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## Common Knowledge and Handshakes in Computer-Mediated Cooperation

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**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 in. • 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.**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 • 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 .**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.**bs: 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 … sss**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.**KBsent() is true at mD.**• But A can’t be sure that KBsent() before mS+. • So KA KBsent() 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 KBsent() 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 KBsent() 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)ksent() 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() EGagree() (**) • By idempotence of , this is equivalent to agreeG() EG (agreeG() agreeG()) • But standard epistemic logic includes the inference rule From 1 EG (21) infer CG2 • Substituting agreeG() for both 1 and 2 in the rule and using (**) for the premise, we infer agreeG() CGagree()**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.