Common Knowledge and Handshakes in Computer-Mediated Cooperation Albert Esterline

Dept. of Computer Science

North Carolina A&T State University

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.

Start with a simple distributed game that displays a common interface. Players collaborate to move proxy agents around a

grid. Requires making agreements—entails common

knowledge. Formal characterization and interpretation of common

knowledge. New common knowledge and simultaneous actions.

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

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.

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 clicking—easy interpretation of communication

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.

Three Approaches to Common KnowledgeIterate Approach Assume n agents named 1, 2, …, n., G={1,…,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.

iGiG KE k

GE

k

GkG EC

Fixed-point Approach

View CG  as a fixed-point of the function

f(x) = EG(  x).

Specifically (derivable in augmented S5),

CG  EG ( CG )

Shared Situation Approach

Assume that A and B are rational.

We may infer common knowledge among A and B that if

1. A and B know that some situation holds.

2. indicates to both A and B that both A and B know that holds.

3. indicates to both A and B that .

Barwise on the Three Approaches Barwise contrasts the 3 approaches within his situation

theory.

An infon is an (n+1)-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).

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

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, there’s a unique fixed-point. But Barwise considers a variant of ZFC giving multiple

fixed-points

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 structure—“forgetful situations.”

I1 generates the cumulative hierarchy characteristic of, e.g., ZFC.

I2 gives a richer universe of sets.

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 it’s neither circular nor ungroundable. Situation s is circular if s … s. s is ungroundable if there’s an infinite sequence

… s s s

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.

Take Aczel’s 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).

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.

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.

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.

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 doesn’t hold for infinite situations. In this sense, iterate approach is weaker than fixed-

point approach.

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.

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.

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.

Barwise’s 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.

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.

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 Devlin’s sense.

Common Knowledge and Simultaneous Action Agents A and B communicate over a channel.

It’s 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 doesn’t specify the sending time.

Let mD denote the message arrival time and sent() the proposition that has been sent.

KB sent() is true at mD.

But A can’t be sure that KB sent() before mS+. So KA KB sent() isn’t 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() doesn’t 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 mS+2.

mS mS+ mS+ 2 mS+ 3

mD mD+ mD+ 2

mS mS+ mS+ 2 mS+ 3

mD mD+ mD+ 2 mD+ 3

A straightforward induction shows that, for any natural number k, before mS+k, (KA KB)k sent() doesn’t hold, while

at mS+k it does.

Common knowledge requires infinitely deep nesting of KA KB. So common knowledge of sent() is never attained no

matter how small .

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+.

Can model the global clock is with another agent. An action by any other agent is always simultaneous

with one of this agent’s 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.

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 surprise—all the agents are involved in the

circularity.

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 there’s 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()

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

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 (n+1)-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.

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.

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.

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).

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.

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.

What’s 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.

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.

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.

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 there’s 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.

Dyadic deontic logic—e.g., O  “Given , it is obligatory that .”

Special obligations, permissions, and prohibitions—e.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 explicit—e.g., A sees to it that operators of dynamic logic.

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.

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 constraints—distinguish between deontic constraints: may be violated necessity constraints: largely analytically true.

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.

AM

B

Figure 1. Physical co-presence

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.

Physical Co-presence Apply the physical co-presence heuristic so that groups

of agents may attain common knowledge perceptually.

Agents must model each others’ perception—requires 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.

Implemented a prototype, coupled with a back propagation neural network.

Agent behavior adapted to trainer’s feedback.

Captured “hidden knowledge” not captured by knowledge engineer.

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 what’s critical: shared situation

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.

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.

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.

Virtual Agents (Clark)

Clark is concerned with how we communicate with virtual partners—e.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 it’s interpreted.

Two main forms: written language mechanized speech (e.g., films, telephone messages)

Disembodied language is a representation of embodied language. We’re 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

Many joint activities divide into layers of joint actions. what is actually happening—a pretense the pretense itself

Two things needed for successful layering Credible characters Props

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.”

Our proxy agents are virtual agents?

But the language (clicking fields) is produced by the player at the moment it’s interpreted.

The 2 layers intermingle.

Equally natural to say the proxy agent moves the player moves the proxy agent

Likewise for obligations.

But common knowledge?

The activities in our game can be automated (some easily)

No way to tell whether we’re communicating with a person or agent behind the proxy agent.

And no way to tell whether there’s one person controlling two proxy agents.

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.

Handshakes—joint actions Process-algebraic agent abstraction

Also need epistemic and deontic effects.

Co-presence heuristics for common knowledge Grounding

Back to coordination game Virtual agents and disembodied language

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.”

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

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 Don’t enforce obligations, but can count violations

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

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