Interpretations of the Growth of Knowledge in Dynamic Learning Situations

1. Motto: “If you have an idea and I have an idea and we exchange these ideas, then will each of us have two ideas…?”(After G.B. Show) Interpretations of the Growth of Knowledge in Dynamic Learning Situations András Benedek Inst. of Philosophy, Research Centre for the Humanities, HAS Cranach, Tree of Knowledge (1472) benedek@webmail.phil-inst.hu

2. A Plausible Thesis A common assumption lurking behind the debates of the 60ies: Knowledge does grow as a result of collaboration and information exchange "If you have an apple and I have an apple and we exchange apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas." Attributed to G.B. Show Trivial(local) counterexamples: Motivation: …to (re)interpret the „Growth of Knowledge” in dynamic (logical) terms

3. Good old-fashioned Questions(to be reconsidered) • What is it that is growing? • Whatconstitutes knowledge? • What kinds of knowledge are growing? • What exactly is ‘growing’ (if anything) in case of the different types of knowledge? • What is ‘Growth’? • What do we mean by (the)‘growth’ (of knowledge)? • What does ‘growth’ consist of? • How is it measured? • How do we detect growth? (Triggers, Statistics, Indicators) • How do we represent growth? (Orderings, Patterns, Measures?) • How could the ‘mechanics’ of growth processes be described? • Temporal dynamics of learning? • What kind of dynamic models we have for the description of changes in epistemic states as growth of knowledge? Does Human Knowledge double in every 5>3>…years?

4. What is growing? Whatkind of knowledge? • Tacit • Propositional • Empirical • Theoretical • Organizational/Institutional/Social/ • Procedural /Strategic, Methodological/ • Knowledge in „object(tive)” forms (Books, Data Stores, Wikis, Scientific Journals, Papers, Cross References, etc.) • The ‘Third World’ (Sets of Propositions, Problems, Theories, Models, Proofs, Methods,…) • The Fields of inquiry (New Questions, Subjects, Frameworks) • Knowledge collectives (Shared understandings, Reflection,Awareness, Common (Global) Knowledge

5. Whatconstitutes knowledge? -Conceptions of knowledge • Theoretically grounded accumulatingevidence / Warranted belief • Justified True Belief, • Defeasable/Undefeasible knowledge • Perceptual(enactive)knowledge, etc. • Any piece of information that promotesthe solution of a task (AI) Whatkind of knowledge? -Types ofpropositional(ly based) knowledge • Individual /Single-agent/Multi-agent/ • Collective /Distributed/Group /Common/Knowledge Networks • Factual /Hypotetical/Normativ/ • Reflexive /Mutual/Higher Order/… • Explicit/Implicit

6. What is ‘Growth’? • Whatdoes Growth consist of? / How is it measured? • Closeness / Convergence to the truth • Elimination of possibilities/ possible worlds / uncertainity • Higher Degree of Belief / Plausibility / • Inductive generalization • Increase in truthlikeness/verisimiltude/factual content, etc. • Higher Measure of Probability / Utility • Changes in Relations bw. Theories and Models / Theory change • More people know it / # ofKnowers/Organizations/CoP/Networks • Higher levels of reflexivity

7. Dynamics of learning Changes in knowledge states are triggered by • incoming semantic information, and • epistemic action(s), including • higher order reflection on the knowledge states of others Formal description of the effects of semantic information on communicating agents require epistemic modelsofchangeinknowledge states, (represented by logic structures) Brookes’ ‘fundamental equation’:K[S] + ΔI= K[S + ΔS] ΔI changes K[S] to K[S+ ΔS] where K[S + ΔS] is the changed knowledge structure, i.e., informationmodifiesknowledge states/structures

8. Dynamic logic models of changing knowledge states as a result of communication A: “B, do you have red?” Bob:“No” „Dynamics”:Temporal development of agents’ knowledge statesrestricted by „rules” Labeled transition systemsrepresented by graphs Other typical examples: ‘100 prisoners and a light bulb’, ‘Russian cards’, etc.

9. Dynamic Logic Models of Information Exchange Current issues: Models of information flow describe meaningful interactions between agents as abstract models of “social software”. Epistemic Logicsemerging from Hintikka’s Knowledge and Belief (1962) set the background of modeling information flowANDknowledgein a common framework Various operations in Dynamic Epistemic Logic (DEL)represent the changes: Updates Upgrades Revisions model the effect of information as a dynamic process.

10. Tools for Modeling Growth Growth process:(iterated) belief revision / upgrades with new (true/reliable) information Group level revision induced by communicationbetweenmembers of the group Assumptions: e.g. sincerity: members already accept the information (beforesharing it). Higher-level (doxastic) information: may refer to the agents' own beliefs, or even to their belief-revision plans.

11. Construction of semantic representations Local epistemic states / states of the the environment (shared statemens + public announcements, e.g.) Representation of communication protocols (e.g. in PAL) Kripke structures in DEL Interpreted scenarios of information flow(transitions of knowledge)

12. Epistemic Models In finite models, any announcement with a proposition ϕ has an update which can be generated equivalently by a proposition which becomes common knowledge after its announcement.

13. Dynamic Consequence Conclusion ϕ follows dynamically from P1, . . . , Pk if, after public announcements of the successive premises, all worlds in the new information state satisfy ϕ :

14. Group Knowledge

15. Combining individual knowledge to explicit Group Level • Summative Collective Attitudes (defined in terms of individual attitudes) • Shared Belief • Mutual Belief • Distributed Knowledge • Common Knowledge • But we also have Non-Summative Collective Attitudes (The fact that all of the group members believe that P is neither sufficient nor necessary for a group belief that P) • .

16. Group Knowledge and Full Communication Instead of being able to communicate with each other, we may say that the G-knowledge is just the knowledge of one distinct agent, (the `wise man') to whom all the agents communicate their knowledge that he combines to end up with the knowledge that previously was implicit. (Growth: implicit knowledge upgraded to explicit knowledge) There is a way of choosing the parameters such that distributivity is not trivialised. Full communication, as a weak variant of distributivity may not be guaranteed. Van der Hoek, van Linder and Meyer gave properties on Kripke models that guarantee that group knowledge does allow full communication. Their results can be extended to models equipped with specific communication structures.

17. Communication Structures Logic Models of the dynamics of information exchange,may depend on - communication structures (CS), and - communication protocols (CP) CS: Relations bw. communicating agents (learners, players of a game, CoP, etc) represented by various relational structures • Communication graphs • Relational algebras • Galois lattices • Hyper graphs N.B.: Applications to social networks

18. Communication Protocols (CP) Rules/Regulations/Patterns/Procedures that governknowledge transfer • E.g.: • Security policies, Secrecy • Rules formaking private information public, • Sincerity conditions, • Orders of epistemic actions, communications, temporal or historical possibilities • Restrictions: e.g., only (hard) factual information is communicated, • Soft (e.g. communicated, non-reliable) information is allowed, • Higher order epistemic information is communicated/restricted, • Bounds on levels of Reflection

19. Communication Protocols in DEL Freedom of Speech: No Hiding: Telling the Truth:

20. PAL The language of public announcement logic PAL can be considered as the prototipical epistemic language, with added expressions of epistemic actions: The modal operator [ϕ] (‘after publicly announcing ’) is interpreted as an epistemic state transformer: the model M |ϕ is the model M restricted so as to only contain worlds in which ϕ is true.

21. Schematic validities

22. Common Knowledge DEL provides the techniques for carryingout the epistemic updates Gives logical means to reasonabout and express common knowledge of groups of agents: ' is common knowledge in group G if ' is true in all worlds that are reachable by a series of g-steps (with g 2 G) from the current world. Example:Modelling what goes on in Card Games Alice (1), Bob(2) and Carol (3) each hold one of cards p, q, r. The actual deal is: 1 holds p, 2 holds q, 3 holds r. After all players have lookedat their own cards they considerwhat the others may know. For common knowledge you have to compute the transitive closure of the union of theaccessibility relations Afixpoint procedure for making a relation transitive goes like this: 1. Check if all two-step transitions can be done in a single step. 2. If so,the relation is transitive, and done. 3. If not, add all two-step transitions as new links, and go back to 1.

24. CoordinatedAttack Problem: Common knowledge cannot be achieved in the absence of a simultaneous event (Public Announcement)

25. Problems Get rid of unrealistic assumptions on critical factors of the growth of knowledge 􀂊 logical omniscience 􀂊 positive and negative introspection 􀂊 unbounded recursion • Formalize higher-order cognition for different agent types • Resource-bounded logics by capabilities • dynamic inference, induction • reflection, • recursion, • update • revision, • upgrade • Extend with realistic components for group reasoning • Common belief • Common knowledge • Collective intention • Collective commitment

26. Upgrades

27. Upgrades that may represent Growth Gierasimczuk, N. (2009) Bridging learning theory and dynamic epistemic logic: the elimination process of learning by erasing can be seen as iterated belief-revision Pacuit. E. and Simon, S. (2011): Reasoning with Protocols under Imperfect Information.

28. DependenciesofReflexive Knowledge(KR) CP CS Ki(CS) Our knowledgeRof what the others knowdependson Asynchronous communication = message sent in serial/temporal order Synchronous communication = message sent to a whole group • Alternative (refined!) solutions to „Coordinated Attac Problems” • There are formulas that the agents may come to know that are not explicitly contained in their communications. • Essentially, these are facts that the agents can derive given their knowledge of the structure of the communication graphand the initial distribution of facts.

29. Thank you! Comments? Questions?

30. References T. Ågotnes, P. Balbiani, H. van Ditmarsch and P. Seban, 2010, Group AnnouncementLogic, Journal of AppliedLogic8(1). van Benthem, J. 2010, Logical Dynamics of Information and Interaction, Cambridge University Press. J. van Benthem, J. van Eijck & B. Kooi, 2006, ‘Logics of Communication and Change’, InformationandComputation204, 1620–1662. van Benthem, J., T. Hoshi, J, Gerbrandy, E. Pacuit, 2009, ‘MergingFrameworksforInteraction’, Journal of PhilosophicalLogic 38(5), 491–526. van Benthem, J., and Pacuit, E. 2006, ‘The tree of knowledgeinaction: Towards a commonperspective.’ InProceedings of AdvancesinModalLogicVolume 6, G. Governatori, I. Hodkinson, and Y. Venema, Eds. King's College Press. H. van Ditmarsch, W. van der Hoek & B. Kooi, 2007, Dynamic-EpistemicLogic, SyntheseLibrary 337, Springer, Berlin. Bird, A. (2008): ‘ScientificProgressasAccumulation of Knowledge—A ReplytoRowbottom’, StudiesinHistory and Philosophy of Science 39, 279–281. Fahrbach, L. (2011): HowtheGrowth of Science Ended TheoryChange. Synthese, 180(2):139-155. J.Y. Halpern and Y.O. Moses. (1990): Knowledge and commonknowledgein a distributedenvironment. Journal of the ACM, 37(3):549-587. R. Fagin and J.Y. Halpern, 1989, ‘Modelling knowledge and actionindistributedsystems. ’ Distrib. Comput. 34, pp. 159–179. Fagin, R, Halpern, J.Y, Moses, Y, and Vardi, M.Y., 1995, Reasoning about knowledge. The MIT Press: Cambridge, MA. Floridi, L, 2004, "Outline of a Theory of StronglySemanticInformation", Minds and Machines, 14(2), 197-222. Floridi, L, 2005, "Is InformationMeaningful Data?" Philosophy and Phenomenological Research, 70(2): 351–370. Hendricks, V .F. and Symons, J., 2006, ‘Where is theBridge? Epistemology and EpistemicLogic’ PhilosophicalStudies,Vol. 128, pp 137-167. T Hoshi & A. Yap, 2009, ‘DynamicEpistemicLogicwithBranchingTemporalStructure’, Synthese169, 259–281. Th. Icard, E. Pacuit & Y. Shoham, 2009, ‘IntentionBasedBeliefRevision’, Departments of Philosophy and Computer Science, Stanford University. D. Israel & J. Perry, 1990, ‘What is Information?’, in P. Hanson, ed., Information, Language and Cognition. University of British Columbia Press, Vancouver. Meyer, Ch. and van derHoek, W., 1995, Epistemic Logic for AI and ComputerScience. Cambridge University Press: Cambridge, England. van Ditmarsch, H, van derHoek, W, and Kooi, B., 2007, Dynamic Epistemic Logic, Springer, Berlin. Fitzgerald, L.A. and van Eijnatten, F.M., 1998, “Letting Go ForControl: The Art of ManagingtheChaothicEnterprise”, The International Journal of Business Transformation, Vol. 1, No. 4, April, pp 261-270. King, Wr. (2006) Knowledge transfer. In Encyclopedia of KnowledgeManagement (SCHWARTZ DG, Ed), pp 538–543, Idea Group Reference,Hershey, PA. S. van Otterloo, 2005, A StrategicAnalysis of Multi-AgentProtocols, Dissertation DS-2005-05, ILLC, University of Amsterdam & University of Liverpool. Nonaka I. and Takeuchi H., (1995) The KnowledgeCreatingCompany. Oxford University Press. Oxford, New York. UK. R. Parikh, 2002, ‘Social Software’, Synthese132, 187–211. Parikh, R., & Ramanujam, R. (2003). A knowledgebasedsemantics of messages. Journal of Logic, Language and Information, 12, 453–467. M. Pauly, 2001, LogicforSocial Software, dissertation DS-2001-10, Institute forLogic, Language and Computation, University of Amsterdam. J. Peregrin (ed.), 2003, Meaning: theDynamicTurn, Elsevier, Amsterdam O. Roy, 2008, ThinkingbeforeActing: Intentions, Logic, and RationalChoice, Dissertation, Institute forLogic, Language and Computation, University of Amsterdam. J. Sack, 2008, ‘TemporalLanguageforEpistemicPrograms’, Journal of Logic, Language and Information17, 183–216.