Traditional work in belief revision deals with the question of what an  agent should believe upon receiving new information.  We will give an  overview about what can be concluded about an agent based on an  observation of its belief revision behaviour. The observation contains  partial information about the revision inputs received by the agent and  its beliefs upon receiving them. We will sketch a method for reasoning  about past and future beliefs of the agent and predicting which inputs it  accepts and rejects. The focus of this talk will be on different degrees  of incompleteness of the observation and variants of the general question we are able to deal with

Booth, Richard

Nittka, Alexander

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A Method for Reasoning about other Agents’Beliefs from Observations(Extended Abstract)Alexander Nittka1, Richard Booth21 Department of Computer Science, University of LeipzigAugustusplatz 10/11, 04109 Leipzig, Germanynittka@informatik.uni-leipzig.de2 Faculty of Informatics, Mahasarakham UniversityMahasarakham 44150, Thailandrichard.b@msu.ac.thKeywords. belief revision, iterated revision, non-prioritised revision,non-monotonic reasoning, rational closure, rational explanation1 IntroductionBelief revision traditionally deals — from a first person perspective — with thequestion of what an agent should believe given an initial state and a revisioninput. This question is approached in two main ways: (i) formulating generalproperties a belief revision operator should satisfy and (ii) constructing specificrevision operators. In this paper, we want to give a brief overview over ourinvestigations of what we can say about another agent’s actual beliefs based onan observation of its belief revision behaviour. This issue is discussed in detail inthe first author’s PhD thesis [9] and in a number of (joint) papers [2,3,4,10,11].The observed agent will be denoted by A and we work in a propositional lan-guage L. We make a number of simplifying assumptions. A employs a particularbelief revision framework introduced in [1]. It can be seen as a non-prioritisedversion of Nayak’s lexicographic revision [7] and is also closely related to Nebel’slinear revision [8]. Further, we are interested only in propositional beliefs, thatis, we will not deal with higher order beliefs. The information about A is given asan observation o containing the revision inputs received during a period of timeand a partial description of beliefs held and not held after receiving an input.We are interested in the following questions. Which inputs are accepted byA and which are rejected? (This need not be explicitly given in the observation.)What did A believe before the observation started? What did it believe duringthe time of observation apart from what o already tells us? What will A believeafter receiving some further input(s)?The general method is to construct a potential initial epistemic state of theagent and progress the inputs recorded in the observation starting in that statein order to generate hypotheses about the beliefs. We call a state an explanationif it verifies the information contained in the observation. There are generallymany possible explanations and one of the tasks is to single out a good one.Dagstuhl Seminar Proceedings 07351Formal Models of Belief Change in Rational Agentshttp://drops.dagstuhl.de/opus/volltexte/2007/12142 Alexander Nittka, Richard Booth2 Belief revision framework, observation, and rationalexplanationDefinition 1. The epistemic state [ρ,N] of an agent consists of a sequence offormulae ρ and a formula N. N is called the agent’s core belief. The revisionoperator ∗ is defined for any epistemic state [ρ,N] and formula ϕ by setting[ρ,N] ∗ ϕ = [ρ · ϕ,N]. The belief set Bel([ρ,N]) in any epistemic state [ρ,N] isBel([ρ,N]) = Cn(f (ρ · N)), wheref (βk, . . . , β1) =β1 , k = 1βk ∧ f (βk−1, . . . , β1) , k > 1 and βk ∧ f (βk−1, . . . , β1) 6` ⊥f (βk−1, . . . , β1) , otherwiseThe agent’s epistemic state [ρ,N] is made up of two components: (i) a se-quence ρ of formulae and (ii) a single formula N, all formulae being elementsof L. N stands for the agent’s set of core beliefs — the beliefs of the agent itconsiders “untouchable” and commits to at all times. One main effect of N isthat revision inputs contradicting it will not be accepted into the belief set. ρ isa record of the agent’s revision history. Iterated revision is handled quite nat-urally. All revision steps are simply recorded and the problem of what A is tobelieve after each revision step, in particular whether the input just receivedis accepted, i.e., is believed, is deferred to the calculation of the beliefs in anepistemic state. The agent’s full set of beliefs Bel([ρ,N]) in the state [ρ,N] isdetermined by a particular calculation on ρ and N which uses the function fmapping a sequence σ of propositional formulae to a formula. The agent startswith its core belief N and then goes backwards through ρ, adding a formula asan additional conjunct if the resulting formula is consistent. If it is not, then theformula is simply ignored and the next element of ρ is considered. The belief setof A then is the set of logical consequences of the formula thus constructed.Definition 2. An observation o = 〈(ϕ1, θ1, D1), . . . , (ϕn, θn, Dn)〉 is a sequenceof triples (ϕi, θi, Di), where for all 1 ≤ i ≤ n: ϕi, θi, and all δ ∈ Di (Di is finite)are elements of a finitely generated propositional language L. [ρ,N] explains o(or is an explanation for o) if and only if the following two conditions hold.1. N 6` ⊥2. for all i such that 1 ≤ i ≤ n :Bel([ρ,N] ∗ ϕ1 ∗ · · · ∗ ϕi) ` θi and∀δ ∈ Di : Bel([ρ,N] ∗ ϕ1 ∗ · · · ∗ ϕi) 6` δWe say N is an o-acceptable core iff [ρ,N] explains o for some ρ.The intuitive interpretation of an observation is as follows. After having re-ceived the revision inputs ϕ1 up to ϕi starting in some initial epistemic state, Abelieved at least θi but did not believe any element of Di. Unless stated other-wise, we assume that during the time of the observation A received exactly therevision inputs recorded in o, in particular we assume that no input was receivedA Method for Reasoning about other Agents’ Beliefs from Observations 3between ϕi and ϕi+1, the observation being correct and complete in that sense.For the θi and Di we assume the observation to be correct but possibly partial,i.e., the agent did indeed believe θi and did not believe any δ ∈ Di, but theremay be formulae ψ for which nothing is known. Note that complete ignoranceabout what the agent believed after a certain revision step can be representedby θi = > and complete ignorance about what was not believed by Di = ∅.An explanation of a given observation o is an epistemic state that verifiesthe information in o and has a consistent core belief. There are infinitely manyexplanations in case o can be explained. This is why our proposed method forreasoning about A is to choose one explanation. A very important propertyof the framework is that A’s beliefs after several revision steps starting in aninitial state can equivalently be expressed as the beliefs after a single revision onthe same initial state: Bel([ρ,N] ∗ ϕ1 ∗ · · · ∗ ϕi) = Bel([ρ,N] ∗ f (ϕ1, . . . , ϕi,N)).Intuitively, the agent merges its core belief and all revision inputs received usingf into a single formula and then conditions its epistemic state using it. Thisallows us to translate the observation into information about a single state —the initial epistemic state we are after. Note however, that N needs to be known asotherwise f (ϕ1, . . . , ϕi,N) cannot be calculated. So, given a core belief N, o yieldsthat A would believe θi (and would not believe any δ ∈ Di) in case it revised itsinitial epistemic state by f (ϕ1, . . . , ϕi,N). This is nothing but conditional beliefsheld and not held by A in its initial state [ρ,N]. That is, o is a partial descriptionof A’s conditional beliefs in [ρ,N]. The above equation also entails that if we hada full description of its conditional beliefs we could calculate the beliefs after anysequence of revision inputs, enabling us to answer the questions we posed in theintroduction. It turns out that the assumed belief revision framework allows us toapply existing work ([6] and in particular [5]) on completing partial informationabout conditional beliefs and to construct a suitable ρ such that [ρ,N] is indeedan explanation for o in case N is o-acceptable. An iterative refinement of the corebelief guarantees that an explanation, which we call the rational explanation, isfound in case there is one.The set of o-acceptable cores is closed under disjunction. We showed the ratio-nal explanation to satisfy a number of desirable properties, e.g., that it yields thelogically weakest o-acceptable core and that the beliefs calculated satisfy a partic-ular minimality property. However, not all conclusions drawn about A based onthe rational explanation need to be correct. This is clear as any other explanationfor o might correspond to the agent’s true initial state. We suggested hypothet-ical reasoning for verifying conclusions. The basic idea is to modify the originalobservation o in a way such that an explanation for o′, which also explains o,would be a counterexample to the conclusion we want to verify. For example,assume that the rational explanation for o = 〈(ϕ1, θ1, D1), . . . , (ϕn, θn, Dn)〉 tellsus that after receiving ϕi the agent believes ψ (which is not entailed by θi). Ifwe now replace the entry (ϕi, θi, Di) in o by (ϕi, θi, Di ∪ {ψ}), the resulting ob-servation o′ expresses that after receiving ϕi the agent does not believe ψ. Soin case o′ has an explanation — which can be tested by calculating the rationalexplanation of o′ — the original conclusion is not safe.4 Alexander Nittka, Richard Booth3 Unknown logical content, missing revision inputs, etc.The assumption that o contains exactly those inputs received by A excludescases where the logical content of the inputs is only partially known or someinputs may have been missed by the observer. However, the original methodscan be extended in order to deal with such cases as well. The idea is to allowany formula in o to contain unknown subformulae which are represented by placeholders χj . The formula p∧χ1, for example, could represent a revision input thatentails p. If the same χj appears in several places, this may carry informationthat can be exploited although the observer does not know the actual formula.The method for dealing with this type of observation is as follows. Each χjis replaced by a new variable xj , transforming an observation with unknownsubformulae into a normal one. Now, the rational explanation construction canbe applied. We showed that this transformation preserves explanations in thefollowing sense. If there is an explanation for some instantiation of the χj thenthere is also one when using new variables. Conversely, if there is no explanationwhen using new variables xj for the χj then no explanation exists using anyinstantiation of the unknown subformulae. Reasoning about A is then done byconsidering the rational explanation of the observation using new variables xjbut restricting the conclusions to the language constructed from those variablesappearing in the observation, i.e., excluding the xj .This does not allow for missing inputs, yet. Note that the transformed ob-servation is assumed to contain an entry for every input received by A. Theseentries can be introduced into the observation by adding a new triple (χj ,>, ∅)for every revision input that may have been missed. This entry states that Areceived a revision input whose logical content is completely unknown and wehave no information about what it believed or did not believe upon receivingthat input. Of course, this methodology is problematic as we may not knowhow many inputs were missed at which position in the observation. However,we can provide results that limit the number of additional inputs that needto be assumed, depending on which information about number and positionsof the missing inputs is available to the observer. The hypothetical reasoningmethodology can still be applied.Extending the work in a yet different direction, it is also possible to constructan initial state that explains several observations in the sense that different re-vision sequences start in the same state. This is reasonable, e.g., when thinkingabout an expert reasoning about different cases (the initial state representingthe expert’s background knowledge) or identical copies of software agents beingexposed to different situations. Our work is focused on reasoning using observa-tions of other agents, but observing oneself can be useful as well. By keeping anobservation of itself an agent may reason about what other agents can concludeabout it, which is important when trying to keep certain information secret.The results can also be applied for slight variations of the assumed belief revi-sion framework. For example, it is possible to allow the core belief to be revisedor to relax the restriction that new inputs are always appended to the end of ρin an epistemic state [ρ,N].A Method for Reasoning about other Agents’ Beliefs from Observations 5References1. Booth, R.: On the logic of iterated non-prioritised revision. In: Conditionals,Information and Inference, Springer’s LNAI 3301 (2005) 86–1072. Booth, R., Nittka, A.: Reconstructing an agent’s epistemic state from observations.In: Proceedings of IJCAI’05. (2005) 394–3993. Booth, R., Nittka, A.: Beyond the rational explanation. In: Belief Change in Ratio-nal Agents: Perspectives from Artificial Intelligence, Philosophy, and Economics.Number 05321 in Dagstuhl Seminar Proceedings (2005)4. Booth, R., Nittka, A.: Reconstructing an agent’s epistemic state from observationsabout its beliefs and non-beliefs. Journal of Logic and Computation (accepted forpublication)5. Booth, R., Paris, J.B.: A note on the rational closure of knowledge bases with bothpositive and negative knowledge. Journal of Logic, Language and Information 7(1998) 165–1906. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Ar-tificial Intelligence 55 (1992) 1–607. Nayak, A.: Iterated belief change based on epistemic entrenchment. Erkenntnis41 (1994) 353–3908. Nebel, B.: Base revision operations and schemes: Semantics, representation andcomplexity. In: Proceedings of ECAI’94. (1994) 342–3459. Nittka, A.: A Method for Reasoning About Other Agents’ Beliefs from Observa-tions. PhD thesis, Leipzig University (submitted)10. Nittka, A.: Reasoning about an agent based on its revision history with missinginputs. In: Proceedings of JELIA’06. (2006) 373–38511. Nittka, A., Booth, R.: A method for reasoning about other agents’ beliefs fromobservations. Texts in Logic and Games (accepted for publication)

A Method for Reasoning about other Agents\u27 Beliefs from Observations

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A Method for Reasoning about other Agents\u27 Beliefs from Observations

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