Does accessibility mean probability?

No. The relation records which alternatives are considered; probabilities require additional structure.

Can something be true without an agent knowing it?

Yes. The worked example has successful transfer without sufficient information for A to establish success.

Must every agent have the same relation?

No. Separate observations and information can produce different accessible alternatives.

AI Glossary / Graphs and symbolic AI / Intermediate

Accessibility relations

Definition

Accessibility relations are connections between possible worlds or states in a formal model, specifying which alternatives are relevant from a given world. In epistemic logic, which studies knowledge, an agent's accessibility relation represents the situations that the agent cannot rule out on the basis of its information. Different agents can have different relations.

What it means

A possible world is a modeled scenario, not necessarily a claim about another physical universe. A scenario can be as modest as “the parcel arrived, but its confirmation message was lost.” Several scenarios may look identical to an observer who sees only part of the system.

An accessibility relation makes that uncertainty explicit. If world v is accessible from world w for agent A, then v remains an alternative for A when the actual situation is w, under the chosen interpretation.

This entry focuses on knowledge representation. Other modal logics interpret accessibility differently, such as possible transitions. It is unrelated to website usability for people with disabilities.

How it works

Start with a set of modeled worlds and identify which basic statements are true in each. For each agent, specify a relation connecting worlds according to the information available to that agent.

In the standard relational account, the agent knows a statement at a world if that statement holds in every world accessible to the agent from there. If even one accessible alternative makes it false, the model does not support that knowledge claim. The relation identifies possibilities; it does not assign their probabilities.

Properties of the relation encode assumptions. An indistinguishability relation is commonly modeled as reflexive, symmetric, and transitive: a world is indistinguishable from itself, indistinguishability works both ways, and linked indistinguishability carries through. Not every application or modal logic imposes the same properties.

Changes in observations may change the modeled alternatives. Designers must specify the observation and update rules rather than assume that receiving any message produces complete knowledge.

Three practical examples

1. A distributed service has sent a request but has not received an acknowledgment. Its model retains both successful delivery and failed delivery as possible situations until additional information distinguishes them.

2. In a cooperative game, one agent sees a hidden resource while another sees only a public signal. Separate accessibility relations capture their different information.

3. An automated workflow checks whether an approval is known to both participants. Recording that approval exists is different from establishing that each participant has received the relevant information.

Think of it this way

Think of a reviewer comparing several sealed case files that all match the facts currently visible on a dashboard. Until another fact becomes available, the reviewer cannot distinguish which file describes the real situation.

The analogy is limited because the formal relation is defined across every modeled world, not just a pile of alternatives at one moment. It also idealizes reasoning. Real people and programs may fail to derive consequences that the formal model counts as known.

Worked example

Consider three original toy scenarios for a transfer: W1 means the transfer failed and no receipt arrived; W2 means it succeeded but the receipt was lost; W3 means it succeeded and the receipt arrived. Let S mean “the transfer succeeded.” S is false in W1 and true in W2 and W3.

Agent A observes only whether a receipt arrived. From W1 or W2, its accessible alternatives are {W1, W2}. From W3, they are {W3}. Assume receipts are reliable in this toy model.

At W2, success is actually true, but A does not know S: accessible world W1 makes S false. A also does not know that the transfer failed, because W2 remains possible. At W3, A knows S because its only accessible world satisfies S.

Now let agent B observe the transfer outcome directly. B can know S at W2 while A cannot. The example separates truth from information without attaching numerical probabilities to either alternative.

Illustrative case study

Fictional case: Cedar Dispatch develops a coordination assistant for two warehouse teams. A status screen shows that a handover was completed, and the first design assumes both teams know it is complete.

Reviewers add a scenario in which completion occurred but one team's notification was delayed. The revised model distinguishes operational completion from shared awareness. The assistant requests acknowledgment before starting a dependent joint task. This invented case illustrates an information-modeling issue; it does not claim that a formal model eliminates communication failures.

Common misconceptions and mistakes

An accessible world is not necessarily likely, reachable by a physical journey, or actually true. Its meaning depends on the model's interpretation.

Failure to know a statement is not the same as knowing its opposite. Both a statement and its negation may remain possible.

Relations for different agents need not match, and information possessed by one participant is not automatically shared knowledge. Similarly, a system should not treat an idealized logic of knowledge as a psychological description of human thought. Its conclusions remain conditional on the worlds, observations, and assumptions included in the model.

Business use

Accessibility relations provide a precise language for reviewing coordination requirements, information flow, and knowledge-dependent actions. They help teams ask what a participant can establish from its own information rather than from an omniscient system designer's view.

A practical review should look for omitted scenarios: delayed notifications, misleading observations, partial failures, or participants with different records. The objective is not to enumerate every imaginable world, but to justify the alternatives relevant to the claimed guarantee. An excluded failure scenario cannot be addressed by reasoning over the remaining model alone.

Check your understanding

A statement is true in two accessible worlds and false in a third. Does the agent know it under the standard all-accessible-worlds rule?

Answer: No. One accessible counterexample is enough to defeat that knowledge claim, even if the statement happens to be true in the actual world.

Related terms

Possible world
Modal logic
Knowledge representation
Multiagent systems
Epistemic logic
belief
common knowledge
multiagent system

Learn next

Next, study the distinction between individual knowledge and common knowledge. Draw a small model with two agents, label their observation-based connections separately, and test simple knowledge claims.

Sources

Eric Pacuit, Dynamic Epistemic Logic I: Modeling Knowledge and Belief, author manuscript, especially the epistemic-model definitions and interpretation discussion: https://ai.stanford.edu/~epacuit/esslli2012/handouts-epgth/del-survey-part1-v2.pdf

Rasmus Rendsvig and John Symons, Epistemic Logic, author reference article: https://plato.stanford.edu/entries/logic-epistemic/

Core relational semantics were checked. Scenarios are original; the entry uses the epistemic sense rather than claiming one interpretation covers every modal logic.

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