Public Announcement Logic in Geometric Frameworks

Public Announcement Logic in Geometric Frameworks Can Bas ¸ kent Abstract In this paper we introduce public announcement logic in different geo- metric frameworks. First, we consider topological models, and then extend our discussion to a more expressive model, namely , subset space models. Furthermore, we prove the completeness of public announcement logic in those frameworks. Moreover , we apply our results to different issues: an- nouncement stabilization, backward induction and persistence. 1 Introduction Public announcement logic is a well-known and well-studied example of dy- namic epistemic logics (Plaza, 1989; van Ditmarsch et al. , 2007). Dynamic epistemic logics are set out to formalize knowledge and knowledge changes in usually multi-agent settings by defining and introducing different ways of up- dates and interaction. The contribution of public announcement logic (P AL, henceforth) to the field of knowledge representation is mostly due to its suc- cinctness and clarity in reflecting the intuition as it does not increase the ex- pressiveness of the basic epistemic logic. P AL updates the epistemic models by the announcements made by a truthful external agent. After the truthful an- nouncement, the model is updated by eliminating the states that do not agree with the announcement. P AL has many applications in the fields of formal ap- proaches to social interaction, dynamic logics, knowledge representation and updates (Balbiani et al. , 2008; Baltag & Moss, 2004; van Benthem, 2006; van Benthem et al. , 2005). Extensive applications of P AL to different fields and frameworks has made P AL a rather familiar framework to many researchers. Moreover , virtually almost all applications of P AL make use of Kripke models for knowledge representation. However , as it is very well known, Kripke models are not the only representational tool for modal and epistemic logics. In this work, we consider P AL in two different geometrical frameworks: topological models for modal logic and subset space logic. T opological mod- els are not new to modal logics, indeed they are the first models for modal logic (McKinsey & T arski, 1944; McKinsey & T arski, 1946). The past decades have witnessed a revival of academic interest towards the topological models for modal logics in many different frameworks (Aiello et al. , 2003; van Ben- 1 them & Bezhanishvili, 2007; van Benthem et al. , 2006; Bezhanishvili & Gehrke, 2005). However , to the best of our knowledge, topological models have not been applied to dynamic epistemic logics. Y et, there have been some influential works on the notion of common knowledge in topological models which has motivated the current paper (van Benthem & Sarenac, 2004). In that work, it was shown that the different definitions of common knowledge diverge in topological models even though these definitions are equivalent in Kripke struc- tures, based on Barwise’s earlier investigation (Barwise, 1988). Nevertheless, the authors did not seem to take the next immediate step to discuss dynamic epistemologies in that framework. This is one of our goals in this paper: to apply topological reasoning to dynamic epistemological cases and present the immediate completeness results. The second framework that we discuss, sub- set space logic, is a rather weak yet expressive geometrical structure dispensing with the topological structure (Moss & P arikh, 1992; P arikh et al. , 2007). Sub- set space logic has been introduced to reason about the topological notion of closeness and the dynamic notion of effort in epistemic situations. In this paper , we also define P AL in subset space logic with its axiomatization and present the completeness of P AL in subset space logics improving the results based on an earlier work (Bas ¸ kent, 2007). There are several reasons that motivate this work. First, topological mod- els can distinguish some epistemic properties that Kripke models cannot (van Benthem & Sarenac, 2004). This is perhaps not surprising as the topological semantics of the necessity modality has Σ 2 complexity , while Kripkean seman- tics offer Π 1 complexity for the same modality , and furthermore topologies deal with infinite cases in a rather special way 1 . Moreover , P AL update procedure is easily defined by using well-defined topological operations giving sufficient reasons to wonder what other different structures one may have in topological models. The present paper is organized as follows. First, we introduce the geometri- cal frameworks that we need: topological spaces and subset spaces. Then, after a brief interlude on P AL, we give the axiomatizations of P AL in such spaces, and their completeness. The completeness proofs are rather immediate - which is usually the case in P AL systems. Then, we make some observations on P AL in geometric models. Our observations will be about the stabilization of updated models, backward induction in games and persistency . 2 Geometric Models In this section, we will briefly recall the geometric models for some modal logics. What we mean by geometric models is topological models and subset space logic models as they inherently are geometrical structures. W e first start with topological models and their semantics, and then discuss subset space models. 1 When we discuss the semantics of topological models, we will see the Σ 2 complexity of the aforementioned definition. Moreover , by definition, topologies do embrace infinite unions. 2 2.1 T opological Semantics for Modal Logic T opological interpretations for modal logic historically precede the relational semantics (McKinsey & T arski, 1944; Goldblatt, 2006). Moreover , as we will observe very soon, topological semantics is arithmetically more complex than relational semantics: the prior is Σ 2 while the latter is Π 1 . Now , let us start by introducing the definitions. Definition 1. A topological space S = h S, σ i is a structure with a set S and a collection σ of subsets of S satisfying the following axioms: 1. The empty set and S are in σ . 2. The union of any collection of sets in σ is also in σ . 3. The intersection of a finite collection of sets in σ is also in σ . The collection σ is said to be a topology on S . The elements of S are called points and the elements of σ are called opens . The complements of open sets are called closed sets. Our main operator in topological spaces is called interior operator I which returns the interior of a given set. The interior of a set is the largest open set contained in the given set. A topological model M is a triple h S, σ, v i where S = h S, σ i is a topological space, and v is a valuation function assigning propositional letters to subsets of S , i.e. v : P → ℘ ( S ) for a countable set of propositional letters P . The basic modal language L has a countable set of proposition letters P , a truth constant > , the usual Boolean operators ¬ and ∧ , and a modal operator  . The dual of  is denoted by ♦ and defined as  ϕ ≡ ¬ ♦ ¬ ϕ . When we are in topological models, we will use the symbol I for  after the interior operator for intuitive reasons, and to prevent any future confusion. Likewise, we will use the symbol C for ♦ . The notation M , s | = ϕ will read the point s in the model M makes the formula ϕ true . W e call the set of points that satisfy a given formula ϕ in model M the extension of ϕ , and denote as ( ϕ ) M . W e will drop the superscript when the model we are in is obvious. In topological models, the extension of a Boolean formula is obtained in the familiar sense. The extension of a modal formula in model M , then, is given as follows ( I ϕ ) M = I (( ϕ ) M ) - namely , the extension of I ϕ is the interior of the extension of ϕ . Now , based on this framework, the model theoretical semantics of modal logic in topological spaces is given as follows. M , s | = p iff s ∈ v ( p ) for p ∈ P M , s | = ¬ ϕ iff M , s 6| = ϕ M , s | = ϕ ∧ ψ iff M , s | = ϕ and M , s | = ψ M , s | = I ϕ iff ∃ U ∈ σ ( s ∈ U ∧ ∀ t ∈ U, M , t | = ϕ ) M , s | = C ϕ iff ∀ U ∈ σ ( s ∈ U → ∃ t ∈ U, M , t | = ϕ ) A few words on the semantics are in order here. The necessity modality I ϕ says that there is an open set that contains the current state and the formula 3 ϕ is true everywhere in this set. Obviously , this is a rather complex statement, first, it requires us to determine the open set, and then check whether each point in this open set satisfies the given formula or not. On the other hand, the possibility modality C ϕ manifests the idea that for every open set that includes the current state, there is point in the same set that satisfies ϕ . This is clearly reflected in the definition: in topological semantics, the definitions of modal satisfaction have the form ∃∀ or ∀∃ . In Kripke models, as it is well-known, the form is either ∃ or ∀ . It is been shown by McKinsey and T arski that the modal logic of topological spaces is S4 (McKinsey & T arski, 1944). Moreover , the logic of many other topo- logical spaces has also been investigated (Aiello et al. , 2003; Cate et al. , 2009; Bezhanishvili et al. , 2005; van Benthem et al. , 2006; van Benthem & Bezhan- ishvili, 2007). Moreover , recently , the topological properties of paraconsistent systems have also been investigated (Bas ¸ kent, 2011c; Mortensen, 2000). The proof theory of the topological models is as expected: we utilize modus ponens and necessitation. Basic modal logic is long to be known to be sound and complete with respect to the well-known axiomatization of topological modal logic. 2.2 Subset Space Logic Subset space logic (SSL, henceforth) was presented in early 90s as a bimodal logic to formalize reasoning about sets and points with an underlying motiva- tion from epistemic logic (Moss & P arikh, 1992). One of the modal operators of SSL is intended to quantify over the sets (  ) whereas the other modal operator was intended to quantify in the current set ( K ). The underlying motivation for the introduction of these two modalities is to be able to speak about the notion of closeness . In this context, K operator is intended for the knowledge operator (for one agent only , as SSL is originally presented for single-agent), and the  modality is intended for the effort modality . Effort can correspond to vari- ous things: computation, observation, approximation - the procedures that can result in knowledge increase. The language of subset space logic L S has a countable set P of propositional letters, a truth constant > , the usual Boolean operators ¬ and ∧ , and two modal operators K and  . A subset space model is a triple S = h S, σ, v i where S is a non-empty set, σ ⊆ ℘ ( S ) is a collection of subsets ( not necessarily a topology), v : P → ℘ ( S ) is a valuation function. Semantics of SSL, then is given inductively as follows. s, U | = p iff s ∈ v ( p ) s, U | = ϕ ∧ ψ iff s, U | = ϕ and s, U | = ψ s, U | = ¬ ϕ iff s, U 6| = ϕ s, U | = K ϕ iff t, U | = ϕ for all t ∈ U s, U | =  ϕ iff s, V | = ϕ for all V ∈ σ such that s ∈ V ⊆ U The duals of  and K are ♦ and L respectively , and defined as usual. The tuple ( s, U ) is called a neighborhood situation if U is a neighborhood of s , i.e. if 4 s ∈ U ∈ σ . The axioms of SSL reflect the fact that the K modality is S5-like whereas the  modality is S4-like. Moreover , we will need an additional axiom to state the interaction between those two modalities: K  ϕ →  K ϕ . Let us now give the complete set of axioms of SSL. 1. All the substitutional instances of the tautologies of the classical proposi- tional logic 2. ( A →  A ) ∧ ( ¬ A →  ¬ A ) for atomic sentence A 3. K ( ϕ → ψ ) → ( K ϕ → K ψ ) 4. K ϕ → ( ϕ ∧ KK ϕ ) 5. L ϕ → KL ϕ 6.  ( ϕ → ψ ) → (  ϕ →  ψ ) 7.  ϕ → ( ϕ ∧  ϕ ) 8. K  ϕ →  K ϕ The rules of inference are as expected: modus ponens and necessitation for both modalities. Therefore, subset space logic is complete and decidable (Moss & P arikh, 1992). Note that SSL is originally proposed as a single-agent system. There have been some attempts in the literature to suggest a multi-agent version of it, but to the best of our knowledge, there is no intuitive and clear presentation of a multi-agent version of SSL (Bas ¸ kent, 2007). 2.3 Public Announcement Logic Public announcement logic is a way to represent changes in knowledge. The way P AL updates the epistemic states of the knower is by “state-elimination”. A truthful announcement ϕ is made, and consequently , the agents updates their epistemic states by eliminating the possible states where ϕ is false (Plaza, 1989; Balbiani et al. , 2007; Balbiani et al. , 2008; van Ditmarsch et al. , 2007). Public announcement logic is typically interpreted on multi-modal (or multi- agent) Kripke structures (Plaza, 1989). Notationwise, the formula [ ϕ ] ψ is in- tended to mean that after the public announcement of ϕ , ψ holds . As usual, K i is the epistemic modality for the agent i . Likewise, R i is the epistemic accessibility relation for the agent i . The language of P AL will be that of multi-agent (multi- modal) epistemic logic with an additional public announcement operator [ ∗ ] where ∗ can be replaced with any well-formed formula in the language of basic epistemic logic. T o see the semantics of P AL, take a model M = h W, { R } i ∈ I , V i where i denotes the agents and varies over a finite set I . F or atomic proposi- tions, negations and conjunction the semantics is as usual. F or modal operators, we have the following semantics. 5 M , w | = K i ϕ iff M , v | = ϕ for each v such that ( w , v ) ∈ R i M , w | = [ ϕ ] ψ iff M , w | = ϕ implies M| ϕ, w | = ψ Here, the updated model M| ϕ = h W 0 , { R 0 i } i ∈ I , V 0 i is defined by restricting M to those states where ϕ holds. Hence, W 0 = W ∩ ( ϕ ) M ; R 0 i = R i ∩ ( W 0 × W 0 ) , and finally V 0 ( p ) = V ( p ) ∩ W 0 . The axiomatization of P AL is the axiomatization of S5 n with additional axioms for dynamic modality . Hence, we give the set of axioms for P AL as follows. 1. All the substitutional instances of the tautologies of the classical proposi- tional logic 2. K i ( ϕ → ψ ) → ( K i ϕ → K i ψ ) 3. K i ϕ → ϕ 4. K i ϕ → K i K i ϕ 5. ¬ K i ϕ → K i ¬ K i ϕ 6. [ ϕ ] p ↔ ( ϕ → p ) 7. [ ϕ ] ¬ ψ ↔ ( ϕ → ¬ [ ϕ ] ψ ) 8. [ ϕ ]( ψ ∧ χ ) ↔ ([ ϕ ] ψ ∧ [ ϕ ] χ ) 9. [ ϕ ] K i ψ ↔ ( ϕ → K i [ ϕ ] ψ ) The additional rule of inference which we will need for announcement modality is called the announcement generalization and is described as expected: From ` ψ , derive ` [ ϕ ] ψ . P AL is complete and decidable. The completeness proof is quite straightfor- ward. Once the soundness of the given axiomatization is proved, then it means that every complex formula in the language of P AL can be reduced to a formula in the basic language of (multi-agent) epistemic logic. Since S5 epistemic logic is long known to be complete, we immediately deduce the completeness of P AL. Notice again that in this section, we have defined P AL in Kripke structures by following the literature. In the next section, we will see how P AL is defined in geometrical models. W e will start with SSL and proceed to topological models with some further observations. 3 Subset Space P AL In SSL, we depend on neighborhood situations (which are tuples of the form ( s, U ) for s ∈ U ∈ σ ) instead of the epistemic accessibility relations. Therefore, if we want to adopt public announcement logic to the context of subset space logic, we first need to focus on the fact that the public announcements shrink the observation sets for each agent. 6 Let us set a piece of notation. F or a formula ϕ , recall that ( ϕ ) S is the ex- tension of ϕ in the model S = h S, σ, v i . In SSL, ( ϕ ) S = { ( s, U ) ∈ S × σ : s ∈ U, ( s, U ) | = ϕ } . Define the projections ( ϕ ) S 1 := { s : ( s, U ) ∈ ( ϕ ) S for some U 3 s } , and ( ϕ ) S 2 := { U : ( s, U ) ∈ ( ϕ ) S for some s ∈ U } . W e will drop the super- script when it is obvious. Now , assume that we are in a subset space model S = h S , σ, v i . Then, after public announcement ϕ , we will move to another subset space model S ϕ = h S | ϕ, σ ϕ , v ϕ i where S | ϕ = ( ϕ ) 1 , and σ ϕ is the reduced collection of subsets after the public announcement ϕ , and v ϕ is the reduct of v on S | ϕ . The crucial point is to construct σ ϕ . As we need to get rid of the refutative states, we eliminate the points which do not satisfy ϕ for each observation set U in σ . W e will disregard the empty set as no neighborhood situations can be formed with empty set. Hence, σ ϕ = { U ϕ : U ϕ = U ∩ ( ϕ ) 2 6 = ∅ , for each U ∈ σ } . Alternatively , σ ϕ := { U ∩ ( ϕ ) 2 : U ∈ σ } − {∅} 2 . But then, how would the neighborhood situations be affected by the public announcements? Consider the neighborhood situation ( s, U ) and the public announcement ϕ . Then the statement s, U | = [ ϕ ] ψ will mean that after the public announcement of ϕ , ψ will hold in the neighborhood situation ( s, U ϕ ) . So, first we will remove the points in U which refute ϕ , and then ψ will hold in the updated set U ϕ which was obtained from the original set U . Then the corresponding semantics can be suggested as follows: s, U | = [ ϕ ] ψ iff s, U | = ϕ implies s, U ϕ | = ψ Before checking whether this semantics satisfies the axioms of public announce- ment logic, let us give the language and semantics of the topologic P AL. The lan- guage of the topologic public announcement logic interpreted in subset spaces is given as follows: p | ⊥ | ¬ ϕ | ϕ ∧ ψ |  ϕ | K ϕ | [ ϕ ] ψ Now , let us consider the soundness of the axioms of basic P AL that we dis- cussed earlier in Section 2.3. W e prove that those axioms are sound in SSL. Theorem 1. Axioms of the basic P AL are sound in subset space logic. Proof . As the atomic propositions do not depend on the neighborhood, the first axiom is satisfied by the subset space semantics of public announcement modal- ity . T o see this, assume s, U | = [ ϕ ] p . So, by the semantics s, U | = ϕ implies s, U ϕ | = p . So, s ∈ v ( p ) . So for any set V where s ∈ V , we have s, V | = p . Hence, s, U | = ϕ implies s, U | = p , that is s, U | = ϕ → p . Conversely , assume s, U | = ϕ → p . So, s, U | = ϕ implies s ∈ v ( p ) . As s, U | = ϕ , s will lie in U ϕ , thus ( s, U ϕ ) will be a neighborhood situation. Thus, s, U ϕ | = p . Then, we conclude s, U | = [ ϕ ] p . The axioms for negation and conjunction are also straightforward formula manipulations and hence skipped. 2 Thanks to the anonymous referee for pointing out this simple reformulation. 7 The important reduction axiom is the knowledge announcement axiom. As- sume, s, U | = [ ϕ ] K ψ . Suppose further that s, U | = ϕ . Then we have the follow- ing. s, U | = [ ϕ ] K ψ iff s, U ϕ | = K ψ iff for each t ϕ ∈ U ϕ , we have t ϕ , U ϕ | = ψ iff for each t ∈ U , t, U | = ϕ implies t, U | = [ ϕ ] ψ iff s, U | = K ( ϕ → [ ϕ ] ψ ) iff s, U | = K [ ϕ ] ψ Thence, the above axioms are sound for the subset space semantics of public announcement logic.  Now , recall that SSL has an indispensable modal operator  . One can won- der whether we can have a reduction axiom for it as well. W e start by consid- ering the statement [ ϕ ]  ψ ↔ ( ϕ →  [ ϕ ] ψ ) . Assume, s, U | = [ ϕ ]  ψ . Suppose further that s, U | = ϕ . Then, we deduce the following. s, U | = [ ϕ ]  ψ iff s, U ϕ | =  ψ iff for each V ϕ ⊆ U ϕ we have s, V ϕ | = ψ iff for each V ⊆ U , s, V | = ϕ implies s, V | = [ ϕ ] ψ iff s, U | =  ( ϕ → [ ϕ ] ψ ) iff s, U | =  [ ϕ ] ψ Now , it is easy to see that the following axiomatize the SSL-P AL together with the axiomatization of SSL: 1. [ ϕ ] p ↔ ( ϕ → p ) 2. [ ϕ ] ¬ ψ ↔ ( ϕ → ¬ [ ϕ ] ψ ) 3. [ ϕ ]( ψ ∧ χ ) ↔ ([ ϕ ] ψ ∧ [ ϕ ] χ ) 4. [ ϕ ] K ψ ↔ ( ϕ → K [ ϕ ] ψ ) 5. [ ϕ ]  ψ ↔ ( ϕ →  [ ϕ ] ψ ) R eferring to the above discussions, the completeness of subset space P AL follows easily . Theorem 2. P AL in subset space models is complete with respect to the axiom system given above. Proof . By reduction axioms we can reduce each formula in the language of topo- logic P AL to a formula in the language of SSL. As SSL is complete, so is P AL in subset space models.  By the same idea, we can import the decidability result. Theorem 3. P AL in subset space models is decidable. 8 4 T opological P AL 4.1 Single Agent T opological P AL W e can use the similar ideas to give an account of P AL in topological spaces. Let T = h T , τ , v i be a topological model and ϕ be a public announcement. W e now need to obtain the topological model T ϕ which is the updated model after the announcement. Define T ϕ = h T ϕ , τ ϕ , v ϕ i where T ϕ = T ∩ ( ϕ ) , τ ϕ = { O ∩ T ϕ : O ∈ τ } and v ϕ = v ∩ T ϕ . W e now need to verify that τ ϕ is a topology , indeed the induced topology . F or the sake of the completeness of our arguments in this paper , let us give the immediate proof here. Proposition 1. If τ is a topology , then τ ϕ = { O ∩ T ϕ : O ∈ τ } is a topology as well. Proof . Clearly , the empty set is in τ ϕ as τ is a topology . As τ is a topology on T , we have T ∈ τ . Thus, T ∩ T ϕ , namely T ϕ , is in τ ϕ . Consider S ∞ i U i where U i ∈ τ ϕ . F or each i , we have U i = O i ∩ T ϕ for some O i ∈ τ . Thus, S ∞ i U i = T ϕ ∩ S ∞ i O i . Since τ is a topology , S ∞ i O i ∈ τ . Thus, T ϕ ∩ S ∞ i O i ∈ τ ϕ yielding the fact that S ∞ i U i ∈ τ ϕ . Similarly , consider T n i U i where U i ∈ τ ϕ for some n < ω . Since U i = O i ∩ T ϕ for some O i ∈ τ , we similarly observe that T n i U i = T n i ( O i ∩ T ϕ ) = T ϕ ∩ T n i O i . Since τ is a topology , T n i O i ∈ τ , thus, T n i U i ∈ τ ϕ .  It is important to notice here that only modal formulas necessarily yield open or closed extensions. The extension of Booleans, then, may or may not be a topological set as it solely depends on the model. Now , when we restrict the carrier set of the topology to a subset of it, we still get a topology immediately and easily . Based on this simple observation, we can give a semantics for the public announcements in topological models. T , s | = [ ϕ ] ψ iff T , s | = ϕ implies T ϕ , s | = ψ In a similar fashion, we can expect that the reduction axioms work in topo- logical spaces. The reduction axioms for atoms and Booleans are quite straight- forward. So, consider the reduction axiom for the interior modality given as follows: [ ϕ ] I ψ ↔ ( ϕ → I [ ϕ ] ψ ) . Let T , s | = [ ϕ ] I ψ which, by definition means T , s | = ϕ implies T ϕ , s | = I ψ . If we spell out the topological interior modality , we get ∃ U ϕ 3 s ∈ τ ϕ s.t. ∀ t ∈ U ϕ , T ϕ , t | = ψ . By definition, since U ϕ ∈ τ ϕ , it means that there is an open U ∈ τ such that U ϕ = U ∩ ( ϕ ) . Under the assumption that T , s | = ϕ , we observe that ∃ U 3 s ∈ τ (as we just constructed it), such that after the announcement ϕ , the non-eliminated points in U (namely , the ones in U ϕ ) will satisfy ψ . Thus, we get T , s | = ϕ → I [ ϕ ] ψ . The other direction is very similar and hence we leave it to the reader . There- fore, the reduction axioms for P AL in topological spaces are given as follows. 1. [ ϕ ] p ↔ ( ϕ → p ) 9 2. [ ϕ ] ¬ ψ ↔ ( ϕ → ¬ [ ϕ ] ψ ) 3. [ ϕ ]( ψ ∧ χ ) ↔ ([ ϕ ] ψ ∧ [ ϕ ] χ ) 4. [ ϕ ] I ψ ↔ ( ϕ → I [ ϕ ] ψ ) As a result, all the complex formulas involving the P AL operator can be reduced to a simpler one. This algorithm directly shows the completeness of P AL in topological spaces by reducing each formula in the language of topological P AL to the language of basic topological modal logic. Thus, the result follows. Theorem 4. P AL in topological spaces is complete with respect to the axiomatiza- tion given. By the same idea, we can import the decidability result. Theorem 5. P AL in topological models is decidable. 4.2 Product T opological P AL There are variety of ways to merge given topological models to express the epistemic interaction between them: products, sums, fusions etc (Gabbay et al. , 2003). In this section, we focus on one of such methods, product topologies, and discuss how public announcements are defined in them. Product topological frameworks for multi-agent epistemic logics have already been discussed in the literature widely (van Benthem et al. , 2006; van Benthem & Sarenac, 2004). Therefore, our treatment of the subject will be based on these works. Based on this basic formalism, we will then introduce public announcement logic. The idea is quite straight-forward. W e are given two topologies (possibly with different spaces) with a modal (epistemic, doxastic etc) model on them. Then, by the standard techniques in the literature, we merge them. After that, we discuss how public announcements work in this unified structure. Let T = h T , τ i and T 0 = h T 0 , τ 0 i be two given topological spaces. Now , we introduce some definitions. Let X ⊆ T × T 0 . W e call X horizontally open ( h- open ) if for any ( x, y ) ∈ X , there is a U ∈ τ such that x ∈ U , and U × { y } ⊆ X . In a similar fashion, we call X vertically open ( v -open ) if or any ( x, y ) ∈ X , there is a U 0 ∈ τ 0 such that y ∈ U 0 , and { x } × U 0 ⊆ X . These notions can be seen as one dimensional projections of openness and closure that we will need soon. Now , given two topological spaces T = h T , τ i and T 0 = h T 0 , τ 0 i , let us asso- ciate two modal operators I and I 0 respectively to these models. Then, we can obtain a product topology in a language with the two aforementioned modal- ities. The product model, then, is of the form h T × T 0 , τ , τ 0 i . Therefore, we consider the cross product × as a way to represent model interaction among epistemic agents which gives us a model with two-dimensional space, and two topologies. The semantics of those modalities are given as follows. 10 ( x, y ) | = I ϕ iff ∃ U ∈ τ , x ∈ U and ∀ u ∈ U , ( u, y ) | = ϕ ( x, y ) | = I 0 ϕ iff ∃ U 0 ∈ τ 0 , y ∈ U 0 and ∀ u 0 ∈ U 0 , ( x, u 0 ) | = ϕ Here, given a tuple ( x, y ) , the modality I ranges over the first component while the modality I 0 ranges over the second. In other words, we localize the product with respect to the given original topologies. It has been shown that the fusion logic S4 ⊕ S4 is complete with respect to products of arbitrary topological spaces (van Benthem & Sarenac, 2004). Then, the question is this: How would a state elimination based dynamic epistemic paradigm work in product topologies? Now , step by step, we will present how to define public announcements in this framework. The difficulty lies in the fact that when we take the product of the given topological models, we increase the dimension of the space. Then, the intuition behind defining public announcements should follow the same idea: the announcement will update the product topology in all dimensions. Let us now be a bit more precise. Before we start, note that here we focus on the product of two topologies representing the interaction between two agents with different spaces and topologies, but it can easily be generalized to n -agents. The language of product topological P AL is given as follows. p | ¬ ϕ | ϕ ∧ ϕ | K 1 ϕ | K 2 ϕ | [ ϕ ] ϕ F or given two topological models T = h T , τ , v i and T 0 = h T 0 , τ 0 , v i , the product topological model M = h T × T 0 , τ , τ 0 , v i has the following semantics. M , ( x, y ) | = K 1 ϕ iff ∃ U ∈ τ , x ∈ U and ∀ u ∈ U , ( u, y ) | = ϕ M , ( x, y ) | = K 2 ϕ iff ∃ U 0 ∈ τ 0 , y ∈ U 0 and ∀ u 0 ∈ U 0 , ( x, u 0 ) | = ϕ M , ( x, y ) | = [ ϕ ] ψ iff M , ( x, y ) | = ϕ implies M ϕ , ( x, y ) | = ψ where M ϕ = h T ϕ × T 0 ϕ , τ ϕ , τ 0 ϕ , v ϕ i is the updated model. W e define all T ϕ , T 0 ϕ , τ ϕ , τ 0 ϕ , and v ϕ as before. Therefore, the following axioms axiomatize the product topological P AL together with the axioms of S4 ⊕ S4. 1. [ ϕ ] p ↔ ( ϕ → p ) 2. [ ϕ ] ¬ ψ ↔ ( ϕ → ¬ [ ϕ ] ψ ) 3. [ ϕ ]( ψ ∧ χ ) ↔ ([ ϕ ] ψ ∧ [ ϕ ] χ ) 4. [ ϕ ] K i ψ ↔ ( ϕ → K i [ ϕ ] ψ ) Theorem 6. Product topological P AL is complete and decidable with respect to the given axiomatization. Proof . Proof of both completeness and decidability is by reduction, and similar to the ones presented before. Thus, we leave the details to the reader .  11 5 Applications Now , we can briefly apply the previous discussions to some issues in P AL, foun- dational game theory and SSL. The purpose of such applications is to give the reader a sense how topological frameworks might affect the aforementioned issues, and in general how dynamic epistemic situations can be represented topologically . 5.1 Announcement Stabilization Muddy Children presents an interesting case for P AL (F agin et al. , 1995). In this game, we assume that a group of children were playing outside in the mud. Then, their father calls them in. Children came back in, and gather around the father in such a way that every children sees all the others, and the father sees them all. W e also assume that there is no mirror in the room, so the children cannot see themselves. Since they were playing in the mud, some got dirty with mud on their forehead. F ather then announces that “ At least one of you has mud on his or her forehead”. If no child steps forward saying that “Y es, I do have mud on my forehead” communicating the fact that she learned it from the announcement, the father keeps repeating the very same announcement (van Ditmarsch et al. , 2007). In that game, the model representing the epistemics of the group (see the Figure) gets updated after each children says that she does not know if she had mud on her forehead. The model keeps updated until the announcement is negated, and then becomes common knowledge (van Benthem, 2007). There- fore, after each update, we get smaller and smaller models up until the moment that the model gets stabilized in the sense that the same announcement does not update the model any longer . As van Benthem pointed out, this is closely related to several issues in modal and epistemic logics (van Benthem, 2007). First, P AL behaves like a fixed-point operator where the fixed point is the model which is stabilized. Second, there seems to be a close relation between game theoretical strategy eliminations, and solution methods based on such approaches. Therefore, it is rather important to analyze announcement stabilization. Here, we will approach the issue from a topological angle. F or a model M and a formula ϕ , we define the announcement limit lim ϕ M as the first model which is reached by successive announcements of ϕ that no longer changes after the last announcement is made. Announcement limits exist in both finite and infinite models (van Benthem & Gheerbrant, 2010). F or in- stance, for any model M , lim p M = M | p for propositional variable p . Therefore, the limit model is the first updated model when the announcement is a ground Boolean formula. In muddy children, the announcement shrinks the model step by step, round by round (van Benthem, 2007). However , sometimes in dialogue games it may take too long to solve such puzzles until the model gets stabilized as shown by P arikh (P arikh, 1991). Similarly , even Zermelo considered similar 12 Figure 1: A model for muddy children played with 3 children a, b, c taken from van Ditmarsch et al . The state n a n b n c for n a , n b , n c ∈ { 0 , 1 } represent that child i has mud on her forehead iff n i = 1 for i ∈ { a, b, c } . The proposition m i means that the child i ∈ { a, b, c } has mud on her forehead. The current state is underlined. approaches in early 20. century to understand as to how long it takes for the game to stabilize (Schwalbe & W alker , 2001). Similar to the discussions of the aforementioned authors, we now analyze how the models stabilize in topological P AL. W e know that topological models do present some differences in epistemic logical structures. F or instance, in topological models, the stabilization of the fixed-point definition 3 version of common knowledge may occur later than ordinal stage ω . However , it stabilizes in ≤ ω steps in Kripke models (van Benthem & Sarenac, 2004). W e also know that there are two possibilities for the limit models. Either it is empty or nonempty . If it is empty , it means that the negation of the an- nouncement has become common knowledge, thus the announcement refuted 3 F ormula ϕ is common knowledge among two-agents 1 and 2 C 1 , 2 ϕ is represented with the (largest) fixed-point definition as follows: C 1 , 2 ϕ := ν p.ϕ ∧ K 1 p ∧ K 2 p where K i , for i = 1 , 2 is the familiar knowledge operator (Barwise, 1988). 13 itself . On the other hand, if the limit model is not empty , it means that the announcement has become common knowledge (van Benthem & Gheerbrant, 2010). Theorem 7. F or some formula ϕ and some topological model M , it may take more than ω stage to reach the limit model lim ϕ M . Proof . The proof is rather immediate for those familiar with the literature. So, we just mention the basic idea here. First, note that it was shown that in multi-agent topological models, stabi- lization of common knowledge with fixed-point definition may occur later than ω stage. However , in Kripke models it occurs before ω stage (van Benthem & Sarenac, 2004). Also note that it was also shown that if the limit model is not empty , the announcement has become common knowledge (van Benthem & Gheerbrant, 2010). Therefore, combining these two observations, we conclude that in some topological models with non-empty limit models, the number of stage for the announcement to be common knowledge may take more than ω steps.  Even if the stabilization takes longer , we can still obtain stable models by taking intersections at the limit ordinals as a general rule (van Benthem & Gheerbrant, 2010). Therefore, we guarantee that the update procedure will terminate. Thus, the following result is now self-evident. Theorem 8. Limit models exist in topological models. Y et another property of topological models is the fact that the topologies are not closed under arbitrary intersection. Then, one can ask the following ques- tion: “How does P AL work in infinite-conjunction announcements?” The follow- ing example illustrates that point. T ake the real closed interval [ − 1 , 1] with the usual Euclidean topology . F or each n ∈ ω , define the valuation for propositions as such v ( p n ) = [ − 1 /n, 1 /n ] . Therefore, p 1 holds in the entire space [ − 1 , 1] , while p 2 holds in [ − 1 / 2 , 1 / 2] . Consider now the announcements  V n ∈ ω p n and V n ∈ ω  p n . The former formula is true in the interior I ( T n ∈ ω p n ) which is equal to empty set while the latter one is true in the intersection T n ∈ ω I ( p n ) which is equal to the singleton { 0 } . Then, clearly these updates will yield the same models in Kripke models. But, in topological models, as the extensions of two formula differ , updated models will clearly differ , too. 5.2 Backward Induction The fact that limit models can be attained in more than ω steps can create some problems in games. Consider the backward induction solution where players trace back their moves to develop a winning strategy . Notice that the Aumann’s 14 backward induction solution assumes common knowledge of rationality (Au- mann, 1995; Halpern, 2001) 4 . Granted, there can be several philosophical and epistemic issues about the centipede game and its relationship with rationality , but we will not pursue this direction here (Artemov , 2009a; Artemov , 2009b). This issue can also be approached from a dynamic epistemic perspective. R ecently , it has been shown that in any game tree model M taken as a P AL model, lim rational M is the actual subtree computed by the backward induction procedure where the proposition rational means that “at the current node, no player has chosen a strictly dominated move in the past coming here” (van Ben- them & Gheerbrant, 2010). Therefore, the announcement of node-rationality produces the same result as the backward induction procedure. Each backward step in the backward induction procedure can then be obtained by the public announcement of node rationality . This result is quite impressive in the sense that it establishes a closer connection between communication and rationality , and furthermore leads to several more intriguing discussions about rationality . In this work, we refrain ourselves from pursuing this line of thought for the time being. However , there seems to be a problem in topological models. The admissi- bility of limit models can take more than ω steps in topological models as we have conjectured earlier . Therefore, the BI procedure can take ω steps or more. Theorem 9. In topological models of games, under the assumption of rationality , the backward induction procedure can take more than ω steps. Proof . Notice that each tree can easily be converted to a topology by taking the upward closed sets as opens. By the previous discussion, we know that backward induction solution can be attained by obtaining the limit models by publicly announcing the proposition rationality . Therefore, by Theorem 7, sta- bilization can take more than ω step. Therefore, the corresponding backward induction scheme can also take more than ω step.  This is indeed a problem about the attainability in infinite games: how can a player continue playing the game when she hit the limit ordinal ω -th step in the backward induction procedure? In order not to diverge from our current focus, we leave this question open for further research. 5.3 P ersistence Let us now discuss stabilization in SSL framework. W e already have a similar notion within the SSL context. Define persistent formula in a model M as the formula ϕ whose truth is independent from the subsets in M . In other words, ϕ is persistent if for all states s and subsets V ⊆ U , we have s, U | = ϕ impies s, V | = ϕ . Clearly , Boolean formulas are persistent in every model. The significance of persistent formulas is the fact that they are independent of the subsets they occupy which means that they are immune to the epistemics 4 Although according to Halpern, Stalnaker proved otherwise (Halpern, 2001; Stalnaker , 1998; Stalnaker , 1994; Stalnaker , 1996). 15 of the model. Therefore, intuitively , they should also be immune to the changes in the model. This is interesting due to the fact that now we have a quite strong way to tell what can and cannot be changed by public announcements in SSL. Theorem 10. Let M be a model and ϕ be persistent in M . Then, for any formula χ and neighborhood situation ( s, U ) , if s, U | = ϕ , then s, U | = [ χ ] ϕ . In other words, true persistent formulas are immune to the public announcements. Proof . Proof follows directly from the definitions and the fact that after the pub- lic announcement of χ , we always have U χ ⊆ U .  In other words, we can have some formulas in SSL framework that are im- mune to the announcements. 6 Conclusion and F uture W ork In this work, without using Kripke structures at all, we discussed P AL in two dif- ferent geometrical systems. In subset space logic, we defined dynamic axioms for both epistemic and dynamic modalities and showed the corresponding com- pleteness theorems. Moreover , we have applied the geometric ideas to model stabilization and persistent formulas. This gave us the connection regarding dy- namic epistemic logics and rationality . W e observed that in topological models, backward induction scheme loses its intuitiveness. There can be some mathe- matical solutions to this problem. F or the backwards induction procedure that takes longer than ω , modal-mu calculus can also be considered with its natural game theoretical semantics. Therefore, this can be a further research to see how > ω -step backward induction scheme gets stabilized. An interesting fact about topological models of modal logic is that only modal formulas can give an open or a closed set. However , one can stipu- late that the extension of any modal formula can be open or dually closed. If that is the case, one can obtain an incomplete or inconsistent logic respectively (Bas ¸ kent, 2011c; Mortensen, 2000). Moreover , some special algebras such as Heyting and co-Heyting algebras, correspond to those logics. Therefore, the topological investigation of P AL can be carried out in these special topological spaces or algebras in such a way that some special logics can further be inves- tigated. This can also lead to the investigation of P AL in paraconsistent, para- complete and dialethic frameworks where announcements and models may not be consistent nor complete. Acknowledgement This paper is an extended version of (Bas ¸ kent, 2011a; Bas ¸ kent, 2011b). W e appreciate the feedback of FLAIRS-24 referees and R o- hit P arikh. 16 R eferences A I E L L O , M A R C O , VA N B E N T H E M , J O H A N , & B E Z H A N I S H V I L I , G U R A M . 2003. R ea- soning About Space: the Modal W ay . Journal of Logic and Computation , 13 (6), 889–920. A R T E M O V , S E R G E I . 2009a. Intelligent Players . T ech. rept. Department of Com- puter Science, The Graduate Center , The City University of New Y ork. A R T E M O V , S E R G E I . 2009b. Rational Decisions in Non-probablistic Setting . T ech. rept. TR-2009012. Department of Computer Science, The Graduate Center , The City University of New Y ork. A U M A N N , R O B E R T J . 1995. Backward Induction and Common Knowledge of Rationality . Games and Economic Behavior , 8 (1), 6–19. B A L B I A N I , P H I L I P P E , B A LT A G , A L E X A N D R U , VA N D I T M A R S C H , H A N S , H E R Z I G , A N - D R E A S , H O S H I , T O M O H I R O , & D E L I M A , T I A G O . 2007. What Can W e Achieve by Arbitrary Announcements? A Dynamic T ake on Fitch’s Knowability . In: S A M E T , D O V (ed), Procedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (T ARK -2007) . B A L B I A N I , P H I L I P P E , B A LT A G , A L E X A N D R U , VA N D I T M A R S C H , H A N S , H E R Z I G , A N - D R E A S , & D E L I M A , T I A G O . 2008. ‘Knowable’ as ‘known after an announce- ment’. Review of Symbolic Logic , 1 (3), 305–334. B A LT A G , A L E X A N D R U , & M O S S , L AW R E N C E S . 2004. Logics for Epistemic Pro- grams. Synthese , 139 (2), 165–224. B A RW I S E , J O N . 1988. Three Views of Common Knowledge. Pages 365–379 of: P U B L I S H E R S , M O R G A N K A U F M A N N (ed), Proceedings of the 2nd conference on Theoretical aspects of reasoning about knowledge . B A S ¸ K E N T , C A N . 2007 (July). T opics in Subset Space Logic . M.Phil. thesis, Institute for Logic, Language and Computation, Universiteit van Amsterdam. B A S ¸ K E N T , C A N . 2011a. Completeness of Public Announcement Logic in T opo- logical Spaces. Bulletin of Symbolic Logic , 17 (1), 142. B A S ¸ K E N T , C A N . 2011b. Geometric Public Announcement Logics. Pages 87–88 of: M U R R AY , R . C H A R L E S , & M C C A R T H Y , P H I L I P M . (eds), Proceedings of the 24th Florida Artificial Intelligence Research Society Conference AAAI Press, for FLAIRS-24. B A S ¸ K E N T , C A N . 2011c. P araconsistency and T opological Semantics. http://arxiv .org/abs/1107.4939 . B E Z H A N I S H V I L I , G U R A M , & G E H R K E , M A I . 2005. Completeness of S4 with re- spect to the Real Line: Revisited. Annals of Pure and Applied Logic , 131 (??), 287–301. 17 B E Z H A N I S H V I L I , G U R A M , E S A K I A , L E O , & G A B E L A I A , D AV I D . 2005. Some R esults on Modal Axiomatization and Definability for T opological Spaces. Studia Logica , 81 (3), 325–55. C A T E , B A L D E R T E N , G A B E L A I A , D AV I D , & S U S T R E T O V , D M I T RY . 2009. Modal Languages for T opology: Expressivity and Definability . Annals of Pure and Applied Logic , 159 (1-2), 146–170. F A G I N , R O N A L D , H A L P E R N , J O S E P H Y . , M O S E S , Y O R A M , & V A R D I , M O S H E Y. 1995. Reasoning About Knowledge . MIT Press. G A B B AY , D . M . , K U R U C Z , A . , W O LT E R , F. , & Z A K H A R YA S C H E V , M . 2003. Many Dimensional Modal Logics: Theory and Applications . Studies in Logic and the F oundations of Mathematics, vol. 145. Elsevier . G O L D B L A T T , R O B E R T . 2006. Mathematical Modal Logic: A V iew of Its Evolution. In: G A B B AY , D O V M . , & W O O D S , J O H N (eds), Handbook of History of Logic , vol. 6. Elsevier . H A L P E R N , J O S E P H Y. 2001. Substantive Rationality and Backward Induction. Games and Economic Behavior , 37 (2), 425–435. M C K I N S E Y , J . C . C . , & T A R S K I , A L F R E D . 1944. The Algebra of T opology . The Annals of Mathematics , 45 (1), 141–191. M C K I N S E Y , J . C . C . , & T A R S K I , A L F R E D . 1946. On Closed Elements in Closure Algebras. The Annals of Mathematics , 47 (1), 122–162. M O RT E N S E N , C H R I S . 2000. T opological Seperation Principles and Logical The- ories. Synthese , 125 (1-2), 169–178. M O S S , L AW R E N C E S . , & P A R I K H , R O H I T . 1992. T opological R easoning and the Logic of Knowledge. P ages 95–105 of: M O S E S , Y O R A M (ed), Proceedings of T ARK IV . P A R I K H , R O H I T . 1991. Finite and Infinite Dialogues. Pages 481–498 of: M O S C H O VA K I S , Y . (ed), Proceedings of a W orkshop on Logic from Computer Science . Springer . P A R I K H , R O H I T , M O S S , L A W R E N C E S . , & S T E I N S V O L D , C H R I S . 2007. T opology and Epistemic Logic. In: A I E L L O , M A R C O , P R AT T - H A R T M A N , I A N E . , & VA N B E N T H E M , J O H A N (eds), Handbook of Spatial Logics . Springer . P L A Z A , J A N A . 1989. Logic of Public Communication. P ages 201–216 of: E M - R I C H , M . L . , P F E I F E R , M . S . , H A D Z I K A D I C , M . , & R A S , Z . W. (eds), 4th International Symposium on Methodologies for Intelligent Systems . S C H WA L B E , U L R I C H , & W A L K E R , P AU L . 2001. Zermelo and the Early History of Game Theory . Games and Economic Behavior , 34 (1), 123–137. 18 S T A L N A K E R , R O B E R T . 1994. On the Evaluation of Solution Concepts. Theory and Decision , 37 (1), 49–73. S T A L N A K E R , R O B E R T . 1996. Knowledge, Belief and Counterfactual R easoning in Games. Economics and Philosophy , 12 (2), 133–163. S T A L N A K E R , R O B E RT . 1998. Belief Revision in Games: F orward and Backward Induction. Mathematical Social Sciences , 36 (1), 31–56. VA N B E N T H E M , J O H A N . 2006. ”One is a Lonely Number”: Logic and Commu- nication. In: C H AT Z I D A K I S , Z . , K O E P K E , P. , & P O H L E R S , W. (eds), Logic Colloquium ’02 . Lecture Notes in Logic, vol. 27. Association for Symbolic Logic. VA N B E N T H E M , J O H A N . 2007. Rational Dynamics and Epistemic Logic in Games. International Game Theory Review , 9 (1), 13–45. VA N B E N T H E M , J O H A N , & B E Z H A N I S H V I L I , G U R A M . 2007. Modal Logics of Space. In: A I E L L O , M A R C O , P R AT T - H A R T M A N , I A N E . , & VA N B E N T H E M , J O H A N (eds), Handbook of Spatial Logics . Springer . VA N B E N T H E M , J O H A N , & G H E E R B R A N T , A M E L I E . 2010. Game Solution, Epis- temic Dynamics and Fixed-P oint Logics. Fundamenta Infomaticae , 100 (1- 4), 19–41. VA N B E N T H E M , J O H A N , & S A R E N A C , D A R K O . 2004. The Geometry of Knowledge. P ages 1–31 of: Aspects of Universal Logic . T ravaux Logic, vol. 17. VA N B E N T H E M , J O H A N , VA N E I J C K , J A N , & K O O I , B A R T E L D . 2005. Logics of Communication and Change . T ech. rept. Institute for Logic, Language and Computation. VA N B E N T H E M , J O H A N , B E Z H A N I S H V I L I , G U R A M , C A T E , B A L D E R T E N , & S A R E N A C , D A R K O . 2006. Modal Logics for Product T opologies. Studia Log- ica , 84 (3), 375–99. VA N D I T M A R S C H , H A N S , VA N D E R H O E K , W I E B E , & K O O I , B A R T E L D . 2007. Dy- namic Epistemic Logic . Springer . 19