The Thirty-First International Florida Artificial Intelligence Research Society Conference (FLAIRS-31) Reasoning with Doxastic Attitudes in Multi-Agent Domains Ben Wright, Enrico Pontelli Department of Computer Science New Mexico State University { bwright, epontell} @cs.nmsu.edu Abstract 2017; Wan et al. 2015). That is, no contextual beliefs would modify an agent’s reasoning to change its belief in the mo- In recent years, we have witnessed a blossoming of research proposals addressing the challenges in reasoning about ac- ment. tion and change in domains that include an agent operating This paper will introduce the concept of doxastic attitudes in a multi-agent setting. In particular, the recent emphasis has to help represent this idea of changing beliefs over time and been on dealing with domains that involve agents reasoning to address issues of false or conflicting beliefs. We intro- not only about the state of the world but also about the knowl- duce some relevant background of Kripke Structures and the edge and beliefs of other agents. An open challenge is the action language mA+ which we build on, then we define management of conflicting and incorrect beliefs. This paper the concept of Doxastic Attitudes. These are then utilized to seeks to introduce a solution to this through the use of dox- modify mA+ to handle attitudes, with intuitions of the tran- mA+ astic attitudes. Built on top of the action language ,we sition functions given. We introduce both dynamic and static extend the transition functions of an agent to include this idea versions of doxastic attitudes. Finally, we work through two of attitudes and showcase how these work in two different examples. different examples to showcase how the attitudes function. Background Introduction & Motivation Kripke Structures Reasoning about action and change has long been a field of study; more recently, a push towards reasoning about multi- A Kripke structure is a formalism commonly used to capture agent actions has been studied. In particular, researchers the possible-world semantics for Logic involving epistemic have emphasized the importance of reasoning about beliefs or doxastic operations (Ditmarsch, van der Hoek, and Kooi and knowledge of agents and the impact that epistemic ac- 2007; Fagin et al. 1995). Given a countable set of proposi- tions P and a finite set of agents A, a Kripke structure is a tions have on them. However, in many of these cases of be- A P lief reasoning, either the system is developed from the per- structure M = S, R ,V , where: spective of a single agent in the system or it is not robust • S is a set of states enough to handle beliefs that conflict or might be false. This • RA ∀a ∈ A RA(a) ⊆ S × S is illustrated in the following example: is a function where • V P : P → 2S ∀p ∈ P Light in the Room Example: There are two agents A is a valuation function, where , V P (p) ⊆ S p and B. Agents A and B are in Room2. In Room2, there is the set of states in which the proposition is a light switch. The switch turns the light on or off is true. for that room. The light cannot be seen from a different In this structure, each state in S is a possible world of our Room1. An agent can look at the light to determine if it domain. The function RA then maps equivalence between is on or not. Agents can announce to a room if the light possible worlds for each agent. That is, if an agent has a re- is on or off. lation between states s and t, then that agent cannot discern Agent A believes the light is on, after seeing Room2. between those two possible worlds. It then moves to Room1. After some time in Room1, Additionally, a pointed Kripke Structure (M,s) is com- Agent B enters Room1 and announces that the light in posed of a Kripke structure, M, and a distinguished state Room2 is off. Agent A now has an invalid belief (light s ∈ S—typically representing the “real” state of the world. on in Room2) that will need to be repaired. It is common to express entailment of the truth of an epis- temic formula w.r.t. a pointed Kripke structure—where an To the best of our knowledge, no current epistemic plan- epistemic formula is built using propositions from P , propo- ner has the ability to change the context in which beliefs sitional connectives, and the operator Ka for any a ∈ A. The are reasoned upon during the reasoning process (Baral et al. following rules intuitively capture such notion of entailment, Copyright c 2018, Association for the Advancement of Artificial where p, q ∈ P, s, t ∈ S and a ∈ A (Ditmarsch, van der Intelligence (www.aaai.org). All rights reserved. Hoek, and Kooi 2007; Fagin et al. 1995): 360 • (M,s) |= p iff s ∈ V (p), of the outcome). For the sake of simplicity, we omit this in- • (M,s) |=(ϕ ∧ ψ) iff (M,s) |= ϕ and (M,s) |= ψ, termediate level of observability in this work. • ( ) |= ¬ ( ) |= The semantics of the action language can be described M,s ϕ iff M,s ϕ, as the composition of separate transition functions, each de- A • (M,s) |= Ka(ϕ) iff ∀(s, t) ∈ R (a).(M,t) |= ϕ. scribing the behavior of a different type of action. All tran- The last operator Ka(ϕ) indicates that agent a knows ϕ sition functions are concerned with ‘How do we transition while in state s of model M. At this point, we use this def- from our current state, to this new state given we performed A+ inition interchangeably with belief (represented as Ba(ϕ)) action a?’. To represent states, m uses pointed Kripke due to the fact that we do not actually check if (M,s) |= ϕ. structures (M,s) as defined above. A+ We utilize the BAϕ operator further in the paper to repre- For m transitions, we provide only an intuition—the sent ‘Agent A believes ϕ’ and use the definition above for reader can find the complete details in (Baral et al. 2015). its connotation with pointed Kripke Structures. A final note There are two facets to each action transition, agents who are about the Ba operator is that it can be nested to discuss other aware of the action and agents who are oblivious. Intuitively, agents beliefs, such as BaBbon which is “Agent A believes the world-changing actions are the simplest to describe. If Agent B believes on”. the action is performed and observed, then something in the world changes—we go to a new state s. Likewise, if an Action Language mA+ observed announcement happens, some relations are modi- fied based on the announced literals. These would be part of In this section we briefly summarize the structure of the ac- tion language mA+ (Baral et al. 2015), used to describe the (M ,s) structure in Fig. 1 on the right side. Oblivious the capabilities of agents operating in a multi-agent setting. agents still believe what was before, both about themselves, mA+ builds on a signature AG, F, A, where AG is a set the state they are in, and the other agents. To do this, we cre- of agents, F is a set of fluents, and A is a set of actions. ate a mirror replica of our pointed Kripke Structure. This is An action theory in mA+ is composed of a set of axioms, a snapshot the oblivious agent stays in and where its reach lives. This can be seen as the (M,s) structure in Fig. 1 on the describing the capabilities of the agents. The syntax for the basic actions of mA+ is as follows: right side. The changed state s of a world changing action or the changed relations of an announced action take place • executable a if ψ in the non-replica section—then we go through and change describes the fact that the property ψ (an epistemic for- all the relations for oblivious agents to point to the mirrored mula) is a pre-condition for the action a. replica (see Fig. 1 for an intuition). • a causes Φ if ψ describes the world-changing effects of action a—it causes a set of fluents Φ to become true if ψ currently (M,s) holds. observant s’ M’ } agents • a determines p if ψ s M describes a sensing action a; it will determine the value of action the fluent p if ψ currently holds. a oblivious agents • a announces if ψ s M } describes an announcement action a; it declares that the fluent literal is true to the other agents if ψ currently holds. Figure 1: Observability and Action Execution In addition to these action types, it is important to know the awareness of an agent. If an agent is aware of an action being performed, that it will change its knowledge. To keep track Doxastic Attitudes of this, mA+ uses the idea of observations: If we are to discuss false beliefs or conflicting beliefs, we • X observes a if ϕ will have to ask the question: “How will the agents decide where X is a set of agents. An agent in X either observes what to do about their current conflict?”. One answer to this, or is aware of action a when the conditions ϕ are met. is the use of attitudes towards beliefs. Aside from this state of awareness, there is also the obliv- Attitudes, or more appropriately propositional attitudes, ious state—which holds when an agent has no awareness or have been used for sometime (Searle 1983) to convey when notion that an action or its effects have occurred.