Common Knowledge and Handshakes in Computer-Mediated Cooperation

1
Common Knowledge and Handshakes in
Computer-Mediated Cooperation

• Albert Esterline
• Dept. of Computer Science
• North Carolina AT State University

2
Introduction

• Goal
• Model human and artificial agents formally and
  uniformly in systems where they collaborate
• Gain insight into the conditions for coordination
  that such modeling offers.

3

• Start with a simple distributed game that
  displays a common interface.
• Players collaborate to move proxy agents around a
  grid.
• Requires making agreementsentails common
  knowledge.
• Formal characterization and interpretation of
  common knowledge.
• New common knowledge and simultaneous actions.

4

• Handshakes and process algebras
• Process-algebraic agent abstraction
• Must add account of common knowledge and deontic
  notions.
• Co-presence heuristics for establishing common
  knowledge
• Grounding (human-computer dialog)
• Back to the simple distributed game
• Virtual agents

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Simple Distributed Cooperative System

• Users move proxy agents on a grid.
• Each player participates at his own workstation.
• But system ensures that grid state is displayed
  in exactly the same way to all players.
• Each agent visits several goal cells specific to
  it in an unspecified order.
• Single-cell moves are made in round-robin
  fashion.
• Object cooperate so as to minimize the total
  number of single-cell moves taken by all proxy
  agents to visit all their goal cells.

6

• Free space on the grid tends to occur in long
  corridors.
• Need agreements to avoid lengthy backtracking
  when two agents travel in opposite directions on
  a corridor.
• Interface has features that allow the players to
  suggest and agree on itineraries.
• All interaction is by clickingeasy
  interpretation of communication

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• A player can make a suggestion when its his/her
  turn.
• All players can negotiate.
• Agreement must be unanimous.
• An agreement is obligates the player of the proxy
  agent in question.
• It must be common knowledge.

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Three Approaches to Common KnowledgeIterate
Approach

• Assume n agents named 1, 2, , n., G1,,n
• Introduce n modal operators Ki, 1 ? i ? n.
• Ki ? is read agent i knows that ?.
• EG ?, read as everyone in group G knows that ?.
• is the EG operator iterated k times.
• CG ? ? is common knowledge in group G.

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Fixed-point Approach

• View CG ? as a fixed-point of the function
• f(x) EG(? ? x).
• Specifically (derivable in augmented S5),
• CG ? ? EG (? ? CG ?)

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Shared Situation Approach

• Assume that A and B are rational.
• We may infer common knowledge among A and B that
  ? if
• A and B know that some situation ? holds.
• ? indicates to both A and B that both A and B
  know that ? holds.
• ? indicates to both A and B that ?.

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Barwise on the Three Approaches

• Barwise contrasts the 3 approaches within his
  situation theory.
• An infon is an (n1)-tuple of a relation and n
  (minor) constituents.
• Its polarity is 1 if the minor constituents are
  related as per the relation.
• A set of infons is a situation (small world).
• An infon with polarity 1 is a fact (of some
  situation, not others).

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• Minor constituents may be situations, even one
  where the infon itself occurs.
• Example
• ?H, pi, 3?? player i has the 3 of clubs
• ?S, pi, s? player i sees situation s
• s ?H, p1, 3??, ?S, p1, s?, ?S, p2, s?
• situation where player 1 has the 3 of hearts and
  this is publicly perceived by both player 1 and
  player 2

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• Define classes INFON (of infons) and SIT of
  (situations) by mutual induction.
• Consider the fixed-points of a monotone
  increasing operator ? corresponding to this
  inductive definition.
• If a standard set theory (e.g., ZFC) is used as
  the metatheory, theres a unique fixed-point.
• But Barwise considers a variant of ZFC giving
  multiple fixed-points

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• Two intuitions about sets
• I1. Sets are collections got by collecting
  together things already at hand to get something
  new (a set).
• I2. Sets arise from independently given
  structured situations by dropping the
  structureforgetful situations.
• I1 generates the cumulative hierarchy
  characteristic of, e.g., ZFC.
• I2 gives a richer universe of sets.

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• b ? s b is a constituent of situation s (a
  minor constituent of some infon in it).
• Reality is wellfounded iff every situation is
  wellfounded.
• A situation is wellfounded iff its neither
  circular nor ungroundable.
• Situation s is circular if s ? ? s.
• s is ungroundable if theres an infinite sequence
• ? s? ? s? ? s

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• These notions also apply to sets.
• The Axiom of Foundation of ZFC
• A set contains no infinitely decreasing
  membership sequence.
• Rules out circular and ungroundable sets.
• Barwise proves
• The universe of sets is wellfounded iff the
  universe of situations is.
• So we must replace the Axiom of Foundation of ZFC
  with something that
• admits non-wellfounded sets and
• supports unique construction of sets.

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• Take Aczels Anti-Foundation Axiom, AFA.
• When this replaces the Axiom of Foundation in
  ZFC, get ZFC/AFA set theory.
• A tagged graph is a directed graph where each
  node without children is tagged with an atom or
  ?.
• A decoration for a tagged graph is a recursive
  function ? mapping a node x to a set.
• If x is childless, then ?(x) is its tag.
• Otherwise ?(x) ?(y) y is a child of x.
• A tagged graph G is wellfounded if the child-of
  relation on G is wellfounded (no circular or
  infinite directed paths).

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• Without AFA, can prove that every wellfounded
  tagged graph has a unique decoration in the
  universe of sets.
• AFA asserts that every tagged graph has a unique
  decoration.

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• With ZFC/AFA as our metatheory, there are many
  fixed-points of ?.
• Least fixed-point gives collection of wellfounded
  infons and situations.
• Interested in greatest fixed-point.
• Includes all the non-wellfounded infons and
  situations as well.

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• Want to compare iterate and fixed-point
  approaches.
• Show how infon ? gives rise to an transfinite
  sequence of wellfounded infons ??, ? a finite or
  infinite ordinal.
• Requires a sequence s? for any situation as well.
• These are sequences of approximations.
• Members of a sequence approximating a
  non-wellfounded situation have increasingly deep
  nestings.
• Corresponds to increasingly deep nestings of
  everyone knows that operator.

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• For circular infon ?, approximations get ever
  stronger but never as strong as ?.
• Yet the totality of all approximations captures
  ?.
• If each ?? holds in a situation, so does ?.
• The finite approximations of a circular infon are
  equivalent to it w.r.t. finite situations.
• But this doesnt hold for infinite situations.
• In this sense, iterate approach is weaker than
  fixed-point approach.

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• In shared-situation approach, characterize common
  knowledge in terms of existence of a real
  situation meeting a certain condition.
• Introduce a second-order language to express the
  existential conditions.
• Variables range over situations, may be bound by
  existential quantifiers.
• Semantics stated in terms of assignment of
  situations to free situation variables in a
  condition.
• A model for a condition is an assignment making
  it true.

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• Two conditions with the same free variables are
  strongly equivalent if they have the same models.
• A condition entails a sequence of infons
• if that sequence is a list of facts, each
  holding in the situation assigned to a given
  variable in any assignment satisfying the
  condition.
• Two conditions with the same free variables are
  informationally equivalent if they entail the
  same sequences of infons.
• A model M of a condition ? is a minimal model of
  ?
• if each situation in M has no more
  information than the corresponding situation in
  any other model of ?.
• A condition generally has several minimal models.

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• Can be shown that 2 conditions are
  informationally equivalent iff they have a
  minimal model in common.
• So, suppose we start with shared-situation
  approach, formulating a condition.
• Situations in a minimal model of this condition
  give a handle for fixed-point approach.
• But 2 conditions can be informationally
  equivalent and not strongly equivalent.
• Conditions are more discriminating than the
  situations that are their minimal models.
• 2 conditions may be different but equally correct
  ways a group comes to have shared information.

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Barwises Conclusions

• Fixed-point approach is correct analysis of
  common knowledge.
• Common knowledge generally arises via shared
  situations.
• Iterate approach characterizes how common
  knowledge is used?
• Progress through sequence of approximations
  corresponds to inferring ever deeper nestings of
  everyone knows that?
• But doubt about a given inference blocks next
  step.

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• Knowing that ? is stronger than carrying the info
  that ?.
• Involves carrying the info in a way relating to
  ability to act.
• Possible-worlds semantics of standard epistemic
  logic requires we know all logical consequences
  of what we know.

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• Common knowledge (per fixed-point approach) is a
  necessary but not sufficient condition for
  action.
• Useful only when arising in a straightforward
  shared situation.
• A situation works not just by giving rise to
  common knowledge.
• It also provides a stage for maintaining common
  knowledge through the maintenance of a shared
  situation.
• The shared interface of our system is a common
  artifact in Devlins sense.

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Common Knowledge and Simultaneous Action

• Agents A and B communicate over a channel.
• Its common knowledge that
• delivery of a message is guaranteed and
• a message A sends to B arrives either immediately
  or after ? time units.
• At time mS, A sends B a message ? that doesnt
  specify the sending time.
• Let
• mD denote the message arrival time and
• sent(?) the proposition that ? has been sent.

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• KB sent(?) is true at mD.
• But A cant be sure that KB sent(?) before mS?.
• So KA KB sent(?) isnt true until mS?.
• And B knows this.
• But ? may have been delivered immediately.
• So B doesn't know that mS? time has elapsed
  until mD?.
• So KB KA KB sent(?) doesnt hold until mD?.
• And A knows this.
• But it may take ? time for ? to be delivered.
• So mD could (for all A knows) be mS?.
• So KA KB KA KB sent(?) does not hold until mS2?.

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mS
mS ?
mS 2?
mS 3?
?
mD
mD ?
mD 2?
mS
mS ?
mS 2?
mS 3?
?
mD
mD ?
mD 2?
mD 3?
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• A straightforward induction shows that, for any
  natural number k,
• before mSk?, (KA KB)k sent(?) doesnt hold,
  while
• at mSk? it does.
• Common knowledge requires infinitely deep nesting
  of KA KB.
• So common knowledge of sent(?) is never attained
  no matter how small ?.

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• But suppose that
• A attaches the sending time mS to ?, giving
  message ??, and
• A and B use the same global clock .
• When B receives ??, he knows it was sent at mS.
• Because of the global clock, it is common
  knowledge at time mS? that it is mS?.
• Since it is also common knowledge that a message
  received at mS? was sent at mS,
• CG sent(??), G A, B,
• holds at mS?.

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• Can model the global clock is with another agent.
• An action by any other agent is always
  simultaneous with one of this agents actions (a
  tick).
• More parsimoniously
• Require that an agent have a different state at
  each point in a run.
• It always knows what time it is.

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• A thesis of standard epistemic logic
• CG ? ? EG CG ?.
• So the transition
• from ? not being common knowledge
• to it being common knowledge
• must involve simultaneous changes in the
  knowledge of all agents in the group.
• I.e., information becomes shared in the required
  sense at the same time for all agents sharing it.
• No surpriseall the agents are involved in the
  circularity.

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Common Knowledge Inherent in Agreement and
Coordination

• Suppose that A and B agree to something ?.
• For there to be an agreement, every party in
  group G A, B must know theres agreement
• agreeG(?)? EG agree(?) ()
• By idempotence of ?, this is equivalent to
• agreeG(?)? EG (agreeG(?) ? agreeG(?))
• But standard epistemic logic includes the
  inference rule
• From ?1 ? EG (?2 ? ?1) infer CG ?2
• Substituting agreeG(?) for both ?1 and ?2 in the
  rule and using () for the premise, we infer
• agreeG(?)? CG agree(?)

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• To show formally that coordination implies common
  knowledge requires extensive development.
• But the result is just as direct.

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Process Algebras and Handshakes

• The standard epistemic-logic framework explicates
  the notion of simultaneous actions.
• But the notion it provides of a joint action
  preformed by n agents is simply
• an (n1)-tuple whose components are the
  simultaneous actions of the environment and the n
  agents.
• One thing critical to a joint action is
• the agents must time their contributions so that
  each contributes only when all are prepared.

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• A handshake in process algebras is a joint
  communication action that happens only when both
  parties are prepared for it.
• A process algebra (e.g., ?-calculus, CCS, CSP) is
  a term algebra.
• Terms denote processes.
• Combinators apply to processes to form more
  complex processes.
• Combinators typically include
• alternative and parallel composition and
• a prefix combinator that forms a process from a
  given process and a name.

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• Names come in complementary pairs.
• A prefix offers a handshake.
• A handshake results in an action identified by
  the prefix of the selected alternative.
• Resulting process consists of only the selected
  alternative with its prefix removed.
• Parallel processes may handshake if they have
  alternatives with complementary prefixes.
• Only way a process can evolve is as result of
  handshakes.

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• Handshakes between parallel components can happen
  only when they have evolved to have alternatives
  beginning with complementary prefixes.
• In this sense, they can handshake only when both
  are prepared.
• Handshakes synchronize the behavior of components
• They thereby coordinate behavior.
• Handshakes are like speech acts.
• Contemporary analysis of face-to-face
  conversation emphasizes the active role of
  addressees (e.g., nods).

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Process-Algebraic Agent Abstraction

• Some of the combinators (and their syntactic
  patterns) persist through transitions
• e.g., parallel composition and restriction (or
  hiding) combinators.
• Other combinators (e.g., alternative composition
  and prefix) don't thus persist.
• Processes corresponding to agents persist
  through transitions.
• So a a multiagent system from is
• a parallel composition.
• Each component models an agent and involves a
  recursively defined process identifier.

43

• This view of agents is simpler than that of
  standard epistemic logic.
• Handshakes are primitives, so no need for
  assumptions about agents states or a global
  clock to support joint actions.
• State of an agent given simply by the current
  form of the term denoting it.
• A process algebra is more concrete than epistemic
  logic.
• A logic lets us assert abstract properties of an
  agent or system of agents.
• Using a process algebra, we specify the behavior
  of agents.

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Whats Missing in the Process-Algebraic Agent
Abstraction

• Tempting to view process-algebraic terms as
  possible plans an agent or a person may
  undertake.
• But the notion that humans execute predefined
  plans in interacting with technology or with each
  other has been heavily criticized by
  ethnomethodologists.
• Emphasize how situated behavior is determined in
  an ongoing way.

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• Certain speech acts occur only to establish
  common knowledge.
• Nearly all contributions in a conversation
  advance our common knowledge.
• So what future actions might be appropriate is
  determined as a joint project unfolds.
• And patterns of joint communication actions have
  nothing to say about behavior that deviates from
  them.

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• What was missing in our agent abstraction was the
  persisting effects of speech acts.
• Within a conversation speech acts can establish
  common knowledge.
• Also, certain speech acts have deontic effects,
  such as obligations, prohibitions, and
  permissions.

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Deontic Logic

• Modal operators of standard deontic logic
• O ?, ? is obligatory,
• P ?, ? is permitted, and
• F ?, ? is forbidden (or prohibited).
• P ? ? ? O ? ? ? ? F ?
• Development driven by certain paradoxes that
  arise when theres a conflict between
• the logical status (valid, satisfiable, etc.) of
  a deontic-logic formula and
• the intuitive understanding of the
  natural-language reading of the formula.

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• Dyadic deontic logice.g.,
• O? ? Given ?, it is obligatory that ?.
• Special obligations, permissions, and
  prohibitionse.g.,
• OA ? It is obligatory for A that ?.
• Directed obligations, etc.e.g.,
• OA,B ? A is obligated to B that ?.
• Deontic operators derived from operators that
  make action explicite.g.,
• A sees to it that ?
• operators of dynamic logic.

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• Deontic notions are appropriate whenever we
  distinguish between
• what is ideal (obligatory) and
• what is actual.
• Reject O ? ? ? as a thesis.
• Obligation may be violated.

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• Some application areas of computer science
• formal specification
• Modern software is so complex, we must cover
  non-ideal cases too in specifications.
• fault tolerance
• Non-ideal behavior introduces obligations to
  correct the situation.
• database integrity constraintsdistinguish
  between
• deontic constraints may be violated
• necessity constraints largely analytically true.

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Co-presence Heuristics

• Clack and Carlson people ordinarily rely on
  special kinds of evidence to which the
  shared-situation induction scheme is applied.
• Co-presence heuristics
• Physical co-presence (cf. Chwe)
• Object is located between agents A and B.
• Both A and B see the object and each other
  simultaneously.
• Gives evidence of the triple co-presence of A,
  B, and the object of common knowledge.

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• Linguistic co-presence
• Triple co-presence of A, B, and the linguistic
  positing of the object of common knowledge
• Community membership
• If A finds that B is in the same community as A,
• then A can conclude that B must have common
  knowledge shared by that community.
• The other 2 heuristics presuppose this one.

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Physical Co-presence

• Apply the physical co-presence heuristic so that
  groups of agents may attain common knowledge
  perceptually.
• Agents must model each others perceptionrequires
  
• shared perceptual abilities and
• common knowledge of these abilities.
• A standard design would have
• rules for classifying perceived objects
  (including other agents) and
• rules for constructing perceptual models of other
  agents.

55

• Implemented a prototype, coupled with a back
  propagation neural network.
• Agent behavior adapted to trainers feedback.
• Captured hidden knowledge not captured by
  knowledge engineer.

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Grounding

• Clark emphasizes the common ground in
  face-to-face conversation.
• Grounding has become a major topic in
  human-computer dialog.
• Two-phase communication presentation,
  acceptance.
• Grounding criterion threshold at which evidence
  for common knowledge is deemed sufficient.
• Diagram (e.g., human-computer) conversations.
• Brennan emphasize private models to avoid
  inconsistencies in a diagram.
• But omits whats critical shared situation

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Back to the Simple Distributed Game

• An agreement is sealed with a handshake in which
  all players take part.
• This joint action establishes the required common
  knowledge.
• The itineraries record obligations.

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• For implementing a handshake mechanism
• each player must be able independently to
  initiate his contribution (this is being
  prepared), and
• there must be feedback indicating to all that all
  have initiated he contributions and persisted
  with them.
• In our implementation, a player
• moves the mouse cursor over a designated area and
  
• holds down the left mouse button.
• If all players participate, the suggestion is
  displayed in the updated display of the
  itinerary.

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• Ways a player may disagree
• By simply not participating in the handshake.
• Then the opportunity times out.
• By offering a counter-suggestion.
• Done in the same way as the original suggestion.
• But done by a player during the turn of the
  player who made the original suggestion.

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Virtual Agents (Clark)

• Clark is concerned with how we communicate with
  virtual partnerse.g.,
• the person speaking to me in the letter
• the person giving me directions via a cook book
• Disembodied language not produced by an actual
  speaker at the moment its interpreted.
• Two main forms
• written language
• mechanized speech (e.g., films, telephone
  messages)

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• Disembodied language is a representation of
  embodied language.
• Were intended to imagine the embodied language
  it represents.
• Salient features
• Virtual speaker
• Producer the person/institution ultimately
  responsible for the disembodied language
• Virtual time
• Pacing

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• Many joint activities divide into layers of joint
  actions.
• what is actually happeninga pretense
• the pretense itself
• Two things needed for successful layering
• Credible characters
• Props

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• When we interpret any form of communication with
  a computer, we communicate with virtual agents.
• Reeves and Nass, The Media Equation
• People equate media and real life.
• This is very common, it is easy to foster, it
  does not depend on fancy equipment, and thinking
  will not make it go away.

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• Our proxy agents are virtual agents?
• But the language (clicking fields) is produced by
  the player at the moment its interpreted.
• The 2 layers intermingle.
• Equally natural to say
• the proxy agent moves
• the player moves the proxy agent
• Likewise for obligations.

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• But common knowledge?
• The activities in our game can be automated (some
  easily)
• No way to tell whether were communicating with a
  person or agent behind the proxy agent.
• And no way to tell whether theres one person
  controlling two proxy agents.

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Conclusion

• Insights from formal modeling into coordination.
• Started with a simple distributed cooperative
  game.
• Agreements, presupposing common knowledge.
• Formal characterization of common knowledge.
• Iterate, fix-point, and shared-situation
  approaches.
• New common knowledge and simultaneous actions.
• Handshakesjoint actions
• Process-algebraic agent abstraction
• Also need epistemic and deontic effects.

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• Co-presence heuristics for common knowledge
• Grounding
• Back to coordination game
• Virtual agents and disembodied language

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• Need conventions to escape dependence on
  simultaneous actions.
• Cf. disembodied language
• In CS, we have languages and protocols.
• Agent communication in the first instance
  (conceptually) is synchronous.
• Need appropriate support for asynchronous
  communication.
• Need the appropriate language game.

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• Agent software development
• Specification
• Modal logics, concepts of common knowledge and
  obligations, etc.
• Design
• Process algebras (or something at comparable
  level of abstraction)
• Implementation
• Appropriate communication primitives, including
  transactional features

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Future WorkCoordination Game

• Agents with path-planning and negotiation
  abilities
• Automated aids for players (e.g., measure
  distances)
• Implementing counters allows operational
  definitions of
• Fairness
• When several players want to make a
  counter-suggestion at the same time, who gets the
  floor?
• Linear combination of inverses of counters gives
  priorities.
• Trustworthiness
• Dont enforce obligations, but can count
  violations

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Formal

• General obligations and some special obligations
  (roles) should be common knowledge.
• Multi-modal logic
• Relation between process algebras and modal
  logics
• Cf. Hennessey-Milner logics

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Conceptual

• A lot packed into the notions of situation,
  situated.
• Also environment, information in the environment
• Not amount of information
• Important for ubiquitous and embedded computing.
• Joint action, joint activity
• Relate to distributed, extended transactions