Social Computing , Social Cognition , Social Networks and Multiagent Systems

Lately, many multi-agent systems (MAS) are designed as multi-modal systems [9, 15, 23, 22, 26, 28, 18]. Moreover, there are different techniques for combining logics, such as products, fibring, fusion, and modalisation, among others [1, 14, 16]. In this paper we focus on the combination of special-purpose logics for building “on demand” MAS. From these engineering point of view, among the most used normal logics for modeling agents’ cognitive states are logics for beliefs, goals, and intentions, while, perhaps, the most well-known non-normal logics for MAS is the logic of agency (and, possibly, ability). We explore combinations of these normal and nonnormal logics. This lead us to handle Scott-Montague structures, (neighbourhood models, in particular) which can be seen as a generalization of Kripke structures [20]. Interested in the decidability of such structures, which is a guarantee of correct systems and their eventual implementations, we give a new presentation for existing theorems that generalize the wellknown results regarding decidability through the finite model property via filtrations for Kripke structures. We understand that the presentation we give, based on neighbourhood models, better fits the most accepted and extended logic notation actually used within the MAS community. 1 Motivation and Aims In [32] Smith and Rotolo adopted [13]s cognitive model of individual trust in terms of necessary mental ingredients which settle under what circumstances an agent x trusts another agent y with regard to an action or state-of-affairs, i.e. under which beliefs and goals an agent delegates a task to another agent. Using this characterization of individual trust, the authors provided a logical reconstruction of different types of collective trust, which for example emerge in groups with multi-lateral agreement, or which are the glue for grounding in solidum obligations raising from a “common front” of agents (where each member of the front can behave, in principle, as creditor or debtor of the whole). These collective cognitive states were characterized in [32] within a multi-modal logic based on [9]s axiomatisation for collective beliefs and intentions combined with a non-normal modal logic for the operator Does for agency. In a subsequent work, the multi-relational model in [32] was reorganized as a fibring, a particular combination of logics which amounts to place one special-purpose normal logics on top of another [31]. In this case, the normal logic was put on top of the nonnormal one. For doing this, authors first obtained two restrictions of 1 FACET, UCALP, Argentina and Facultad de Informática, UNLP, Argentina 2 Facultad de Informática, UNLP, Argentina and CONICET 3 FACET, UCALP, Argentina the original logics. By exploiting results in regard to some techniques for combining logics, it was proved that [32]s system is complete and decidable. Hence, the sketch for an appropriate model checker is there outlined. One motivation regarding a further combination of those special purpose logics for MAS is the aim to have an expressive enough system for modelling interactions between a behavioural dimension and a cognitive dimension of agents, and testing satisfiability of the corresponding formulas. For example, for modelling expressions such as Doesi (Belj A ) which can be seen as a form of persuasion or influence: agent i makes agent j have A as belief. This formula cannot be written in the fibred language in [31] neither in the original language in [32] because such languages have a restriction over the form of the wffs: no modal operator can appear in the scope of a Does. In [31], authors outlined a combination of the normal and the non-normal counterparts of the base logics. That combination lead to an ontology of pairs of situations allowing a structural basis for more expressiveness of the system. That combination is the result of (again) splitting of the original structure, which is a multi-relational frame of the form [32, 17]: F = 〈A,W, {Bi}i∈A, {Gi}i∈A, {Ii}i∈A, {Di}i∈A〉 where: A is a set of agents, W is a set of posible worlds, and {Bi}, {Gi}, {Ii}, {Di} are the accessibility relations for beliefs, goals, intentions, and agency respectively. The underlying set of worlds of the combination is an ontology of pairs of worlds (wN , wD). There are two structures where to respectively test the validity of the normal modalities and the non-normal modalities. The former is a Kripke model; the latter a neighbourhood model. The definition of a formula being satisfied in the combined model at a state (wN , wD) amounts to a scan through the combined structure, done according to which operator is being tested. Normal operators move along the first componentwN , and non-normal operators move along the second component of the current world wD . Regarding the application to agents, it is also common that the cognitive modalities are extended with temporal logics. For example, Schild [29] provides a mapping from Rao and Georgeff’s BDI logic [27] to μ-calculus [24]. The model of Rao and Georgeff is based on a combination of the branching time logic CTL∗ [8] and modal operators for beliefs, desires, and intentions. Schild collapses the (original) two dimensions of time and modalities onto a one dimensional structure. J. Broersen [5] presents an epistemic logic that incorporates interactions between time and action, and between knowledge and action. Correspondingly, H. Wansing in [2] points out that (i) agents act in time, (ii) obligations change over time as a result of our actions and the actions of others, and (iii) obligations may depend on the future course of events. In ([2], Section 10.3) he adopts a semantics reflecting the non-determinism of agency: models are based on trees of moments of time branching to the future. Agentive sentences are history dependent, formulas are not evaluated at points in time but rather at pairs (moment, history), where history is a linearly ordered set of moments. Cohen and Levesque [7, 21] embed, using function mappings, a modal logic of beliefs and goals with a temporal logic with nondeterministic and parallel features. In this paper we define a combination of logics for MAS as a special case of neighbourhood structures. Previously, we give a new presentation of decidability results which apply to a particular kind of models: neighbourhood models. In the literature, the analysis of transfer of logical properties from special purpose logics to combined ones is usually based on properties of normal logics. It is claimed that the proof strategies in the demonstration of transference of properties of normal logics could in principle be applied to non-normal modal logics [12]. In a mono-modal logic with a box modality, normality implies that the following formulas are valid: 2(p → q) → (2p → 2q) and 2(p ∧ q) ↔ (2p ∧ 2q), as well as the admission of the rule from ` A infer ` 2A [3, 12]. None of this is assumed to hold for a non-normal logics. We indeed use a non normal modal logic for agency, as developed by Elgesem [11, 17]; and aim to put it to work with normal logics for, e.g, beliefs and goals. The logic of agency extends classical propositional logic with the unary symbol Does satisfying the following axioms: ¬(Does>), (Does A ) ∧ (Does B) ⇒ Does(A ∧ B) and Does A ⇒ A together with the rule of Modus Ponens and the rule saying that from A ⇔ B you can conclude Does A ⇔ Does B. The intended reading of Does A is that ‘the agent brings it about that A ’. (See Section 2.1 in [11].) A detailed philosophical justification for this logic is given in [11] and neighborhood and selection function semantics are discussed in [11, 17]. One advantage regarding the choice of a logic of agency such as Does relies on the issue of action negation. For Does, and for other related logics of action such as the one in [5], action negation is wellunderstood: given that the logic for Does is Boolean, it is easy to determine what ¬Does A means. This allows providing accurate definitions for concepts such as e.g. “refrain”, especially useful in normative MAS: I have the opportunity and ability to do something, but I do not perform it as I have the intention not to. Up to now, although addressed, there are no outstanding nor homogeneous solutions for the issue on action negation in other relevant logics for MAS such as dynamic logics (see e.g. [4, 5, 25]). We organize the work as follows. In Section 2 we directly adapt for neighbourhood models the strategy in [3] regarding the finite model property (FMP) via filtration. This includes: (i) establishing conditions for finding a filtration of a neighbourhood model, (ii) the demonstration of a filtration theorem for the neighbourhood case, (iii) guaranteeing the existence of a filtration, and (iv) the proof of the FMP Theorem for a mono-modal neighbourhood model. In Section 3 we show how the results in Section 2 can be applied for proving decidability of a neighbourhood model with more than one modality. We also devise examples for a uni-agent mono-modal non-normal system, a uni-agent multi-modal system and a multi-modal multiagent system. In Section 4 we concentrate on a combined MAS, with an underlying neighbourhood structure. Conclusions end the paper. 2 Decidability for the neighbourhood case through the extension of the FMP strategy for the Kripke case. We mentioned that normal logics can be seen as a platform for the study of transference of decidability results for non-normal logics and combination of logics. We rely on well-studied results and existing techniques for Kripke structures, which are usual support of normal logics, to provide a new presentation of existing decidability results for a more general class of structures supporting non-normal logics. We start from the definitions given by P. Blackburn et. al. [3]. In [3](Defs. 2.36, 2.38 and 2.40), the construction of a finite model for a Kripke structure is supported in: (i) the definition of a filtration, (ii) the Filtration Theorem, (iii) the existence of a filtration for a model and a subformula closed set of formulas, and (iv) the Finite Model Property Theorem via Filtrations. B. Chellas, in its turn, defined filtrations for minimal models in [6] (Section 7.5). Minimal models are a generalization of Kripke ones. A minimal model is a structure 〈W,N,P 〉 in whichW is a set of possible worlds and P gives a truth value to each atomic sentence at each world. N , is a function that associates with each world a collection of sets of worlds. The notation used throughout is one based on truth sets (‖A ‖ is the set of points in a model where the wff A is true). Truth sets are a basic ingredient of selection function semantics. In what follows we give a definition of filtration for ScottMontague models using a neighbourhood approach and notation. Neighbourhood semantics is the most important (as far as we consider) generalization of Kripke style (relational) semantics. The set of possible worlds is replaced by a Boolean algebra, then the concept of validity is generalized to the set of true formulas in an arbitrary subset of the Boolean algebra, but (generally for every quasi-classical logics) the subset must be a filter. This ‘neighbourhood approach’ focuses on worlds, which directly leads us to the underlying net of situations that ultimately support the system: relative to a worldw we are able to test whether agents believe in something or carry out an action. The neighbourhood semantics better adapts to the specification of most prevailing modal multi-agent systems, which lately tend to adopt the Kripke semantics with a notation given as in [3]. This because, probably, that notation is more intuitive for dealing with situations and agents acting and thinking according to situations, rather than considering formulas as ‘first class’ objects. This is crucial in current practical approaches to agents; in a world an agent realises its posibilities of succesful agency of A , its beliefs, it goals, all relative to the actual world w, In this perspective, situations are a sort of “environmental support” for agent’s internal configuration and visible actions. Worlds are, therefore, in a MAS context, predominantly, abstract descriptions of external circumstances of an agent’s community that allow or disallow actions, activate or nullify goals. That is why we prefer to work with neighbourhood models as models for MAS, keeping in mind that, while it is possible to devise selection function models for MAS, this is not nowadays usual practice. Also, as it is well-known, the difference between selection function semantics and neighbourhood semantics is merely at the intuitive level (their semantics are equivalent, and both known as ScottMontague semantics [17]). P. Schotch has already addressed the issue of paradigmatic notation and dominating semantics for modalities. In his work [30] he points out that the necessity truth condition together with Kripkean structures twistedly “represent” the model-theoretic view of the area, given that -among other reasonsmany “nice” logics can be devised with those tools. Moreover, due to this trend, he notes that previous complex and important logics (due to Lewis, or to the “Pennsylvania School”) have become obsolete or curiosities just because their semantics is less elegant. We adopt an eclectic position in this paper: we choose a structure that allows non-normal semantics and we go through it with the notation as given in [3], which is currently well-accepted and wellunderstood for modal MAS. Next we outline some tools for finding a filtration of a neighbourhood model. We generalize the theorems for Kripke structures given in [3]. Definition 1 (Neighbourhood Frame). A neighbourhood frame [20, 6] is a tuple 〈W, {Nw}w∈W 〉 where: 1. W is a set of worlds, and 2. {Nw}w∈W is a function assigning to each elementw inW a class of subsets of W , the neighbourhoods of w. We will be working with a basic modal language with a single unary modality, let us say ‘#’. We asume that this modality has a neighbourhood semantics. For example, ‘#’ may be read as the Does operator, or an ability operator, as proposed by Elgesem [11]; or represent a “refrain” operator based on Does and other modalities such as ability, opportunity and intentions. Definition 2 ((Recall Def. 2.35 in [3]) Closure). A set of formulas Σ is closed under subformulas if for all formulas φ, if φ∨φ′ ∈ Σ then so are φ and φ′; if ¬φ ∈ Σ then so is φ; and if #φ ∈ Σ then so is φ. (For the Does modality, for example, if Doesφ ∈ Σ so is φ). Definition 3 (Neighbourhood Model). We define M = 〈W, {Nw}, V 〉 to be a model, where 〈W, {Nw}〉 is a neighbourhood frame, and V is a valuation function assigning to each proposition letter p in Σ a subset V (p) of W (i.e. for every propositional letter we know in which worlds it is true). Given Σ a subformula closed set of formulas and given a neighbourhood model M, let ≡Σ be a relation on the states of M defined by w ≡Σ v iff ∀φ ∈ Σ (M, w |= φ iff M, v |= φ). That is, for all wff φ, φ is true inw iff is also true in v. Clearly≡Σ is an equivalence relation. We denote the equivalence class of a state w of M with respect to ≡Σ by [w]Σ (or simply [w] when no confusion arises). Let WΣ = {[w]Σ /w ∈W}. Next we generalize for neighbourhood models the concept of filtration given in [3]. Definition 4 (Filtrations for the neighbourhood case). Suppose M is any model 〈W f , {Nw} , V f 〉 such that W f = WΣ and: 1. If U ∈ Nw then {[u]/u ∈ U} ∈ N [w] , 2. For every formula #φ ∈ Σ, if U ∈ N [w] and (∀[u] ∈ U)(M, u |= φ), then M, w |= #φ, 3. V f (p) = {[w] /M, w |= p}, for all proposition letter p in Σ. Condition (1) requires that for every neighbourhood ofw there is a corresponding neighbourhood of classes of equivalences for the class of equivalence of w (i.e. [w]) in the filtration. Condition (2) settles, among classes of equivalences, the satisfaction definition regarding a world and its neighbourhoods. We use U for the neighbourhoods in the original model M, and U for the neighbourhoods of [w] in the filtration M . Theorem 1 (Filtration Theorem for the neighbourhood case.). Consider a unary modality ‘#’. Let M be a filtration of M through a subformula closed set Σ. Then for all φ in Σ and all w in M, M, w |= φ iff M, [w] |= φ. That is, filtration preserves satisfiability. Proof. We show that M, w |= φ iff M , [w] |= φ. As Σ is subformula closed, we use induction on the structure of φ. We focus on the case φ = #γ. Assume that #γ ∈ Σ, and that M, w |= #γ. If M, w |= #γ then there is a neighbourhood U such that U ∈ Nw and (∀u ∈ U)(M, u |= γ), that is, for every world in that neighbourhood, γ holds. Thus, by application of the induction hypothesis, for each of those u we have that M , [u] |= γ. By condition (1) above, {[u]/u ∈ U} ∈ N [w]. Hence M , [w] |= #γ. Conversely we have to prove that if M , [w] |= φ then M, w |= φ. Assume that φ = #γ and M , [w] |= #γ. By truth definition, there exists U neighbourhood of [w] such that (∀[u] ∈ U)(M , [u] |= γ). Then by inductive hypothesis (∀[u] ∈ U)(M, u |= γ). Then by condition (2) M, w |= #γ. Note that clauses (1) and (2) above are devised to make the neighbourhood case of the induction step straightforward. Existence of a filtration. Notation. [U ] = {[u]/u ∈ U} i.e. [U ] is a set of classes of equivalences. Define N [w] as follows: [U ] ∈ N [w] iff (∃w ≡Σ w′/U ∈ Nw′). That is, [U ] is a neighbourhood of [w] if there exists a neighbourhood U in the original model reachable through a world w′ which is equivalent to w (under ≡Σ). This definition leads us to the smallest filtration. Lemma 1 (See Lemma 2.40 in [3]). Let M be any model, Σ any subformula closed set of formulas,WΣ the set of equivalence classes of W induced by ≡Σ, V f the standard valuation on WΣ. Then 〈WΣ, N [w], V f 〉 is a filtration of M through Σ. Proof. It suffices to show thatN [w] fulfills clauses (1) and (2) in Definition 4. Note that it satisfies (1) by definition. It remains to check that N [w] fulfills (2). Let #φ ∈ Σ, we have to prove that (∀U ∈ N [w]) (∀[u] ∈ U)(M, u |= φ) → (M, w |= #φ). We know that U = [U ] for some U ∈ Nw′ such that w ≡Σ w′. Recall that (∀[u] ∈ U)(M, u |= φ) means that (∀u ∈ U)(M, u |= φ). By truth definition M, w′ |= #φ, then becausew ≡Σ w′ we get M, w |= #φ. Theorem 2 (Finite Model Property via Filtrations). Assume that φ is satisfiable in a model M as in Definition 3; take any filtration M through the set of subformulas of φ. That φ is satisfiable in M is immediate from the Filtration Theorem for the neighbourhood case. Being ≡Σ an equivalence relation, and using Theorem 1 it’s easy to check that, a model M and any filtration M are equivalent modulus φ. This result is useful to understand why the original properties of the frames in the models are preserved. This results are provided in [Chellas] for the preservation of frames clases through filtrations. Example 1 (uni-agent mono-modal system). A simple system can be defined with structure as in Definition 3, where we can write and test situations like the one following: Bus stop scenario ([13], revisited). Suppose that agent y is at the bus stop. We can test whether y raises his hand and stops the bus by testing the validity of the formula: Doesy(StopBus). This simple kind of systems are proved decidable via FMP through Definition 4, Theorem 1 and Lemma 1 in this Section. They are powerful enough to monitor a single agent’s behaviour. Note that Doesy(StopBus) holds in a worldw in a model M, that is, M, w |= Doesy(StopBus) iff (∃U ∈ Nyw ) such that (∀u ∈ U) (M, u |= StopBus). 3 Extension to the multi-agent multi-modal case Recall that the original base structure discussed in [32] is a multirelational frame of the form: F = 〈A,W, {Bi}i∈A, {Gi}i∈A, {Ii}i∈A, {Di}i∈A〉