which is given here (I took them from section 3, page 296 of [11]). (agree) xyz(r j xy R i yz R i xz) (propagation) xyz(rxy R i yz R i xz) (elimination) x(s i x R i xx) (ref) xy(r i xy S i y) (self-dual) x!y(r i xy) Lemma (labelled lemma 3.3 in [11]) A point-generated non-standard frame F is nice iff F = h. We will use this similar lemma as well. Lemma A nice frame validates Proof. The proof is easy, considering the first order correspondents of formulas of. Before studying this new kind of frames we just introduced, let us give some notations and conventions. Notation The class of all standard frames is noted M. Notation Let K be a class of frames. We call ML(K) (resp. HL(K)) the modal (resp. hybrid) logic of K. In other words: ML(K) = {modal formulas ϕ; K = m ϕ} HL(K) = {hybrid formulas ϕ; K = h ϕ} 14

15 2.2. HYBRID EXTENSION OF MODAL CLASSES 2.2 Hybrid Extension of Modal Classes One problem we have with the previous definition of nice frames is that the nominals jump from the model to the frame. Thus, if we say that a formula is valid in a nice frame, we only consider the particular valuation on nominals defined by the modalities (R i ) and (S i ) of this nice frame. In this section, as an attempt to solve this problem, we define the hybrid extension of a class of standard frames. Having a standard frame F, instead of considering a nice frame based on F, representing only one possible valuation on the nominals, we will consider the class of all nice frames based on F. Thus, when dealing with this latter class, we will consider, at the same time, all the possible valuations on the nominals in F. Then, to build the hybrid extension of a class K of standard frames, we apply the previous operation over each of the frames of the class K. This will be described more precisely in section As we will see in the sections and 2.2.3, this construction is sound in the sense that it preserves some class properties Definition In this section, we define the hybrid extension, as well as the modal reduct, which allows us to perform the inverse operation, i.e. to retrieve the standard frame which a non-standard one was built over. Definition Let K be a class of standard frames (with an unique binary modality R). The hybrid extension of K is the class H(K) of nice frames that can be built over K. H(K) = {nice frames W, R, (R i ) i N, (S i ) i N such that W, R K} Definition Let F = W, R, (R i ) i N, (S i ) i N be a non-standard frame. The modal reduct of F, noted F M is the standard frame F M = W, R Validity of Formulas in the Hybrid Extension The goal of this section is to prove that for any class of frames K, the hybrid logic of K and the modal logic of H(K) are equal. First, we extend the result of lemma to the validity in a model. Proposition Let M m = W, R, (R i ) i N, (S i ) i N, V m be a nice non-standard model, and M h = W, R, V h (where V h = V m {(i, S i ) i N }) be its corresponding hybrid model. For any hybrid formula φ, we have: M m = m φ iff M h = h φ Proof. The proof is straightforward. ( ) Suppose M m = m φ. Let w be a world of M h (which is a world of M m as well). As φ is valid in M m, we have M m, w = m φ. By lemma 2.1.5, we get M h, w = h φ. So, φ is valid in M h. ( ) The proof is exactly the same. Proposition Let K be a class of frames, and ϕ be a formula. K = h φ iff H(K) = m φ 15

16 CHAPTER 2. NOMINALS AS MODALITIES Proof. ( ) Suppose K = h φ. Let F = W, R, (R i ) i N, (S i ) i N be a frame of H(K), and V a valuation over F. By definition, W, R K, so W, R, V {(i, S i ) i N } = h φ. By proposition 2.2.3, F, V = m φ. Therefore, H(K) = m φ. ( ) Suppose H(K) = m φ and K h φ. So, there exists a frame W, R K, and an hybrid valuation such that W, R, V h φ. We built the corresponding nice model W, R, (R i ) i N, (S i ) i N, V of W, R, V the following way. S i = {u W : V (i) = {u}} R i = {(u, v) : u W and v S i } V = V V (remind that V is the set of propositional variables). In this line, we remove the valuation of nominals. Hence, by proposition 2.2.3, W, R, (R i ) i N, (S i ) i N, V m φ. Eventually, as by definition, we have W, R, (R i ) i N, (S i ) i N H(K), we get H(K) m φ, which is a contradiction. Theorem For all class K of frames, the hybrid logic of K is the modal logic of H(K). ML(H(K)) = HL(K) Proof. This is a direct application of the previous proposition Elementary Classes We will now prove that a class K is elementary (defined below) if and only if H(K) is elementary. In this section, all the evaluations are first-order evaluations, and will be noted = f. Definition A class of frames is elementary if it is defined by a set of first-order formulas (this set is called a theory). Lemma Let K and L be two classes of frames. H(K) H(L) iff K L Proof. The right to left direction is immediate. Suppose H(K) H(L). Let F be a frame of K. By definition of H(K), for all (R i ), (S i ) such that F, (R i ), (S i ) is nice, F, (R i ), (S i ) H(K) So, since H(K) H(L), for all (R i ), (S i ) such that F, (R i ), (S i ) is nice, F, (R i ), (S i ) H(L) Therefore, F L. Notation L(R) is the first order-language with relation symbol R. 2. L(R, (R i ), (S i )) is the first order-language with relation symbols R, R i and S i (for i N). We define an operation which converts a L(R, R i, S i ) sentence into a L(R) sentence. We will later see that for any formula φ, φ and φ are equivalent, in a sense that will be defined in lemma Definition Let φ be a L(R, R i, S i ) sentence. We define by induction: it commutes with any operator but R i and S i, and for those operators, we have: (R i (x, y)) = (S i (y)) = (i = y). Here, i is a new variable corresponding to the nominal i. We do not distinguish them notationally. We have nearly the same definition with a theory: we apply on each sentence in the theory, and we put universal quantifiers on every nominal occurring in the sentence, not to get free variables. 16

17 2.2. HYBRID EXTENSION OF MODAL CLASSES Definition Let T be a L(R, R i, S i ) theory. T is a L(R) theory, defined by the following formula. T = { ijk φ : φ T and i, j, k... are the nominals occurring in φ} We will now prove that φ and φ are equivalent (in a sense that will be defined in lemma ). Definition Let F = W, R, (R i ), (S i ) be a frame. An assignment h : {i, j, k...x, y, z...} W is nice with respect to F if {h(i)} = S i, for all nominals i. Lemma Let φ be a formula and h be a nice assignment, with respect to a nice frame F. We have: F, h = f φ iff F M, h = f φ Proof. We prove this by induction. The proof is obvious for every operator but (R i ) and (S i ). F, R, (R i ), (S i ), h = f S i (x) iff h(x) S i (by definition of S i ) iff h(x) {h(i)} (h is nice) iff h(x) = h(i) iff F, h = f x = i (definition of equality in first-order logic.) F, R, (R i ), (S i ), h = f R i (x, y) iff F, R i, S i,h = f S i (y) (F is nice, so S i (y) R i (x, y)) iff F, h = f y = i Lemma Let K be any class of frames. For any L(R, R i, S i ) sentence φ, we have (here again, ijk are the nominals occurring in φ): H(K) = f φ iff K = f ijk...φ Proof. ( ) Suppose H(K) = f φ and let F = W, R be a frame of K, and h be any assignment {i, j, k...x, y, z...} W. There exists an unique nice frame F = W, R, (R i ), (S i ) such that h is nice with respect to F. By hypothesis, F = f φ, so F, h = f φ and, by the previous lemma, F, h = f φ. φ is a sentence, so the only free variables in φ are the nominals, so, as h can be any assignment, we get F = f ijk φ. ( ) Suppose K = f ijk φ. Let F = W, R, (R i ), (S i ) be a frame of H(K). Let h be any assignment {i, j, k...} W such that h is nice with respect to F. F M = f ijk φ by hypothesis. F M, h = f φ F, h = f φ, by the previous lemma. F = f φ, since there is no free variables in φ. Corollary For any theory T, we have: H(K) = f T iff K = f T Proof. This is a straightforward application of the previous proposition. Theorem For any class of standard frames K, K is elementary iff H(K) is elementary. 17

18 CHAPTER 2. NOMINALS AS MODALITIES Proof. ( ) Suppose K = {F : F = f T }, and let K be the class of non-standard models of T = T H, where: H = { xy(r i (x, y) S i (y)), xy((s i (x) S i (y)) x = y), xs i (x)} i N Let us show that K = H(K). ( ) By definition, frames of H(K) are nice, so they validate H. By definition, K validates T, so, H(K) validates T as well (H(K) only adds relations to K, that do not occur in T). So H(K) K. ( ) Frames of K validate H, so they are nice. They validate T by definition. Let F = W, R, (R i ), (S i ) be a frame of K. K = f T, so F = f T. But formulas of T are first-order formulas that do not use R i and S i, so F M = f T as well. Thus, F M K, by definition, and, as F is nice, F H(K). ( ) Suppose H(T) is elementary. Let T be the theory of H(K), i.e. H(K) = {F : F = f T }. Let K be the standard models of T. Let us show that K = K. ( ) K validates T so, by previous corollary, H(K ) validates T. This means that H(K ) H(K). Therefore, by lemma 2.2.7, we get: K K. ( ) Let F be a frame of K. T is the theory of H(K), so H({F}) validates T. By the previous corollary, F validates T, so F is in K. 18

19 Chapter 3 Fairly Nice Frames The nice frames we described in the previous chapter are interesting because as they allow us to simulate hybrid logic using modal logic, one can apply theorems of modal logic to hybrid logic. But they have a big limitation. The four fundamental frame constructions (described in section 1.3), which preserve modal validity (i.e. a modal formula valid in a class of frame will be valid in the image of this class by any of these frame constructions), do not even preserve the niceness of a frame: the image of a nice frame by these operations may be not nice. For example, consider the frame F = {w},, (R i ), (S i ), which has a single world w, an empty relation R, and relations (R i ) and (S i ) defined by: R i = {(w, w)} and S i = {w}, for all i. We build the disjoint union of two copies of F, and we get F 2 = {w 1, w 2 },, (R 2 i ), (S2 i ), where R 2 i = {(w 1, w 1 ), (w 2, w 2 )} and S 2 i = {w 1, w 2 }, for all i. This frame is no longer nice; for instance, the (S 2 i ) are not singletons. In this chapter, we define fairly nice frames as the union of nice frames. These are interesting for several reasons. First, we will prove in lemma that a non-standard frame is nice iff it validates the axioms. This gives us a modal characterisation of these frames. Then, the class of fairly nice frames is the class of all frames that validate the H(@)-logic of the class of nice frames (in the non-standard sense, i.e. by considering nominals and hybrid operators as modalities). So, since we want to use the hybrid logic with modally (so that we can use modal methods), we need to understand fairly nice frames. And, obviously, as we already said, they have some advantages on nice frames, and the class of fairly nice frames is the closure of nice frames under the four fundamental operations. In a first section, we study the behaviour of nice frames with the four fundamental frame construction. We then define (in section 3.2) fairly nice frames, and we show in the preservation lemma (lemma 3.2.4) how they behave with these frame constructions. At last, in section 3.3, we give some equivalent characterisations of fairly nice frames. 3.1 Nice Frames and Fundamental Frames Constructions In this part, we study how the four fundamental frame constructions behave with nice frames. This is important, as the fairly nice frames we will define in the next section are closely related to these frame constructions: they are built to behave well with them. Lemma A nice frame is weakly connected. Proof. Take any nominal i. Any two worlds x and y of a nice frame are related in two steps by the world named i: by definition of a nice frame, there exists a world z such that R i xz and R i yz hold. 19

20 CHAPTER 3. FAIRLY NICE FRAMES Proposition Niceness is preserved by bounded morphic images, generated subframes and ultraroots. Proof. H Bounded morphic image Let F = W, R, (R i ), (S i ) be a nice frame, and F = W, R, (R i ), (S i ) be the bounded morphic image of F by the bounded morphism f. We have to show that for all i, S i is a singleton, and for all x, y in W, R i xy iff S i y. By definition of a bounded morphism, for all x in W, we have: S i (x) iff S i (f(x)). So S i is not empty. Now, suppose there exist x and y in W such that S i (x ) and S i (y ). Let x and y be any antecedents of x and y : f(x) = x and f(y) = y. As f is a bounded morphism, we get S i (x) and S i (y). As F is nice, x and y are equal, and so are x and y. S i is a singleton. Suppose we have R i (x, y ), for x, y in W. As F is a bounded morphic image of F, there exists x in W such that f(x) = x. So, we have R i (f(x), y ). By definition of a bounded morphism, there exists y in W such that f(y) = y and R i (x, y). So, as F is nice, S i (y) holds, and S i (y ) holds as well. For the other direction, let x, y be in W, with S i (y ). By definition of a bounded morphism, there is y in W such that f(y) = y and S i (y). Let x be any antecedent of x : f(x) = x. As F is nice and S i (y) holds, we get R i (x, y). At last, we have R i (x, y ). S Generated subframe Let F = W, R, (R i ), (S i ) be a nice frame, and F = W, R, (R i ), (S i ) be a generated subframe of F (R = R W W, R i = R i W W, S i = S i W ). Let x be one of the worlds that generated F. As F is nice, there exists y in W such that R i (x, y) and S i (y). So, y is in W, and S i (y) holds. S i is not empty. Then, as S i = S i W, S i cannot have more worlds than S i, i.e. no more than one world. S i is a singleton. Suppose we have R i (x, y), for x and y in W. We have R i (x, y), thus, S i (y) holds. And as y is in W, S i (y) holds. For the other direction, let x, y be two worlds of W such that S i (y) holds. This means that S i(y) holds as well, so we get R i (x, y), and R i (x, y). Ru Ultraroot Niceness can be defined by first-order sentences, so, by Loś s theorem (see theorem B.1.9), it is preserved by ultraroots. Remark Niceness is (in general) not preserved by disjoint union. Proof. We give a counter example. Let F = {w},, (R i ), (S i ) be a nice frame with a single worlds and an empty relation R (for all i, R i = {(w, w)} and S i = {w}). Let F 2 = {w 1, w 2 },, (R 2 i ), (S2 i ) be the disjoint union of isomorphic copies of F. We have, for all i, R 2 i = {(w 1, w 1 ), (w 2, w 2 )} and S 2 i = {w 1, w 2 }. This frame is not nice (for instance, S 2 i is not a singleton). We now give some relations between H(XK) and X H(K), where X {Ud, Ru, H, S, Sc} and K is a class of standard frames. Lemma K being an arbitrary class of standard frames, we have the following relations. Proof. (i) H H(K) H(H K) (ii) Ru H(K) = H(Ru K) (iii) S H(K) = H(S K) (iv) Sc H(K) H(Sc K) 20

21 3.1. NICE FRAMES AND FUNDAMENTAL FRAMES CONSTRUCTIONS (i) H H(K) H(H K) Let G = W, R, (R i ), (S i ) be a frame of H H(K). There exist F H(K) and a bounded morphism f such that F is the image of G by f. By definition, F M is in K. G M is equal to f(f M ), and f is a bounded morphism from G M onto F M. Therefore, G = G M, (R i ), (S i ) is in H(H K). (ii) Ru H(K) = H(Ru K) ( ) Let G = W, R, (R i ), (S i ) be a frame of Ru H(K). There exist F H(K) such that G is an ultraroot of F. Obviously, G M is an ultraroot of F M. So, G M is in Ru K, and G belongs to H(Ru K). ( ) Let G = W, R, (R i ), (S i ) be a frame of H(Ru K), and F = W, R K be an ultrapower of G M (W = W I /U, where I is a set, and U an ultrapower on I). For all i, we define R i as: for (x n), (y n ) in W (n I): R i((x n ) n I /, (y n ) n I / ) iff {n I; R i (x n, y n )} U The S i are defined in a similar way to match the definition of ultrapowers. Thus, F, (R i ), (S i ) is an ultrapower of G. The frame F, (R i ), (S i ) is nice, as niceness is preserved by ultraroots (proposition 3.1.2), so G is a frame of Ru H(K). (iii) S H(K) = H(SK) ( ) Let G = W, R, (R i ), (S i ) be a frame of S H(K). There exists a F in K such that G is a subframe of F and F M is in K. Let F be the subframe of F generated by the worlds of G. As G equals F, (R i ), (S i ), we eventually get G H(S K). ( ) Let G = W, R, (R i ), (S i ) be a frame of H(S K). By definition, G M is a generated subframe of a frame of K. Let F = W, R be this frame. Define, for all i, S i = S i, and R i = {(x, y); x W, y S i }. The frame W, R, (R i ), (S i ) is nice. Therefore, G is a subframe of W, R, (R i ), (S i ), and G S H(K). (iv)sc H(K) H(Sc K) Let F be a frame of H(Sc K). As a connected generated subframe is a subframe, F is in H(S K) as well. We just proved that it is in S H(K). But we know that F is nice, thus, it is weakly connected. F is a frame of Sc H(K). Remark We now show that the inverse relations of points (i) and (iv) of the previous proof are not always true, and that there is no general relation of inclusion between Ud H(K) and H(Ud K). Proof. In general, we do not have H(H K) H H(K) Consider the class K containing only one frame, the one drawn in figure 3.1(a). Let F be the frame of H(H K) represented in figure 3.1(b). Suppose F is a frame of H H(K). That means that there is a G H(K) and a bounded morphism f such that F is the image of G by f, and G M is the only frame of K. This is not possible, since G would have two worlds named i. 21

22 CHAPTER 3. FAIRLY NICE FRAMES i (a) The only frame of K Relations (R i ) and (S i ) have not been represented. (b) A frame F in H(H K) Figure 3.1: H(H K) H H(K) is i j i (a) The only frame of K (b) A frame F in Sc K (c) A frame F in S H(K) Figure 3.2: Sc H(K) H(Sc K) is false In general, we do not have Sc H(K) H(Sc K) Let K be a class of frames containing the only frame presented in figure 3.2(a). It is easy to see that Sc K contains only one frame, which is the frame F of the figure 3.2(b). However, every subframe of the frame presented in figure 3.2(c) will have two worlds. This frame is in Sc H(K), but not in H(Sc K). In general, we do not have Ud H(K) H(Ud K) Let K be a class of frames containing the only frame with a single world w and an empty relation R. Frames of H(K) have only one world w. However, with disjoint union, as we did in the introduction, one can build a frame of Ud H(K) with two worlds, e.g. {w 1, w 2 },, (R i ), (S i ), where for all i, R i = {(w 1, w 1 ), (w 2, w 2 )} and S i = {w 1, w 2 }. This frame is not nice, thus, it is not included in H(Ud K) In general, we do not have H(Ud K) Ud H(K) Let K be a class of frames containing only {w},. The frame F = {w 1, w 2 },, (R i ), (S i ), where R i = {(w 1, w 1 ), (w 2, w 1 )} and S i = {w 1 }, for all i, is in H(Ud K). Suppose F is in Ud H(K). As it has two worlds, we have two solutions. (i) F is the disjoint union of one frame, i.e. F H(K). This is not possible, for F have two worlds, and the only frame of K has a single world. (ii) F is the disjoint union of two frames, each of them having one worlds. But in this case, for any i, one of these frames would have an empty S i. Thus, at least one of these frames would not be nice, and F would not be in H(Ud K). 22

23 3.2. PRESERVATION UNDER THE FOUR OPERATIONS Definition For a class of frames K, K is the closure of K under the four fundamental frames constructions. 3.2 Preservation Under H, S, Ru and Ud We recall that connected components were defined in Here is the definition of fairly nice frames. These frames are closely related to nice frames, but are defined to have better preservation properties than nice frames. These preservation results are presented in lemma below, and the fairly nice frames are characterised in the next section. Definition A non-standard frame F is said to be fairly nice if every connected component of F is nice. Remark A nice frame is fairly nice. First, remember (from lemma 2.1.8) that a nice frame validates. Here is a kind of converse property. Lemma A non-standard frame F is fairly nice iff F = m. Proof. ( ) A fairly nice frame is nice, so by lemma 2.1.8, it validates. ( ) Let F be a non-standard frame validating. Each of its connected component validates as well (each connected component can be seen as a generated subframe, which preserves modal validity). Claim 1. A non-standard connected frame that validates is nice. Proof of Claim 1. Let F = W, R, (R i ), (S i ) be such a connected frame. Let us show that it is nice. We use here the first order correspondents of the axioms, which can be read after definition First, we want to show that for all x, y, we have R i xy iff S i y. The left-to-right direction is the axiom (ref). The right-to-left direction is less obvious. Let x and y be two worlds such that S i y. As x and y are connected, there exist a sequence (t m ) m n, for a finite n, where x = t 0, y = t n and R m (t m 1, t m ) where 1 m n and R m is one of the relations R, R 1, R j, Rj 1 for any nominal j. We show by induction that for all m, we have R i (t m, y). The base case is given by the elimination axiom: R i (t n, y). Then, suppose we have R i (t m, y). The previous element t m 1 in the sequence can be related to t m in four different ways. R(t m 1, t m ) By the axiom (propagation), we get R i (t m 1, y). R(t m, t m 1 ) By the axiom (self-dual), there exists y such that R i (t m 1, y ). With (propagation), we get R i (t m, y ). But by (self-dual) again, y = y, and so, we have R i (t m 1, y). R j (t m 1, t m ) By the axiom (agree), we get R i (t m 1, y). R j (t m, t m 1 ) By the axiom (self-dual), there exists y such that R i (t m 1, y ). With (agree), we get R i (t m, y ). But by (self-dual) again, y = y, and so, we have R i (t m 1, y). At last, by induction, we get R i (x, y). We now have to show that S i is a singleton. By (self-dual), S i is not empty. Suppose now that there exist y and z such that S i (y) and S i (z). Let x be any world. We just proved that we have R i (x, y) and R i (x, z). Therefore, by (self-dual), y and z are the same world. S i is a singleton. of claim 1 Now, as each connected component of F validates, each of them is nice, and F is fairly nice. 23