An algorithmic approach to knowledge evolution

Intelligent agents must update their knowledge base as they acquire new information about their environment The modal logic S n has been designed for representing knowledge bases in societies of agents Halpern and Vardi have proposed the notion of re nement of S n Kripke models in order to solve multi agent problems in which knowledge evolves We argue that there are some problems with their proposal and attempt to solve them by moving from Kripke models to their corresponding trees We de ne re nement of a tree with a formula show some properties of the notion and illustrate with the muddy children puzzle We show how some diagnosis problems in engineering can be modelled as knowledge based multi agent systems and hence how our approach can address them Introduction Temporal epistemic modal logics and their potential for applications In the last few years there is been a growing trend towards applying logical theories and Multi Agent theories in general to the speci cation and analysis of engineering products The reason behind this trend is that logic is a precise and unambiguous language and it is increasingly seen a useful tool for specifying reasoning about and validating complex systems Agent theories see Wooldridge and Jennings for a review aim to represent key prop erties of an intelligent entity such as its knowledge beliefs intentions desires actions and most importantly its temporal evolution in a changing environment Although much literature has been published in all of these areas there is a general consensus about using epistemic and temporal modal logics in which much progress has been made Epistemic modal logics see Hintikka Fagin Halpern Moses and Vardi Meyer and van der Hoek aim to represent the state of knowledge of an agent and to study what properties the state of knowledge should satisfy This is formally done by using a classical modal operators de ned on a Kripke style semantics Kripke Temporal modal logics see Gabbay Hodkinson and Reynolds Clarke and Emerson Emerson Lamport Pnueli Shoham use a similar technical tool to represent the temporal evolution of a system and investigate properties of this evolution In this paper we try to develop further part of this tool and we suggest that this can be proven useful in a practical example speci cally fault diagnosis in a distributed robotics situation We examine a quite well known puzzle studied in computer science known as the muddy children puzzle and demonstrate that this example is conceptually equivalent to integrity self checking in a robotics plant We propose a general algorithm that can be applied in similar situations involving distributed knowledge among a group of agents The theoretical background The logic S n models a community of ideal knowledge agents Ideal knowledge agents have among others the properties of veridical knowledge everything they know is true positive introspection they know what they know and negative introspection they know what they do not know The modal logic S n see for example Popkorn and Goldblatt can be axiomatised by taking all the propositional tautologies the schemas of axioms i i i i i i i i i i where i A represents an agent in the set of agents A f ng and the inference rules Modus Ponens and Necessitation The logic S n has also been extended to deal with properties that arise when we investigate the state of knowledge of the group Subtle concepts like common knowledge and distributed knowledge have been very well investigated as in Fagin et al The logic S n is a successful tool for the agent theorist also because even in its extensions to common knowledge and distributed knowledge it has important meta properties like closure under substitution completeness and decidability see for example Meyer and van der Hoek The standard consequence relation approach to using S n is to describe a situation as a set of formulas and to attempt to show that the situation satis es a property by establishing or Establishing involves nding a proof of from while establishing involves reasoning about all usually in nitely many Kripke models satisfying to show that they also satisfy The completeness of S n shows that these two notions are equivalent However experience has shown that this approach is computationally very expensive In order to overcome the intractability of this approach Halpern and Vardi have proposed to use model checking as an alternative to theorem proving Halpern and Vardi In the model checking approach the situation to be modelled is codi ed as a single Kripke model M rather than as a set of formulas The task of verifying that a property holds boils down to checking that M satis es written M This task is computationally much easier than the theorem proving task being linear in the size of M and the size of Halpern and Vardi Halpern and Vardi informally illustrate their approach by modelling the muddy children puzzle In that puzzle there are n children and n atomic propositions p p pn representing whether each of the children have mud on their faces or not Various announcements are made rst by the father of the children and then by the children themselves The children thus acquire information about what other children know and after some time the muddy ones among them are able to conclude that they are indeed muddy We describe the problem in greater detail below Halpern and Vardi propose the following way of arriving at the model M to be checked They start with the most general model for the set of atomic propositions at hand In order to deal with the announcements made they successively re ne the model with formulas expressing the announcements made This re nement process consists of removing some links from the Kripke model At any time during this process they can check whether child i knows pi for example by checking whether the current model satis es ipi This method is illustrated in the paper Halpern and Vardi and the book Fagin et al but a precise de nition of the re nement operation is not given Our original aim for this paper was to provide such a de nition and explore its properties However we soon came to the opinion that there is no de nition of model re nement on arbitrary S n Kripke structures that will have intuitively acceptable properties We explain our reasons for this view in section We believe the re nement and model checking ideas can still be made to work however In section we introduce a structure derived from a Kripke model which we call a Kripke tree and de ne the re nement operation on Kripke trees We illustrate this notion using the muddy children example in section We prove some some properties of the re nement operation on Kripke trees in section and we conclude with some discussion in section This is mainly a theoretical paper However we argue that scenarios conceptually equivalent to the muddy children puzzle can occur in robotics We describe one of these scenarios in section and we solve it in section by applying the technical machinery we develop in section Syntax and semantics We assume nite sets P of propositional atoms and A of agents Formulas are given by the usual grammar p j j j i j C where p P and i A Intuitively the formula i represents the assertion that the agent i knows the fact represented by the formula The other propositional connectives can be de ned in the usual way The modal connectives i E and B are de ned as i means i E means V i A i B means C i means it is consistent with i s knowledge that E means that everyone knows while C is the much stronger statement that is common knowledge In a multi agent setting a formula is said to be common knowledge if it is known by all the agents and moreover that each agent knows that it is known by all the agents and moreover each agent knows that fact and that one etc An announcement of results in common knowledge of among the hearers because as well as hearing they also see that the others have heard it too we assume throughout that all the agents are perceptive intelligent truthful If one agent secretly informs all the others of the result will be that everyone knows but will not be common knowledge B is the dual of C Although not particularly useful intuitively we will need it for technical reasons We will also need the following de nitions De nition A formula is universal if it has only the modalities C E i and no negations outside them Formally take p j j j and de ne a formula to be universal if it follows the following syntax j j j i j E j C De nition A formula is safe if it is universal and after negations are pushed inwards no i and no C appears in the scope of Formally take p j j j and de ne a formula to be safe if it follows the following syntax j j i j E j C De nition A formula is disjunction free if it is universal and has no Formally take p j j and de ne a formula to be disjunction free if it follows the following syntax j j i j E j C De nition An equivalenceKripke model M W w of the modal language over atomic propositions P and agents A is given by A set W whose elements are called worlds An A indexed family of relations f igi A For each i n i is an equivalence relation on W i W W called the accessibility relation A function W P P called the assignment function A world w W the actual world See Figure for an illustration Let x W We de ne the relation of satisfaction of by M at x written M j x in the usual way M j x p i p x M j x i M j x M j x i M j x andM j x M j x i i for each y W x i y implies M j y M j x C i for each k and i i ik A we have M j x i ik We say that y is reachable in k steps from x if there are w w wk W and i i ik in A such that x i w i w ik wk ik y We also say that y is reachable from x if there is some k such that it is reachable in k steps The following fact is useful for understanding the technical di erence between E and C Theorem Fagin et al M j x E k i for all y that are reachable from x in k steps we have M j y M j x C i for all y that are reachable from x we have M j y Re ning Kripke models Halpern and Vardi propose to re ne Kripke models in order to model the evolution of knowl edge They illustrate their method with the muddy children puzzle This example is particularly important in the literature We report it in the following The muddy children puzzle There is a large group of children playing in the garden A certain number say k get mud on their foreheads Each child can see the mud if present on others but not on his own forehead If k then each child can see another with mud on its forehead so each one knows that at least one in the group is muddy The father rst announces that at least one of them is muddy which if k is something they know already and then he repeatedly asks them Does any of you know whether you have mud on your own forehead The rst time they all answer no Indeed they go on answering no to the rst k questions but at the kth those with muddy foreheads are able to answer yes At rst sight it seems rather puzzling that the children are eventually able to answer the father s question positively The clue to understanding what goes on lies in the notion of common knowledge Although everyone knows the content of the father s initial announcement the father s saying it makes it common knowledge among them so now they all know that everyone else knows it etc Consider a few cases of k k i e just one child has mud That child is immediately able to answer yes since she has heard the father and doesn t see any other child with mud k say a and b have mud Everyone answers no the rst time Now a thinks since b answered no the rst time he must see someone with mud Well the only person I can see with mud is b so if b can see someone else it must be me So a answers yes the second time b reasons symmetrically about a and also answers yes k say a b c Everyone answers no the rst two times But now a thinks if it was just b and c with mud they would have answered yes the second time So there must be a third person with mud since I can only see b c having mud the third person must be me So a answers yes the third time For symmetrical reasons so do b c And similarly for other cases of k To see that it was not common knowledge before the father s announcement that one of the children was muddy consider again k say a b Of course a and b both know someone is muddy they see each other but for example a doesn t know that b knows that someone is dirty For all a knows b might be the only dirty one and therefore not be able to see a dirty child An engineering example The muddy children puzzle together with its many variants like the three wise men puzzle etc is popular among computer scientists The reason is that it encodes subtle properties about reasoning while also being applicable to real life scenarios We can imagine an example in which an engineering system could bene t from being able to cope with muddy children like situations Consider a factory in which similar robots collectively manufacture an object while moving in group in a large space The robots can roughly be thought of being made of two components the reasoning module and the mechanical actuators e ectively operating on the object We want to design a fault detection system for the actuators Given the large area the robots can be in the installation of cameras to monitor the operational status of the robots arms is not an option Let us suppose that the robots have a visual system directed towards the other robots that can detect faults in their mechanical arms Note this is quite a reasonable assumption since it is often problematic to have visual systems that can do self monitoring as well as monitoring the environment Suppose now that the factory has a quality control mechanism that can detect if something went wrong during the production of the object and assume this device broadcasts an alarm every time it notices a defect in the production This robotic scenario complies with the muddy children example the children are now robots the role of the father is taken by the fault detection system Note that the assumption of communi cation being common knowledge is not violated because messages are assumed to be broadcasted to all the agents The task of the robots is then to reason about their status and stop their operation in case they come to know that their mechanical arm is faulty The evolution of their knowledge proceeds exactly as the case of the muddy children example where we assume the robots to operate synchronously Assuming the robots have a reasoning module able to handle the muddy children problem the group of robots is then e ectively able to do collective diagnosis In the following we refer our discussion to muddy children but the above scenario can serve equally well Halpern and Vardi s formulation Suppose A f ng and P fp png pi means that the ith child has mud on its forehead Suppose n The assumption of this puzzle is that each child can see the other children but cannot see itself so each child knows whether the others have mud or not but does not know about itself Under these assumptions Halpern and Vardi propose the Kripke structure of Figure to model the initial situation Let w be any world in which there are at least two muddy children i e w is one of the four upper worlds In w every child knows that at least one of the children has mud However it is not the case that it is common knowledge that each child has mud since the world at the bottom of the lattice is reachable cf Theorem To model the father s announcement Halpern and Vardi re ne the model M in Figure arriving at M in Figure these gures also appear in Halpern and Vardi Fagin et al The re nement process is not precisely de ned in Halpern and Vardi Fagin et al