Announcement as effort on topological spaces

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and prove soundness and completeness results for these systems.


Introduction
In [15], Moss et al. introduce a bi-modal logic with language called subset space logic (SSL), in order to formalize reasoning about sets and points together in one modal system.The main interest in their investigation lies in spatial structures such as topological spaces and using modal logic and the techniques behind for spatial reasoning, however, they also have a strong motivation from epistemic logic.While the modality K is interpreted as knowledge, 2 intends to capture the notion of effort, i.e., any action that results in increase in knowledge.They propose subset space semantics for their logic.A subset space is defined to be a pair (X, O), where X is a non-empty domain and O is a collection of subsets of X (not necessarily a topology), wherein the modalities K and 2 are evaluated with respect to pairs of the form (x,U), where x ∈ U ∈ O.According to subset space semantics, given a pair (x,U), the modality K quantifies over the elements of U, whereas 2 quantifies over all open subsets of U that include the actual world x.Therefore, while knowledge is interpreted 'locally' in a given observation set U, effort is read as open-set-shrinking where more effort corresponds to a smaller neighbourhood, thus, a possible increase in knowledge.The schema 3Kϕ states that after some effort the agent comes to know ϕ where effort can be in the form of measurement, observation, computation, approximation [15,8,16,5], or announcement [17,1,10].
The epistemic motivation behind the subset space semantics and the dynamic nature of the effort modality suggests a link between SSL and dynamic epistemic logic, in particular dynamics known as public announcement [4,5,3,19,6].The works [4,5,3] propose modelling public announcements on subset spaces by deleting the states or the neighbourhoods falsifying the announcement.This dynamic epistemic method is not in the spirit of the effort modality: dynamic epistemic actions result in global model change, whereas the effort modality results in local neighbourhood shrinking.Hence, it is natural to search for an 'open-set-shrinking-like' interpretation of public announcements on subset spaces.To best of our knowledge, Wang and Ågotnes [19] were the first to propose semantics for public announcements on subset spaces in the style of the effort modality, although this is not necessarily on topological spaces.Bjorndahl [6] then proposed a revised version of the [19] semantics.In contrast to the aforementioned proposals, Bjorndahl uses models based on topological spaces to interpret knowledge and information change via public announcements.He considers the language where int(ϕ) means 'ϕ is true and can be announced', and where [ϕ]ψ means 'after public announcement of ϕ, ψ.' In [1], Balbiani et al. introduce a logic to quantify over announcements in the setting of epistemic logic based on the language (with single-agent version here) In this case, unlike above, 2ϕ means 'after any announcement, ϕ (is true)' so that 2 quantifies over epistemically definable subsets (2-free formulas of the language) of a given model.In this case, 3Kϕ again means that the agent comes to know ϕ, but in the interpretation that there is a formula ψ such that after announcing it the agent knows ϕ.What becomes true or known by an agent after an announcement can be expressed in this language without explicit reference to the announced formula.
Clearly, the meaning of the effort 2 modality and of the arbitrary announcement 2 modality are related in motivation.In both cases, interpreting the modality requires quantification over sets.Subsetspace-like semantics provides natural tools for this.In [10], we extended Bjorndahl's proposal [6] with an arbitrary announcement modality and provided topological semantics for the 2 modality, and proved completeness for the corresponding single-agent logic APAL int .
In the current proposal we generalize this approach to a multi-agent setting.Multi-agent subset space logics have been investigated in [13,14,4,18].There are some challenges with such a logic concerning the evaluation of higher-order knowledge.The general setup is for any finite number of agents, but to demonstrate the challenges, consider the case of two agents.Suppose for each of two agents i and j there is an open set such that the semantic primitive becomes a triple (x,U i ,U j ) instead of a pair (x,U).Now consider a formula like K i Kj K i p, for 'agent i knows that agent j considers possible that agent i knows proposition p'.If this is true for a triple (x,U i ,U j ), then Kj K i p must be true for any y ∈ U i ; but y may not be in U j , in which case (y,U i ,U j ) is not well-defined: we cannot interpret Kj K i p.Our solution to this dilemma is to consider neighbourhoods that are not only relative to each agent, as usual in multi-agent subset space logics, but that are also relative to each state.This amounts to, when shifting the viewpoint from x to y ∈ U i , in (x,U i ,U j ), we simultaneously have to shift the neighbourhood (and not merely the point in the actual neighbourhood) for the other agent.So we then go from (x,U i ,U j ) to (y,U i ,V j ), where V j may be different from U j .If they are different, their intersection should be empty.
In order to define the evaluation neighbourhood for each agent with respect to the state in question, we employ a technique inspired by the standard neighbourhood semantics [7].We use a set of neighbourhood functions, determining the evaluation neighbourhood relative to both the given state and the corresponding agent.These functions need to be partial in order to render the semantics well-defined for the dynamic modalities in the system.
In Section 2 we define the syntax, structures, and semantics of our multi-agent logic of arbitrary public announcements, APAL int , interpreted on topological spaces equipped with a set of neighbourhood functions.Without arbitrary announcements we get the logic PAL int , and with neither arbitrary nor public announcements, the logic EL int .In this section we also show some typical validities of the logic, and give a detailed example.In Section 3 we give axiomatizations for the logics: PAL int extends EL int and APAL int extends PAL int .In Section 4 we demonstrate completeness for these logics.The completeness proof for the epistemic version of the logic, EL int , is rather different from the completeness proof for the full logic APAL int .We then compare our work to that of others (Section 5) and conclude.

The logic APAL int
We define the syntax, structures, and semantics of our logic.From now on, Prop is a countable set of propositional variables and A a finite and non-empty set of agents.

Syntax
Definition 1 The language L APAL int is defined by where p ∈ Prop and i ∈ A .Abbreviations for the connectives ∨, → and ↔ are standard, and ⊥ is defined as abbreviation by p ∧ ¬p.We employ Ki for ¬K i ¬ϕ, and 3ϕ for ¬2¬ϕ.We denote the non-modal part of L APAL int (without the modalities K i , int, [ϕ] and 2) by L Pl , the part without 2 by L PAL int , and the part without 2 and [ϕ] by L EL int .
Necessity forms [12] allow us to select unique occurrences of a subformula in a given formula (unlike in uniform substitution).They will be used in the axiomatization (Section 3).Definition 2 Let ϕ ∈ L APAL int .The necessity forms are inductively defined as It is not hard to see that each necessity form ξ ( ) has a unique occurrence of .Given a necessity form ξ ( ) and a formula ϕ ∈ L APAL int , the formula obtained by replacing by ϕ is denoted by ξ (ϕ).
In the completeness proof (Section 4) we use a complexity measure on formulas based on the size and 2-depth of formulas where the size of a formula is a weighted count of the number of symbols and 2-depth counts the number of the 2-modalities occurring in a formula.The measure was first introduced in [2].
The factor 4 in the clause for [ϕ]ψ is to ensure Lemma 7.Although the choice of the number 4 might seem arbitrary, it is the smallest natural number guaranteeing the desired result (see the proof of Lemma 7).
We now define three order relations on L APAL int based on the size and 2-depth of the formulas.
We let Sub(ϕ) denote the set of subformulas of a given formula ϕ.
Lemma 6 For any ϕ, ψ ∈ L APAL int , 1. < S , < d , < S d are well-founded strict partial orders between formulas in Lemma 7 For any ϕ, ψ, χ ∈ L APAL int and i ∈ A , Proof We only prove Lemma 7.4.The proof demonstrates why in the [ϕ]ψ clause of Definition 3, 4 is the smallest natural number guaranteeing the result.By Definition 3, we have that (This is similar in the first three items.)

Background
In this section, we introduce the topological concepts that will be used throughout this paper.All the concepts in this section can be found in [11].
Definition 8 A topological space (X, τ) is a pair consisting of a non-empty set X and a family τ of subsets of X satisfying / 0 ∈ τ and X ∈ τ, and closed under finite intersections and arbitrary unions.
The set X is called the space.The subsets of X belonging to τ are called open sets (or opens) in the space; the family τ of open subsets of X is also called a topology on X.If for some x ∈ X and an open The set of all interior points of A is called the interior of A and denoted by Int(A).We can then easily observe that for any A ⊆ X, Int(A) is the largest open subset of A. Definition 9 A family B ⊆ τ is called a base for a topological space (X, τ) if every non-empty open subset of X can be written as a union of elements of B.
Given any family Σ = {A α | α ∈ I} of subsets of X, there exists a unique, smallest topology τ(Σ) with Σ ⊆ τ(Σ) [11,Th. 3.1].The family τ(Σ) consists of / 0, X, all finite intersections of the A α , and all arbitrary unions of these finite intersections.Σ is called a subbase for τ(Σ), and τ(Σ) is said to be generated by Σ.The set of finite intersections of members of Σ forms a base for τ(Σ).

Structures
In this section we define our multi-agent models based on topological spaces.
Definition 10 Given a topological space (X, τ), a neighbourhood function set Φ on (X, τ) is a set of partial functions θ : X A → τ such that for all x, y ∈ Dom(θ ), for all i ∈ A , and for all U ∈ τ: We call the elements of Φ neighbourhood functions.
Definition 11 A topological model with functions (or in short, a topo-model) is a tuple M = (X, τ, Φ,V ), where (X, τ) is a topological space, Φ a neighbourhood function set, and V : Prop → X a valuation function.We refer to the part X = (X, τ, Φ) without the valuation function as a topo-frame.

Semantics
Definition 13 Given a topo-model M = (X, τ, Φ,V ) and a neighbourhood situation (x, θ ) ∈ M , the semantics for the language L APAL int is defined recursively as: for all topo-models M we have M |= ϕ.Soundness and completeness with respect to topo-models are defined as usual.
Let us now elaborate on the structure of topo-models and the above semantics we have proposed for L APAL int .Given a topo-model (X, τ, Φ,V ), the epistemic neighbourhoods of each agent at a given state x are determined by (partial) functions θ : X A → τ assigning an open neighbourhood to the state in question for each agent.We allow for partial functions in Φ, and close Φ under taking restricted functions θ | U where U ∈ τ (see Definition 10, condition 5), so that updated neighbourhood functions are guaranteed to be well-defined elements of Φ.As in the standard subset space semantics, by picking a neighbourhood situation (x, θ ), we first localize our focus to an open subdomain, in fact to Dom(θ ), including the state x and the epistemic neighbourhood of each agent at x determined by θ .Then the function θ (x) designates an epistemic neighbourhood for each agent i in A .It is guaranteed that every agent i is assigned a neighbourhood by θ at every state x in Dom(θ ), since each θ (x) is defined to be a total function from A to τ.Moreover, condition 2 of Definition 10 ensures that / 0 cannot be an epistemic neighbourhood, i.e., θ (x)(i) = / 0 for all x ∈ Dom(θ ).Finally, conditions 2 and 4 of Definition 10 make sure that the S5 axioms for each K i are sound with respect to all topo-models.
We now provide some semantic results.As usual in the subset space setting, truth of non-modal formulas only depends on the state in question.

Example
We illustrate our logic by a multi-agent version of Bjorndahl's convincing example in [6] about the jewel in the tomb.Indiana Jones (i) and Emile Belloq (e) are both scouring for a priceless jewel placed in a tomb.The tomb could either contain a jewel or not, the tomb could have been rediscovered in modern times or not, and (beyond [6]), the tomb could be in the Valley of Tombs in Egypt or not.The propositional variables corresponding to these propositions are, respectively, j, d, and t.We represent a valuation of these variables by a triple xyz, where x, y, z ∈ {0, 1}.Given carrier set X = {xyz | x, y, z ∈ {0, 1}}, the topology τ that we consider is generated by the base consisting of the subsets {000, 100, 001, 101}, {010}, {110}, {011}, {111}.The idea is that one can only conceivably know (or learn) about the jewel or the location, on condition that the tomb has been discovered.Therefore, {000, 100, 001, 101} has no strict subsets besides empty set: if the tomb has not yet been discovered, no one can have any information about the jewel or the location.
A topo-model M = (X, τ, Φ,V ) for this topology (X, τ) has Φ as the set of all neighbourhood functions that are partitions of X for both agents, and restrictions of these functions to open sets.A typical θ ∈ Φ describes complete ignorance of both agents and is defined as θ (s)(i) = θ (s)(e) = X.This corresponds most to the situation described in [6].A more interesting neighbourhood situation in this model is one wherein Indiana and Emile have different knowledge.Let us assume that Emile has the advantage over Indiana so far, as he knows the location of the tomb but Indiana doesn't.This is the θ such that for all x ∈ X, θ (x)(i) = X whereas the partition for Emile consists of sets {101, 100, 001, 000}, {111, 011}, {110, 010}, i.e., θ (111)(e) = {111, 011}, etc.
We now can evaluate what Emile knows about Indiana at 111, and confirm that this goes beyond Emil's initial epistemic neighbourhood.This situation however does not create any problems in our setting since Indiana's epistemic neighbourhoods will be determined relative to the states in Emile's initial neighbourhood.Firstly, Emile knows that the tomb is in the Valley of Tombs in Egypt M , (111, θ ) |= K e t and he also knows that Indiana does not know that The latter involves verifying M , (111, θ ) |= Ki t and M , (111, θ ) |= Ki ¬t.And this is true because θ (111)(i) = X, and 000, 001 ∈ X, and while M , (001, θ ) |= t, we also have M , (000, θ ) |= ¬t.We can also check that Emile knows that Indiana considers it possible that Emile doesn't know the tomb's location M , (111, θ ) |= K e Ki ¬(K e t ∨ K e ¬t) Announcements will change their knowledge in different ways.Consider the announcement of j.This results in Emile knowing everything but Indiana still being uncertain about the location.
Model checking this involves computing the epistemic neighbourhoods of both agents given by the updated neighbourhood function (θ ) j at 111.Observe that Int[[ j]] θ = {111, 110}.Therefore, (θ ) j (111 There is an announcement after which Emile and Indiana know everything (for example the announcement of j ∧ t): As long as the tomb has not been discovered, nothing will make Emile (or Indiana) learn that it contains a jewel or where the tomb is located: 3 Axiomatization We now provide the axiomatizations of EL int , PAL int , and APAL int , and prove their soundness and completeness with respect to the proposed semantics.

Definition 18
The axiomatization APAL int is given in Table 1.The axiomatization PAL int is the one without (DR5) and (R7).We get EL int if we further remove axioms (R1)-(R6) and the rule (DR4).
(P) all instantiations of propositional tautologies  Proof In the above, χ[p/ϕ] means uniform substitution of ϕ for p.The proof is not trivial but proceeds along similar lines as for public announcement logic, see [9].
Proposition 21 APAL int is sound with respect to the class of all topo-models.

Corollary 22
The axiomatizations EL int and PAL int are sound with respect to the class of all topomodels.

Completeness
We now show completeness for EL int , PAL int , and APAL int with respect to the class of all topo-models.Completeness of EL int is shown in a standard way via a canonical model construction and a Truth Lemma that is proved by induction on formula complexity.Completeness for PAL int is shown by reducing each formula in L PAL int to an equivalent formula of L EL int .The proof of the completeness for APAL int becomes more involved.Reduction axioms for public announcements no longer suffice in the APAL int case, and the inductive proof needs a subinduction where announcements are considered.Moreover, the proof system of APAL int has an infinitary derivation rule, namely the rule (DR5), and given the requirement of closure under this rule, the maximally consistent sets for that case are defined to be maximally consistent theories (see, Section 4.2).Lastly, the Truth Lemma requires the more complicated complexity measure on formulas defined in Section 2. There, we need to adapt the completeness proof of [2] to our setting.

Completeness of EL int and PAL int
For L EL int we define consistent and maximally consistent sets in the usual way, see e.g.[6] for details, and the multi-agent aspect does not complicate the definition.Let X c be the set of all maximally consistent sets of EL int .We define relations Lemma 23 (Lindenbaum's Lemma) Each consistent set can be extended to a maximally consistent set.
Definition 24 We define the canonical model X c = (X c , τ c , Φ c ,V c ) as follows: • X c is the set of all maximally consistent sets; • τ c is the topological space generated by the subbase Observe that, since int( ) = X c , we have [x] i ∩ int( ) = [x] i ∈ Σ for each i.Therefore, each [x] i is an open subset of X c .Moreover, the elements of Φ c satisfy the required properties given in Definition 10.
Proof Cases for the propositional variables and Booleans are straightforward.We only show the cases for K i and int.
Case ϕ := int(ψ) Then, by (int -T), since y is maximal consistent, we have ψ ∈ y.Thus, by IH, we have (y, θ * ) |= ψ.Therefore, Recall that the set of finite intersections of the elements of Σ forms a base, which we denote by B Σ , for where I 1 , . . ., I n are finite subsets of A , x 1 . . .x k ∈ X c and Form fin is a finite subset of L EL int .Since int is a normal modality, we can simply write where η∈Form fin η := γ.Since x is in each [x j ] i with 1 ≤ j ≤ k, we have [x j ] i = [x] i for all such j.Therefore, we have x where This implies, for all y ∈ ( i∈I [x] i ), if y ∈ int(γ) then ψ ∈ y.From this, we can say Then, by (int-K), (DR1) and since int(int(γ)) ↔ int(γ) and x ∈ int(γ) (i.e., int(γ) ∈ x) , we obtain int(ψ) ∈ x.
Theorem 26 EL int is complete with respect to the class of all topo-models.
Theorem 27 PAL int is complete with respect to the class of all topo-models.
Proof This follows from Theorem 26 by reduction in a standard way.The occurrences of the modality int on the right-hand-side of the reduction axioms (axioms (R1)-(R6)) should not lead to any confusion: extending the complexity measure defined in [9, Definition 7.21 p. 187] to the language L PAL int by adding the same complexity measure for the modality int as for K i gives us the desired result.

Completeness of APAL int
We now reuse the technique of [2] in the setting of topological semantics.Given the closure requirement under derivation rule (DR5) it seems more proper to call maximally consistent sets of APAL int maximally consistent theories, as further explained below.Definition 28 A set x of formulas is called a theory iff APAL int ⊆ x and x is closed under (DR1) and (DR5).A theory x is said to be consistent iff ⊥ ∈ x.A theory x is maximally consistent iff x is consistent and any set of formulas properly containing x is inconsistent.
Observe that APAL int constitutes the smallest theory.Moreover, maximally consistent theories of APAL int posses the usual properties of maximally consistent sets: Proposition 29 For any maximally consistent theory x, ϕ ∈ x iff ¬ϕ ∈ x, and ϕ ∧ ψ ∈ x iff ϕ ∈ x and ψ ∈ x.
In the setting of our axiomatization based on the infinitary rule (DR5), we will say that a set x of formulas is consistent iff there exists a consistent theory y such that x ⊆ y.Obviously, maximal consistent theories are maximal consistent sets of formulas.Under the given definition of consistency for sets of formulas, maximal consistent sets of formulas are also maximal consistent theories.Definition 30 Let ϕ ∈ L APAL int and i ∈ A .Then x + ϕ := {ψ | ϕ → ψ ∈ x} and K i x := {ϕ | K i ϕ ∈ x}.
Lemma 31 For any theory x of APAL int and ϕ ∈ L APAL int , x + ϕ is a theory and it contains x and ϕ, and K i x is a theory.
Lemma 33 (Lindenbaum's Lemma [1]) Each consistent theory can be extended to a maximal consistent theory.
Lemma 34 If K i ϕ ∈ x, then there is a maximally consistent theory y such that K i x ⊆ y and ϕ ∈ y.
Proof Let ϕ ∈ L APAL int and x be such that K i ϕ ∈ x.Thus, ϕ ∈ K i x.Hence, by Lemma 32, K i x + ¬ϕ is consistent.Then, by Lemma 33, there exists a maximally consistent set y such that K i x + ¬ϕ ⊆ y.Therefore K i x ⊆ y and ϕ ∈ y.
Proof Let ϕ ∈ L APAL int and x be a maximally consistent theory.
For further research, we envisage a finitary axiomatization for APAL int wherein the infinitary derivation rule (DR5) is replaced by a finitary rule.The obvious derivation rule would derive something after any announcement if it can be derived after announcing a fresh variable [1].Under subset space semantics, it is unclear how to prove that this rule is sound.
We are still investigating expressivity and (un)decidability.If the logic APAL int is undecidable, this would contrast nicely with the undecidability of arbitrary public announcement logic.Otherwise, there may be interesting decidable versions when restricting the class of models to particular topologies.
The logic APAL int is also axiomatizable on the class where the K modalities have S4 properties, a result we have not reported in this paper for consistency of presentation.This class is of topological interest.
In our setup all agents have the same observational powers.If agents can have different observational powers, we can associate a topology with each agent and generalize the logic to an arbitrary epistemic action logic.
Furthermore, we would like to explore the exact difference between the effort modality and the arbitrary announcement modality (in the single agent case, see [10]) by constructing a topological model which distinguishes the two: a topological model might have more than epistemically definable opens with respect to the proposed semantics.
The parts (DR1) to (DR5) are the derivation rules and the other parts are the axioms.A formula is a theorem of APAL int , notation ϕ, if it belongs to the smallest set of formulas containing the axioms and closed under the derivation rules.(Similarly for EL int and PAL int .)Lemma 19 Axiomatization APAL int satisfies substitution of equivalents.If ϕ ↔ ψ, then χ[p/ϕ] ↔ χ[p/ψ].