Dynamic Term-Modal Logic for Epistemic Social Network Dynamics (Extended Version)
aa r X i v : . [ c s . M A ] A ug Dynamic Term-Modal Logic for Epistemic SocialNetwork Dynamics (Extended Version)
Andrés Occhipinti Liberman and Rasmus K. Rendsvig DTU Compute [email protected] Center for Information and Bubble Studies, University of Copenhagen [email protected]
Abstract.
Logics for social networks have been studied in recent liter-ature. This paper presents a framework based on dynamic term-modallogic ( DTML ), a quantified variant of dynamic epistemic logic (DEL). Incontrast with DEL where it is commonly known to whom agent namesrefer,
DTML can represent dynamics with uncertainty about agent iden-tity. We exemplify dynamics where such uncertainty and de re / de dicto distinctions are key to social network epistemics. Technically, we showthat DTML semantics can represent a popular class of hybrid logic epis-temic social network models. We also show that
DTML can encode pre-viously discussed dynamics for which finding a complete logic was leftopen. As complete reduction axioms systems exist for
DTML , this yieldsa complete system for the dynamics in question.
Keywords: social networks, term-modal logic, dynamic epistemic logic
Over recent years, several papers have been dedicated to logical studies of socialnetworks, their epistemics and dynamics [2, 10–14, 18–22, 24, 25]. The purpose ofthis literature typically is to define and investigate some social dynamics withrespect to e.g. long-term stabilization or other properties, or to introduce formallogics that capture some social dynamics, or both.This paper illustrates how dynamic term-modal logic ( DTML , [1]) may beused for the second purpose. In general, term-modal logics are first-order modallogics where the index of modal operators are first-order terms. I.e., the operatorsdouble as predicates to the effect that e.g. ∃ xK x N ( x, a ) is a formula—read, inthis paper, as “there there exists an agent that knows of itself that it is a socialnetwork neighbor of a ”. The dynamic term-modal logic of [1] extends term-modal logic with suitably generalized action models that can effectuate bothfactual changes (e.g. to the network structure) as well as epistemic changes. Forall the DTML action model encodable dynamics, [1] presents a general soundand complete reduction axiom-based logic in the style of dynamic epistemiclogic (DEL, [3, 4]). Hence, whenever an epistemic social network dynamics isencodable using
DTML , completeness follows. With this in mind, the main goalf this paper is to introduce and illustrate
DTML as a formalism for representingepistemic social network dynamics, and to show how it may be used to obtaincompleteness results.To this end, the paper progresses as follows. Sec. 2 sketches some commonthemes in the logical literature on social networks before introducing
DTML andits application to epistemic social networks. Sec. 2 contains the bulk of the paper,with numerous examples of both static
DTML models and action models. Theexamples are both meant to showcase the scope of
DTML and to explain the morenon-standard technical details involved in calculating updated models. In Sec. 3,we turn to technical results, where it is shown that
DTML may encode popularstatic hybrid logical models of epistemic networks, as well as the dynamics of [12],for which finding a complete logic was left open. Sec. 4 contains final remarks.
To situate
DTML in the logical literature on social networks, we cannot butdescribe the literature in broad terms. We omit both focus, formal details andmain results of the individual contributions in favor of a broad perspective. Thatsaid, then all relevant literature in one way or other concern social networks . Ingeneral, a social network is a graph ( A, N ) where A is a set of agents and N ⊆ A × A is represents a social relation, e.g., being friends on some socialmedia platform. Depending on interpretation, N may be assumed irreflexiveand symmetric. Social networks may be augmented with assignments of atomicproperties to agents, representing e.g. behaviors, opinions or beliefs. One set ofpapers investigates such models and their dynamics using fully propositionalstatic languages [13, 20, 24, 25].A second set of papers combines social networks with a semantically repre-sented epistemic dimension in the style of epistemic logic. In these works, thefundamental structure of interest is (akin to) a tuple ( A, W, { N w } w ∈ W , ∼ ) with agents A and worlds W , with each world w associated with a network N w ⊆ A × A , and ∼ : A → P ( W × W ) associating each agent with an indistinguishability(equivalence) relation ∼ a . Call such a tuple an epistemic network structure .The existing work on epistemic network structures may be organized in termsof the static languages they work with: propositional modal logic [2,14] or hybridlogic [9–12, 18, 19, 21, 22]. In the former, the social network is described usingdesignated atomic propositions (e.g., N ab for ‘ b is a neighbor of a ’). To producea model, an epistemic network structure is augmented with a propositional val-uation V : P → P ( W ) . Semantically, N ab is then true at w iff ( a, b ) ∈ N w .Knowledge is expressed using operators { K a } a ∈ A as in standard epistemic logicwith K a the Kripke modality for ∼ a .In the hybrid case, the network is instead described using modal operators.The hybrid languages typically include a set of agent nominals N om (agentnames), atoms P and indexical modal operators K and N , read “I know that”nd “all my neighbors”. Some papers additionally include state nominals, hy-brid operators ( @ x , ↓ x ) and/or universal modalities U (“for all agents”). A hy-brid network model is an epistemic network structures extended with twoassignments: a nominal assignment g : N om → A that names agents, and atwo-dimensional hybrid valuation V : P → P ( W × A ) , where ( w, a ) ∈ V ( p ) represents that the indexical proposition p holds of agent a at w . The satis-faction relation is relative to both an epistemic alternative w and an agent a ,where the noteworthy clause are: M, w, a | = p iff ( w, a ) ∈ V ( p ) ; M, w, a | = Kϕ iff M, v, a | = ϕ for every v ∼ a w ; and M, w, a | = N ϕ iff M, w, b | = ϕ for every b such that N w ( a, b ) . With these semantics, formulas are read indexically. E.g. KN p reads as “I know that all my neighbors are p ”.In relation to these two language types, the term-modal approach of this pa-per lies closer to the former: By including a binary ‘neighbor of’ relation symbol N in the signature of a term-modal language, the social network component ofmodels is described non-modally. This straightforwardly allows expressing e.g.that that all agents know all their neighbors ( ∀ x ∀ y ( N ( x, y ) → K x ( N ( x, y )) ) orthat an agent has de re vs. de dicto knowledge of someone being a neighbor( ∃ xK a N ( a, x ) vs. K a ∃ xN ( a, x ) ). Moreover, hybrid languages can be translatedinto DTML , in such a way that hybrid formulas such as @ a p (“agent a has prop-erty p ”) become equivalent to P ( a ) , if a is the name of a . In general, term-modal languages may be based on any first-order signature, byfor the purposes of representing social networks and factual properties of agents,we limit attention to the following: Definition 1. A signature is a tuple Σ = ( V , C , P , N, . =) with V a countablyinfinite set of variables, C and P countable sets of constants and unary predicates, N a binary relation symbol and . = for identity. The terms of Σ are T := V ∪ C .With t , t ∈ T , x ∈ V and P ∈ P , the language L ( Σ ) is given by ϕ := P ( t ) | N ( t , t ) | ( t . = t ) | ¬ ϕ | ϕ ∧ ϕ | K t ϕ | ∀ xϕ Standard Boolean connectives, ⊤ , ∃ and ˆ K t are defined per usual. With ϕ ∈L , t ∈ T , x ∈ V , the result of replacing all occurrences of x in ϕ with t is denoted ϕ ( x t ) . Formulas from the first three clauses are called atoms ; if an atomcontains no variables, it is ground .Throughout, a, b , etc. are used for constants and the relation symbol N de-notes a social network. The reading of N ( t , t ) depends on application. K t ϕ isa term-indexed epistemic operator which read as “agent t knows that ϕ ”. L ( Σ ) neither enforces nor requires a fixed-size agent set A , in contrast with standardepistemic languages, where the set of operators is given by reference to A . Hencethe same language may be used to describe networks of varying size. The defined are special cases of the setting in [1], which allows general signaturesand non-agent terms. [1] also reviews the term-modal literature. o interpret L ( Σ ) , we use constant-domain models (the same number ofagents in each world) with non-rigid constants (names, like predicates and re-lations, may change extension between worlds; this allows for uncertainty aboutagent identity). See Figs. 1 and 2 for examples of such models. a b c w : a b c v : a b c u : a, b, c a, b, c Fig. 1. Example 1, pt. 1 (Server Error).
Three agents a , b and c work in a companywith a hierarchical command structure, −→ : a is the direct boss of b , who is the directboss of c . The server has thrown an error after both b and c tampered with it. Either w ) the server failed spontaneously, v ) b made a mistake (marked by gray) or u ) c made amistake. Lines represent indistinguishability with reflexive and transitive links omitted.There is no uncertainty about the hierarchy, but nobody knows why the server failed.In fact, c made a mistake: the actual world has a thick outline. Definition 2. An L ( Σ ) - model is a tuple M = ( A, W, ∼ , I ) where A is a non-empty domain of agents , W is a non-empty set of worlds , ∼ : A → P ( W × W ) assigns to each agent a ∈ A an equivalence relation on W denoted ∼ a , and I is an interpretation satisfying, for all w ∈ W , 1. for c ∈ C , I ( c, w ) ∈ A ; 2. for P ∈ P , I ( P, w ) ⊆ A ; 3. I ( N, w ) ⊆ A × A . A pointed model is a pair ( M, w ) with w ∈ W called the actual world .A variable valuation of Σ over M is a map g : V → A . The valuation identicalto g except mapping x to a is denoted g [ x a ] . The extension of the term t ∈ T at w in M under g is J t K I,gw = g ( t ) for t ∈ V and J t K I,gw = I ( t, w ) for t ∈ C .Given the inclusion of N in the signature Σ , each L ( Σ ) -model embeds an epis-temic network structure ( A, W, ( ∼ a ) a ∈ A , ( I ( N, w )) w ∈ W ) . Formulas are evaluated over pointed models using a direct combination offirst-order and modal semantics:
Definition 3.
Let Σ , M and g be given. The satisfaction of formulas of L ( Σ ) is given recursively by M, w (cid:15) g P ( t ) iff J t K I,gw ∈ I ( P, w ) , for P ∈ P . M, w (cid:15) g N ( t , t ) iff ( J t K I,gw , J t K I,gw ) ∈ I ( N, w ) . M, w (cid:15) g ( t . = t ) iff J t K I,gw = J t K I,gw . M, w (cid:15) g ¬ ϕ iff not M, w (cid:15) g ϕ . M, w (cid:15) g ϕ ∧ ψ iff M, w (cid:15) g ϕ and M, w (cid:15) g ψ . M, w | = g ∀ xϕ iff M, w | = g [ x a ] ϕ for all a ∈ A . M, w (cid:15) g K t ϕ iff M, w ′ (cid:15) g ϕ for all w ′ such that w ∼ J t K I,gw w ′ . .2 Knowing Who and Knowledge De Dicto and
De Re
First-order modal languages can represent propositional attitudes de dicto (aboutthe statement) and de re (about the thing) in principled manners. For example, K a ∃ xP ( x ) is a de dicto statement: knowledge is expressed about the proposi-tion that a P -thing exists. In contrast, ∃ xK a P ( x ) is a de re statement: it isexpressed that of some thing x , that x is known to be a P -thing. In general, dere statements are stronger than de dicto statements. The difference has been ap-preciated in epistemic logic since Hintikka’s seminal [16], where he argues that ∃ xK a ( x . = b ) expresses that a knows who b is (see Fig. 2). Semantically, theformula ensures that the constant b refers to the same individual in all a ’s epis-temic alternatives (i.e., b is locally rigid ). Both de dicto and de re statementsmay partially be expressed in propositional languages (e.g. de dicto K a ( p b ∨ p c ) vs. de re K a p b ∨ K a p c ; see [2] for such a usage), but not in a principled manner:the required formulas will depend on the specific circumstances. t bh i c w : t t, b b, a h t bh i c w : t b, b t, a h t bh i c w : t t, b b, a i t bh i c w : t b, b t, a i c cc c Fig. 2. Example 2, pt.1 (Knowing Who).
Two thieves, t and b , hide in a buildingwith hostages h and i . Outside, a cop, c , waits. To communicate safely, the thievesuse code names ‘Tokyo’ and ‘Berlin’ for each other and ‘The Asset’ for the speciallyvaluable hostage h . Agents t, b, h and i all know whom the code names denote (thenames are rigid for them), but the cop does not. The code names are t for t , b for b and a for h . Known by all, h and i are in fact called h and i . The thief network (—) isassumed symmetric and transitive. The case is modeled using four worlds, identical upto code name denotation, (shown by ). E.g., in the actual world is w , t names t , butin w , it names b . Hence the cop does not know who Tokyo is: M, w (cid:15) g ¬∃ xK c ( x . = t ) . To code operations on static models, we use a a variant of DEL-style actionmodels, adapted to term-modal logic (see Fig. 3). They include (adapted versionsof) preconditions specifying when an event is executable ( [3, 4]), postconditions describing the factual effects of events ( [5, 7, 15]) as well as edge-conditions representing how an agent’s observation of an action depends on the agent’scircumstances ( [6])—for example their position in a network, cf. Fig. 3. Edge-conditions are non-standard and deserve a remark. With E the set of events,edge-conditions are assigned by a map Q . For each edge ( e, e ′ ) ∈ E × E , Q ( e, e ′ ) is a formula with a single free variable x ⋆ . Given a model M , an agent i cannotistinguish e from e ′ iff the edge-condition Q ( e, e ′ ) is true in M when the freevariable x ⋆ is mapped to i . Intuitively, if the situation described by the edge-condition is true for i , the way in which i is observing the action does not allowher to tell whether e or e ′ is taking place. See Figure 4 for an example. See [1] fora comparison of this approach to that of [6] and the term-modal action modelsof [17]. ¬∃ xM ( x ) ⊤ M ( b ) ⊤ M ( c ) ⊤ ∃ xM ( x ) ⊤ ϕ ϕ ϕϕ := ∃ xN ( x, x ⋆ ) Fig. 3. Example 1, pt. 2 (Edge-Conditions: Announcement to Subgroup).
Tolearn what happened to the server, the top boss a requests its log file. The log holdsone of four pieces of information: Nobody made a mistake, b made a mistake ( M ), c made a mistake or somebody made a mistake. Each box represents one of theseevents: top lines are preconditions, bottom lines postconditions ( ⊤ means no factualchange). In fact, the log rats on c . N denotes the hierarchy. The log is send only to thetop boss: the others cannot see its content. This is represented by the edge-condition ϕ : If you, x ⋆ , have a boss, then you cannot tell from nor from etc. Forunillustrated edges, Q ( e, e ) = ( x ⋆ . = x ⋆ ) and Q ( e, e ′ ) = ϕ when e = e ′ . For simplicity, we here only define action models that take pre-, post, andedge-conditions in the static language L ( Σ ) . However, dynamic conditions areneeded for completeness; we refer to [1] for details. Definition 4. An action model for L ( Σ ) is a tuple ∆ = ( E, Q, pre , post ) where ✄ E is a non-empty, finite set of events . ✄ Q : ( E × E ) → L ( Σ ) where each edge-condition Q ( e, e ′ ) has exactly onefree variable x ⋆ . ✄ pre : E → L ( Σ ) where each precondition pre ( e ) has no free variables. ✄ post : E → ( GroundAtoms ( L ( Σ )) → L ( Σ )) assigns to each e ∈ E a post-condition for each ground atom.To preserve the meaning of equality, let post ( e )( t . = t ) = ⊤ for all e ∈ E .With no general restrictions on Q , to ensure that all agents’ indistinguishabilityrelations continue to be equivalence relations after updating, Q must be cho-sen with care. Throughout, we assume Q ( e, e ) = ( x ⋆ . = x ⋆ ) for all e ∈ E . Toupdate, product update may be altered to fit the edge-condition term-modal set-ting as below. Fig. 4 illustrates the product update of Figs. 1 with 3. The use ofpostconditions is illustrated in Figs. 7 and 8. efinition 5. Let M = ( A, W, ∼ , I ) and ∆ = ( E, Q, pre , post ) be given. The product update of M and ∆ is the model M ⊗ ∆ = ( A ′ , W ′ , ∼ ′ , I ′ ) where A ′ = A W ′ = { ( w, e ) ∈ W × E : ( M, w ) (cid:15) g pre ( e ) } for any g , ( w, e ) ∼ ′ i ( w ′ , e ′ ) iff w ∼ i w ′ and M, w (cid:15) g [ x ⋆ i ] Q ( e, e ′ ) , I ′ ( c, ( w, e )) = I ( c, w ) for all c ∈ C , and I ′ ( X, ( w, e )) = ( I ( X, w ) ∪ X + ( w )) \ X − ( w ) , for X = { P, N } , P ∈ P , where: P + ( w ) := { J t K I,vw : (
M, w ) (cid:15) g post ( e )( P ( t )) } ; P − ( w ) := { J t K I,vw : (
M, w ) (cid:15) g post ( e )( P ( t )) } ; N + ( w ) := { ( J t K I,vw , J t K I,vw ) : (
M, w ) (cid:15) g post ( e )( N ( t , t )) } ; N − ( w ) := { ( J t K I,vw , J t K I,vw ) : (
M, w ) (cid:15) g post ( e )( N ( t , t )) } If ( M, w ) | = pre ( e ) , then ( A, e ) is applicable to ( M, w ) , and the product updateof the two is the pointed model ( M ⊗ ∆, ( w, e )) . Else it is undefined. a b c w a b c v a b c u a b c v a b c u b, c b, cb, c a, b, cb, c b, c Fig. 4. Example 1, pt. 3 (Product Update: Edge-Conditions).
The productupdate of Fig. 1 and Fig. 3. After checking the logs, the boss has learned that c madea mistake, while both b and c are now both uncertain about this, as well as about theboss’ information. Worlds are named using by the world-event pair they represent: w isthe child of w and , etc. The pair w is not a world: w did not satisfy the preconditionof . We have w ∼ ′ b v as w ∼ b v and M, w (cid:15) g [ x ⋆ a ] Q (1 , —as M, w (cid:15) g ∃ xN ( x, b ) .Likewise, v ∼ ′ b w as v ∼ b w and M, v (cid:15) g ∃ xN ( x, b ) . That w ′ a v follows as M, w (cid:15) g ¬∃ xN ( x, b ) , but v ∼ ′ a u as M, v (cid:15) g ( a . = a ) . The same reason, reflexiveloops are preserved. The boss now knows that c made a mistake: K a M ( c ) . De Dicto and
De Re
With de dicto and de re statements expressible in
DTML , they may be usedto define principled announcements, as exemplified in Fig. 5 and 6. The actionmodels are applicable to any
DTML model for a signature that includes theconstant a and the predicate M , irrespective of the size of the set of agents. Thislevel of general applicability is not mirrored in standard DEL action models. a ∃ xM ( x ) ⊤ e : a b c v e : a b c u e : a b c v e : a b c u e : b, ca, b, cb, c b, c Fig. 5. Example 1, pt. 3 (De Dicto Announcement).
The boss breaks the newsfrom the log to b and c piecemeal. Left:
First, a makes a de dicto announcement: a knows that somebody made a mistake. Right:
The effect on Fig. 4. Only w does notsurvive. In u e , everybody knows de dicto that somebody messed up: ∀ xK x ∃ yM ( y ) .The boss also knows de re , i.e., knows who : u e (cid:15) g ∃ xK a M ( x ) , as u e (cid:15) g [ x c ] K a M ( x ) . The employees do not know that a knows de re : u e (cid:15) g ∀ x ( ∃ yN ( y, x ) → ˆ K x ¬∃ zK a M ( z )) —since v e (cid:15) g M ( x ) iff g ( x ) = b , but then u e (cid:15) g M ( x ) . I.e., there isno one object to serve as valuation for x such that v e and u e satisfy M ( x ) simulta-neously). The employees are held in suspense! ∃ xK a M ( x ) ⊤ σ : a b c v eσ : a b c u eσ : b, c Fig. 6. Example 1, pt. 4 (De Re Announcement).
Following a dramatic pause,the boss reveals a stronger piece of information: the boss knows who messed up. This de re announcement is on the left, with Q ( e, e ) = ( x ⋆ = x ⋆ ) ; its result on Fig. 5 (Right)on the right. In u eσ , everybody knows that a has de re knowledge: ∀ xK x ∃ yK a M ( y ) ,but b and c still only have de dicto knowledge: ∀ x (( x = b ∨ x = c ) → K x ∃ yM ( y ) ∧¬∃ zK x M ( z )) . Action models with postconditions allows
DTML to represent changes to thesocial network. Such changes may be combined with the general functionalityof action models such that some agents may know what changes occur whileothers remain in the dark. Fig. 7 provides a simple example, including the detailscalculating the updated network. Fig. 8 presents an example of how de re/dedicto knowledge affects what is learned by a publicly observed network change. ⊤ N ( a, b ) , N ( b, c )
7→ ⊥ , N ( a, c )
7→ ⊤ † : a b c v eσ † : a b c u eσ † : b, c Fig. 7. Example 1, pt. 5 (Getting Fired).
The employees are dying to know whomessed up the server. But the boss just proclaims: ‘ b , you are fired! c , you are pro-moted!’ Left:
Action with three instructions for factual change: post ( † )( N ( a, b )) = ⊥ , post ( † )( N ( b, c )) = ⊥ and post ( † )( N ( a, c )) = ⊤ (illustrated by ). Else post = id . As u eσ (cid:15) ⊥ , the first two instructions entail that ( a, b ) , ( b, c ) ∈ N − ( u eσ ) , while the lat-ter implies that ( a, c ) ∈ N + ( u eσ ) . Right:
The network is updated to I ′ ( N, u eσ † ) =( I ( N, u eσ ) ∪ N + ( u eσ )) \ N − ( u eσ ) = ( { ( a, b ) , ( b, c ) } ∪ { ( a, c ) } ) \{ ( a, b ) , ( b, c ) } = { ( a, b ) } . In u eσ † , neither b nor c know who made the mistake. Unrepresented, a thinks that only bad superiors let their employees make mistakes. N ( · , a ) , N ( a, · )
7→ ∃ xN ( · , x ) e : t bh i c w e : t t, b b, a h t bh i c w e : t b, b t, a h t bh i c w e : t t, b b, a i t bh i c w e : t b, b t, a i c cc c Fig. 8. Example 2, pt.2 (Becoming Criminal)
Left:
The thieves convince TheAsset to cooperate with them, in exchange for stolen goods. For simplicity, assume thatthe action of a joining the thief network is noticed by everyone. We model this withthe action model, with post ( e )( N ( · , a )) = ∃ xN ( · , x ) and post ( e )( N ( a, · )) = ∃ xN ( x, · ) for · ∈ { t, b, a, h, i, c } . Informally, these say: “If you are a member of the network, then a becomes your neighbor”. Right:
The effect of event e on Fig. 2: The network haschanged in all worlds, but differently. E.g., in w , we had ¬ N ( b, a ) ; in ( w , e ) , we have N ( b, a ) as ( b, h ) ∈ N + (( w , e )) since w (cid:15) g post ( e )( N ( b, a )) —i.e., ∃ xN ( b, x ) . Now allthieves and hostages know the new network, as they know whom a refers to. E.g.: Tokyoknows all her neighbors, ( w , e ) (cid:15) g ∀ x ( N ( t, x ) → K t N ( t, x )) . The cop only learns that some hostage has joined the network, but can’t tell whom: ( w , e ) (cid:15) g K c ∃ x ( x . = t ∧ x . = b ∧ N ( t, x )) but ( w , e ) (cid:15) g ∃ xK c ( x . = t ∧ x . = b ∧ N ( t, x )) . Allowing for the possibility of non-rigid names has the consequence that pub-lic announcements of atomic propositions may differ in informational contentdepending on the epistemic state of the listener. This can be exploited by thethieves of Example 2 to enforce a form of privacy —as code names should. Thenotion of privacy involved is orthogonal to the notion of privacy modeled in DELusing private announcements. Though the message is public in the standard senseof everyone being aware of it and its content, as it involves non-rigid names, itsepistemic effects are not the same for all agents. This is in contrast with standardpublic announcements, which yield the same information to everyone. a . = h ⊤ σ : t bh i c w eσ : t t, b b, a h t bh i c w eσ : t b, b t, a h c Fig. 9. Example 2, pt.4 (Revealing the Asset)
In the model in Fig. 8 (Right),even a public announcement of N ( t, a ) would not inform the cop about who joined thenetwork. To know who joined the network, the cop must learn who The Asset is . Asthe cop knows who h is, learning that h is The Asset suffices. Left:
The event model σ for the public announcement that a . = h , revealing the identity of The Asset. Right:
The product update of Fig. 8 (Right) and event σ . The cop now knows the structureof the network, as a result of the removal of w e and w e . Embedding Dynamic Social Network Logics in
DTML
This section examines relations between the hybrid network models and theirlanguages to
DTML . As hybrid languages corresponds to fragments of first-orderlogic with equality (
FOL = ), which term-modal logic extends, it stands to reasonthat the hybrid languages and models mentioned in Sec. 2 may be embedded interm-modal logic. A precise statement and a proof sketch follows below. Turningto dynamics, things are more complicated. [22] presents a very flexible hybridframework expressing network dynamics using General Dynamic Dynamic Logic ( GDDL , [23]). We leave general characterizations of equi-expressive fragments of
GDDL and
DTML as open question, but remark that all
GDDL action-examplesof [22] may be emulated using
DTML action models, and in many cases via fairlysimple ones. More thoroughly, we show that the logic of
Knowledge, Diffusionand Learning ( KDL , [12]) has a complete and decidable system, a question leftopen in [12]. This is shown by encoding
KDL in DTML . The static hybrid languages of [9–12,19,21,22] are all sub-languages of L ( P, N om ) ,defined and translated into DTML below. [18] also includes state nominals, whichour results do not cover. L ( P, N om ) is read indexically, as described in Sec. 2. Definition 6.
With p ∈ P and x ∈ N om , the language L ( P, N om ) is given by ϕ := p | ¬ ϕ | ϕ ∧ ϕ | @ x ϕ | Kϕ | N ϕ | U ϕ
Denote the fragments without U and @ x by L − U ( P, N om ) and L − @ ( P, N om ) . Hybrid logics may be translated into
FOL = ; our translation resembles thatof [8]. We identify agent nominals with first-order variables, translate the modaloperator N to the relation symbol N ( · , · ) , and relativize the interpretation ofthe indexical K to the nominal/variable x by using the term-indexed operator K x . Formally, the translation is defined as follows. Definition 7.
Let Σ n ( P, N om ) = ( V , C , P , N, ˙=) be the signature with V = N om , C = { a , . . . , a n } and P = P . Translations T x , T y both mapping L ( P, N om ) to L ( Σ n ( P, N om )) are defined by mutual recursion. It is assumed that two nominals x and y are given which do not occur in the formulas to be translated. For p ∈ P and i ∈ N om , define T x by: T x ( p ) = p ( x ) T x (@ i ϕ ) = T x ( ϕ )( x i ) T x ( i ) = x ˙= i T x ( N ϕ ) = ∀ y ( N ( x, y ) → T y ( ϕ )) T x ( ϕ ∧ ψ ) = T x ( ϕ ) ∧ T x ( ψ ) T x ( Kϕ ) = K x T x ( ϕ ) T x ( ¬ ϕ ) = ¬ T x ( ϕ ) T x ( U ϕ ) = ∀ xT x ( ϕ ) The translation T y is obtained by exchanging x and y in T x . To show the translation truth-preserving, we embed the class of hybrid net-work models into a class of term-modal models: efinition 8.
Let M = ( A, W, ( N w ) w ∈ W , ∼ , g, V ) be a hybrid network modelfor L ( P, N om ) . Then the TML image of M is the L ( Σ n ( P, N om )) TML model T ( M ) = ( A, W, ∼ , I ) sharing A, W and ∼ with M and with I given by1. ∀ c ∈ C , ∀ w, v ∈ W, ∀ a, b ∈ A, ( I ( c, w ) = a and w ∼ b v ⇒ I ( c, v ) = a ) I ( p, w ) = { a : ( w, a ) ∈ V ( p ) } I ( N, w ) = { ( a, b ) ∈ A × A : ( a, b ) ∈ N w } The model T ( M ) has the same agents, worlds and epistemic relations as M .The interpretation 1. encodes weak rigidity : if ( w, v ) ∈ S a ∈ A ∼ a , then anyconstant denotes the same in w and v , emulating the rigid names of hybridnetwork models; 2. ensures predicates are true of the same agents at the sameworlds, and 3. ensures the same agents are networked in the same worlds.With the translations T x , T y and the embedding T , it may be shown that DTML can fully code the static semantics of L ( P, N om ) hybrid network logics: Proposition 1.
Let M = ( A, W, ( N w ) w ∈ W , ∼ , g, V ) be a hybrid network model.Then for all ϕ ∈ L ( P, N om ) , M, w, g ( • ) | = ϕ iff T ( M ) , w | = g T • ( ϕ ) , • = x, y . KDL
Dynamic Transformations and Learning Updates in
DTML
We show that
KDL [12] dynamics may be embedded in
DTML , for finite agent sets(as assumed in [12]). Given Prop. 1, we argue that each
KDL model transformeris representable by a
DTML action model and that the dynamic
KDL language istruth-preservingly translatable into a
DTML sublanguage. The logic of the classof
KDL models is, up to language translations, the logic of its correspondingclass of
DTML models. We show that the logic of this class of
DTML modelscan be completely axiomatized, and the resulting system is decidable . Thus, byembedding
KDL in DTML , we find a complete system for the former.In
KDL , agents are described by feature propositions reading “for feature f ,I have value z ”. With F a countable set of features and Z f a finite set of possiblevalues of f ∈ F , the set of feature propositions is FP = { ( f + z ) : f ∈ F , z ∈ Z f } .The static language of [12] is then L − U ( FP , N om ) . The dynamic language L KDL extends L − U ( FP , N om ) with dynamic modalities [ d ] and [ ℓ ] for dynamic trans-formations d and learning updates ℓ : ϕ ::= ( f + z ) | i | ¬ ϕ | ϕ ∧ ϕ | @ i ϕ | N ϕ | Kϕ | [ d ] ϕ | [ ℓ ] ϕ A dynamic transformation d changes feature values of agents: each is a pair d = ( Φ, post ) where Φ ⊆ L KDL is a non-empty finite set of pairwise inconsistentformulas and post : Φ × F → ( Z n ∪ { ⋆ } ) is a KDL post-condition. Encoded by post ( ϕ, f ) = x is the instruction: if ( w, a ) (cid:15) ϕ , then after d , set f to value x at ( w, a ) , if x ∈ Z n ; if x = ⋆ , f is unchanged. A learning update cuts accessibilityrelations: the update with finite ℓ ⊆ L KDL keeps a ∼ a link between worlds w and v iff, for all ϕ ∈ ℓ , ( w, b ) (cid:15) ϕ ⇔ ( v, b ) (cid:15) ϕ for all neighbors b of a . Notation here is equivalent but different to fit better with the rest of this paper. efinition 9.
Given a
KDL model M = ( A, W, ( N w ) w ∈ W , ∼ , g, V ) , the modelreached after applying d is M d = ( A d , W d , ( N dw ) w ∈ W , ∼ d , g d , V d ) where only V d is different, and is defined as follows: ( w, a ) ∈ V d ( f + z ) iff (a) post ( ϕ, f ) = x for some ϕ ∈ Φ such that M, w, a | = ϕ , where x = ⋆ ; or (b) condition (a) doesnot hold and ( w, a ) ∈ V ( f + z ) . Definition 10.
A learning update is a finite set of formulas ℓ ⊆ L KDL . Givena
KDL model M = ( A, W, ( N w ) w ∈ W , ( ∼ a ) a ∈ A , g, V ) , the model after ℓ is M ℓ =( A, W, ( N w ) w ∈ W , ( ∼ ′ a ) a ∈ A , g, V ) where: w ∼ ′ a v iff w ∼ a v and ∀ b ∈ A ( N w ( a, b ) ⇒ ∀ ϕ ∈ ℓ ( M, w, b | = ϕ iff M, v, b | = ϕ )) Let D and L be the sets of dynamic transformations and learning updates. Theresult of applying † ∈ D ∪ L to M is denoted M † , and the [ † ] modality hassemantics M, w, a | = [ † ] ϕ iff M † w, a | = ϕ .As we show below, for every † ∈ D ∪ L , there is a pointed DTML actionmodel ∆ † with identical effects. As KDL operations may involve formulas with [ † ] -modalities, we must use DTML action models that allow [ ∆, e ] -modalities intheir conditions, and translate L KDL into the general
DTML language that results,denoted L ( Σ n ( FP , N om )+[ ∆ ]) . This language is interpreted over
DTML modelswith standard action model semantics: ( M, w ) (cid:15) g [ ∆, e ] ϕ iff M ⊗ ∆, ( w, e ) (cid:15) ϕ. We define now the action models ∆ † . For a dynamic transformation d ∈ D ,[11] provide reduction axioms showing d ’s instructions statically encodable in L KDL ) . The reduction axiom for atoms is as follows: [ d ] f + z ↔ _ ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ ∨ ¬ _ ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ ∧ f + z As d changes atomic truth values under a definable instruction, its effectsmay be simulated by an action model with a matching post-condition (i.e., thetranslation of the definable instruction). More specifically, the action model ∆ d is defined as follows. Definition 11.
For dynamic transformation d = ( Φ, post ) , the action model ∆ d = ( E, Q, pre , post ) is defined by E = { e d } , Q ( e d , e d ) = pre ( e d ) = ⊤ and foreach constant a , post ( e )( T x ( f + z )( x a )) = T x _ ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ ∨ ¬ _ ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ ∧ f + z ( x a ) For a learning update ℓ ∈ L , ∆ ℓ has events e X , e Y for any consistent subsets X, Y of { ϕ ( c ) , ¬ ϕ ( c ) : ϕ ∈ ℓ, c ∈ C } with edge-condition Q ( e X , e Y ) satisfied foragents for whom all neighbors agree on X and Y . Unsatisfied edge-conditionsthereby capture the link cutting mechanism of ℓ. The detailed definition of ∆ ℓ is as follows. Defined using double recursion as standard; see [1] for details. efinition 12.
Let ℓ = { ϕ , . . . , ϕ m } be a learning update. Let T x ( ℓ ) := { T x ( ϕ i ) | i = 1 , . . . , n } and let G ℓ := { T x ( ϕ )( x a ) | T x ( ϕ ) ∈ T x ( ℓ ) , a ∈ C } be the ground-ing of T x ( ℓ ) obtained by replacing each free occurrence of x in T x ( ϕ ) for eachpossible constant a ∈ C . Define a G ℓ -valuation as a function val : G ℓ → { , } and let V ℓ be the set of all such valuations. Definition 13.
Let ℓ be a learning update. The corresponding DTML actionmodel ∆ ℓ = ( E ℓ , Q ℓ , pre ℓ , post ℓ ) is defined by letting ✄ E ℓ = { e val | val ∈ V ℓ } , ✄ pre ℓ ( e val ) = V { ϕ | val ( ϕ ) = 1 } ∪ {¬ ϕ | val ( ϕ ) = 0 } ✄ Q ℓ ( e val , e val ) = ⊤ ✄ Q ℓ ( e val , e val ′ ) = V { a ∈ C |∃ ϕ ∈ ℓ s.t. val ( T x ( ϕ )( x a )) = val ′ ( T x ( ϕ )( x a )) } ¬ N ( x ⋆ , a ) , forany two distinct events e val , e val ′ ✄ post ℓ ( e ) = id for all e ∈ E ℓ Note that the signature Σ n ( F P, AN om ) is defined to have finitely many con-stants C = { a , . . . , a n } , and hence both E , the preconditions and the edge-conditions in ∆ ℓ are finite, as required. The action model ∆ ℓ works as follows.Each event e val corresponds to one way the agents can be with respect to G ℓ ,as indicated by val . The edge conditions control how links get cut. Two worlds ( w, e val ) and ( v, e val ′ ) in the updated model will keep a link for the agent named a , if any disagreement between val and val ′ does not concern a neighbor of a .Or, equivalently, if all neighbors of a are identical with respect to G ℓ . Preciselythis condition is encoded in Q ( e val , e val ′ ) .To formally state that the dynamics of † ∈ D ∪ L are simulated by ∆ † , thefollowing clauses are added to translation T • , for • = x, y : T • ([ d ] ϕ ) = [ ∆ d , e d ] T • ( ϕ ) ,T • ([ ℓ ] ϕ ) = ^ e ∈ E ℓ ( pre ℓ ( e ) → [ ∆ ℓ , e ] T • ( ϕ )) where ( ∆ † , e † ) is an action model implementations of † ∈ D ∪ L . Then KDL statics and dynamics can be shown performable in
DTML : Proposition 2.
For any finite agent hybrid network model M with nominalvaluation g and ϕ ∈ L KDL : M, w, g ( • ) | = ϕ iff T ( M ) , w | = g T • ( ϕ ) , for • = x, y .Proof. By induction on ϕ . We include the cases for the dynamic modalities.Let ϕ = [ d ] ψ , where d = ( Φ, post ) . We need to show that M, w, g ( x ) | = [ d ] ψ iff T ( M ) , w | = g [ ∆ d , e d ] T x ( ψ ) (the case for T y is analogous). Note that M, w, g ( x ) | = [ d ] ψ iff M d , w, g ( x ) | = ψ iff(by i.h.) T ( M d ) , w | = g T x ( ψ ) . We will show that T ( M d ) and T ( M ) ⊗ ∆ d satisfythe same formulas. To prove this, we will show that there is a bounded morphismlinking these two models (it is straightforward to show that term-modal formulasare preserved when this is the case, as in the propositional modal setting). Define b : T ( W d ) → T ( W ∆ d ) by w ( w, e d ) . We show that b is a bounded morphism.. w and ( w, e d ) satisfy the same basic formulas: T ( M d ) , w | = g T x ( f + z ) iff (i.h.) M d , w, g ( x ) | = f + z iff M, w, g ( x ) | = [ d ] f + z iff (reduction axiom for [ d ] f + z ) M, w, g ( x ) | = (cid:16)W ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ (cid:17) ∨ (cid:16) ¬ (cid:16)W ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ (cid:17) ∧ f + z (cid:17) iff (i.h., where we let g ( x ) = a for some a ∈ A named a ) T ( M ) , w | = g T x ( (cid:16)W ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ (cid:17) ∨ (cid:16) ¬ (cid:16)W ϕ ∈ Φ : post ( ϕ, f )= z , z ∈ Z f ϕ (cid:17) ∧ f + z (cid:17) )( x a ) iff (by definition of ∆ d ) T ( M ) , w | = g post ( e )( T x ( f + z )( x a )) iff T ( M ) ⊗ ∆ d , ( w, e d ) | = g T x ( f + z )( x a ) iff (since g ( x ) = a and a is named a ) T ( M ) ⊗ ∆ d , ( w, e d ) | = g f + z .2. if ( w, v ) ∈ T ( ∼ da ) then (( w, e d ) , ( v, e d )) ∈ T ( ∼ ∆ d a ) : ( w, v ) ∈ T ( ∼ da ) iff ( w, v ) ∈∼ da iff ( w, v ) ∈∼ a iff ( w, v ) ∈ T ( ∼ a ) iff ( w, v ) ∈ T ( ∼ ∆ d a ) (since ∆ d does not change the accessibility relations).3. if (( w, e d ) , ( v ′ , e d )) ∈ T ( ∼ ∆ d a ) then there is v such that ( w, v ) ∈ T ( ∼ da ) and b ( v ) = ( v ′ , e d ) :Reasoning as in step 2, (( w, e d ) , ( v ′ , e d )) ∈ T ( ∼ ∆ d a ) iff ( w, v ′ ) ∈ T ( ∼ da ) , and b ( v ′ ) = ( v ′ , e d ) .Hence, b is a bounded morphism, and T ( M d ) and T ( M ) ⊗ ∆ d satisfy the sameformulas. Thus, M, w, g ( x ) | = [ d ] ψ iff M d , w, g ( x ) | = ψ iff (by i.h.) T ( M d ) , w | = g T x ( ψ ) iff (bounded morphism) T ( M ) ⊗ ∆ d , ( w, e d ) | = g T x ( ψ ) iff T ( M ) , w | = T x ([ d ] ψ ) .Next, let ϕ = [ ℓ ] ψ . We need to show that M, w, g ( x ) | = [ ℓ ] ψ iff T ( M ) , w | = g ^ e ∈ E ℓ ( pre ℓ ( e ) → [ ∆ ℓ , e ] T x ( ψ )) (the case for T y is analogous). Note that M, w, g ( x ) | = [ ℓ ] ψ iff M ℓ , w, g ( x ) | = ψ iff(by i.h.) T ( M ℓ ) , w | = g T x ( ψ ) . As in the previous case, we will show that T ( M ℓ ) and T ( M ) ⊗ ∆ ℓ satisfy the same formulas by defining a bounded morphismlinking the two. Note that the preconditions in ∆ ℓ are pairwise inconsistent andjointly exhaustive, since each precondition corresponds to one way of assigningtruth values to the formulas in G ℓ . Hence, for each w ∈ T ( W ) , there is exactlyone event e val such that T ( M ) , w | = pre ℓ ( e val ) . Define b : T ( W ℓ ) → T ( W ∆ ℓ ) by w ( w, e val ) . We show that b is a bounded morphism.1. w and ( w, e val ) satisfy the same basic formulas:This is clear from the fact that learning updates do not change the acces-sibility relations. T ( M ℓ ) , w | = g T x ( f + z ) iff (i.h.) M ℓ , w, g ( x ) | = f + z iff M, w, g ( x ) | = f + z iff (i.h.) T ( M ) , w | = g T x ( f + z ) iff T ( M ) ⊗ ∆ ℓ , ( w, e val ) | = g T x ( f + z ) .. if ( w, v ) ∈ T ( ∼ ℓa ) then (( w, e val ) , ( v, e val ′ )) ∈ T ( ∼ ∆ d a ) :As T ( M ) is weakly rigid, each agent has the same name in each equivalenceclass [ w ] ∼ a of ∼ a . In what follows, we let the name of any agent o ∈ A inworlds of [ w ] ∼ a be o . Now, ( w, v ) ∈ T ( ∼ ℓa ) iff w ∼ ℓa v iff w ∼ a v and ∀ b ∈ A ( N w ab ⇒ ∀ ϕ ∈ ℓ ( M, w, b | = ϕ iff M, v, b | = ϕ )) iff (contrapositive) w ∼ a v and ∀ b ∈ A ( ∃ ϕ ∈ ℓ (( M, w, b | = ϕ and M, v, b | = ¬ ϕ ) or ( M, w, b | = ¬ ϕ and M, v, b | = ϕ )) ⇒ ¬ N w ab ) iff (by i.h.) ( w, v ) ∈ T ( ∼ a ) and (by def. of T ( M ) ⊗ ∆ ℓ ) T ( M ) , w | = g pre ( e val ) and T ( M ) , v | = g pre ( e val ′ ) for some val, val ′ ∈ V ℓ , andfor all b ∈ C :if there is a ϕ ∈ ℓ such that (cid:0) T ( M ) , w | = g T x ( ϕ )( x b ) and T ( M ) , v | = g T x ( ¬ ϕ )( x b ) (cid:1) or (cid:0) T ( M ) , w | = g T x ( ¬ ϕ )(( x b )) and M, v | = g T x ( ϕ )( x b ) (cid:1) then T ( M ) , w | = g ¬ N ( a, b ) iff ( w, v ) ∈ T ( ∼ a ) and T ( M ) , w | = g pre ( e val ) and T ( M ) , v | = g pre ( e val ′ ) forsome val, val ′ ∈ V ℓ and (by def. of ∆ ℓ ) T ( M ) , w | = g [ x ⋆ a ] Q ( e val , e val ′ ) iff (( w, e val ) , ( v, e val ′ )) ∈ T ( ∼ ∆ ℓ a ) .3. if (( w, e val ) , ( v ′ , e val ′ )) ∈ T ( ∼ ∆ ℓ a ) then there is v such that ( w, v ) ∈ T ( ∼ ℓa ) and b ( v ) = ( v ′ , e val ′ ) :Reasoning as in step 2, (( w, e val ) , ( v ′ , e val ′ )) ∈ T ( ∼ ∆ d a ) iff ( w, v ′ ) ∈ T ( ∼ ℓa ) ,and b ( v ′ ) = ( v ′ , e val ′ ) .Hence, b is a bounded morphism, and T ( M ℓ ) and T ( M ) ⊗ ∆ ℓ satisfy the sameformulas. Thus, M, w, g ( x ) | = [ ℓ ] ψ iff M d , w, g ( x ) | = ψ iff (by i.h.) T ( M ℓ ) , w | = g T x ( ψ ) iff (bounded morphism) for the unique event e val such that T ( M ) , w | = g pre ℓ ( e val ) , we have T ( M ) ⊗ ∆ ℓ , ( w, e val ) | = g T x ( ψ ) iff T ( M ) , w | = g V e ∈ E ℓ ( pre ℓ ( e ) → [ ∆ ℓ , e ] T x ( ϕ ) iff T ( M ) | = T x ([ ℓ ] ψ ) .This completes the proof.With Prop. 2 embedding KDL in DTML , it remains to show that there is acomplete and decidable system for the image of
KDL . Up to translation, such alogic is then a logic for the class of
KDL models. To state the result, denote the
TML image of the class of n -agent KDL models by T ( KDL n ) . We now define aset of formulas, F n , which can be shown to characterise the class T ( KDL n ) . Definition 14.
Let F n ⊆ L ( Σ n ( FP , N om )+[ ∆ ]) be the logic extending the term-modal S5 logic with the reduction axioms for action models ( ∆ † , e † ) , † ∈ D ∪ L (defined in [1]), as well as the following static axioms:. There are n agents and they are all named: Named n := ∃ x , ..., x n ( ^ i,j ≤ n,i = j x i = x j ∧ ∀ y _ i ≤ n y = x i ∧ ^ i,j ≤ n,i = j c i = c j ∧ ^ i ≤ n x i = c i )
2. Weak rigidity (Def. 8):
Rig n := ^ c ∈ C ∀ x (( c = x ) → ∀ y ( K y ( c = x )))
3. The neighbour relation is irreflexive and symmetric:
Neigh := ∀ x ∀ y ( ¬ N ( x, x ) ∧ ( N ( x, y ) ↔ N ( y, x )))
4. Agents know their neighbors:
KnowNeigh := ∀ x ∀ y ( N ( x, y ) ↔ K x N ( x, y )) We then obtain the result:
Proposition 3. F n statically characterizes T ( KDL n ) .Proof. By model-checking of the formulas in F n . Which we can use to state completeness:
Theorem 1.
For any n ∈ N , the logic F n is sound, strongly complete and de-cidable w.r.t. T ( KDL n ) .Proof (sketch). By Prop. 3, F n statically characterizes T ( KDL n ) . The result thenfollows from three results from [1]: 1. Any extension of the term-modal logic K with axioms A is strongly complete with respect to the class of frames charac-terized by A , and 2. If A characterizes a class with finitely many agents, then thelogic is also decidable, and 3. any dynamic DTML formula is provably equivalentto a static
DTML formula using reduction axioms.Thus, since F n characterizes T ( KDL n ) , which is a class with finitely manyagents, and all dynamic axioms in F n are probably equivalent to static DTML , itfollows that K + F n is strongly complete and decidable with respect to T ( KDL n ) . This paper has showcased
DTML as a framework for modeling social networks,their epistemics and dynamics, including examples in which uncertainty aboutname reference and de dicto/de re distinctions are key to modelling informationflow and network change correctly. It was shown that
DTML may encode thepopular hybrid logical models of epistemic networks, and that
DTML may beused to obtain completeness for an open-question dynamics through emulation.We are very interested in learning how
DTML relates to
GDDL with respectto the encodable dynamics. We have been able to emulate the updates usedn the examples of [22], but the general question is open. Further, the staticsof frameworks that describe networks using propositional logic [2, 14] must be
DTML encodable, and we expect the name about their updates, where reduc-tion axioms exist. This raises two questions: if we can show this by a generalresults instead of piecemeal, and whether principled
DTML action models existfor classes of updates. E.g., the threshold update of [2] gives an agent’s property P if a given fraction of neighbors are P ; for a fixed agent set, this is DTML encod-able by using the reduction axioms of [2] to provide pre- and postconditions. Fora principled update, however, seemingly we need a generalized quantifier (e.g.,a Rescher quantifier). If so, the general update form is not
DTML encodable.Classification results like these would add valuable insights on network logics.
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