Uncertainty

Uncertainty is multiplicity in the states, histories, causes, outcomes, or interpretations that an observer cannot yet distinguish at a declared boundary.

An observer's information state can be represented as a set of possible worlds or system states:

knowledge(observer) subset-of PossibleStates

An observation refines uncertainty when it rules out possibilities. It need not reveal one complete or perfectly current state. Stale, partial, probabilistic, conflicting, or differently authorized observations can leave several possibilities open.

Forms of Uncertainty

State uncertainty: the current state or version is not known precisely.
Outcome uncertainty: several future results remain possible.
Causal uncertainty: the dependencies or explanation for an occurrence are not known.
Measurement uncertainty: an observation is noisy, approximate, sampled, or interval-valued.
Timing and failure uncertainty: delay cannot be distinguished from pause, loss, partition, overload, or crash within the available evidence.
Model uncertainty: the transition rules, environment, probability model, or relevant boundary are themselves incomplete or disputed.
Authority uncertainty: an observer does not know which source, lease, term, quorum certificate, or decision currently has authority.

Probability, possibility sets, intervals, belief states, evidence lattices, and provenance graphs preserve different information. A probability distribution can encode belief about hidden state, stochastic behavior in the system, or both. Its interpretation must therefore be stated rather than inferred from the representation alone.

Relationship to Nondeterminism

Nondeterminism is multiplicity in the continuation relation: one state admits several possible next outcomes. Uncertainty is multiplicity in an observer's information: several states, histories, or outcomes remain compatible with what has been observed.

For state space S:

nondeterministic transition : S -> Set<S>
observer information        : Set<S>

The two interact without being identical:

An unobserved nondeterministic transition can enlarge uncertainty.
A deterministic system can still be uncertain to an observer because relevant state or input is hidden.
A later observation can reduce uncertainty without changing the system's transition relation.
A scheduler or policy can resolve a choice while an observer remains uncertain about which branch was selected.
Several internal paths can remain uncertain but irrelevant when they are confluent or observationally equivalent at the boundary.

Distributed Decisions

Distributed systems make uncertainty operationally important. A participant often cannot determine whether a remote action failed, is delayed, committed but unacknowledged, or will later be retried. Consistency models, versions, causal metadata, acknowledgments, leases, and consensus certificates constrain which conclusions are justified; they do not remove all uncertainty at every boundary.

Uncertainty should not be converted directly into rejection, retry, or acceptance. A policy determines how evidence and risk affect a decision. Authority determines whose assertion can settle a question. Safety and liveness determine whether waiting, proceeding, compensating, or exposing tentative state is permitted.

Modeling Checks

Which observer is uncertain about which subject or occurrence?
Which possibilities remain compatible with its observations?
Is the uncertainty epistemic, stochastic, causal, temporal, measurement-based, or structural?
Which observation, version, evidence, or authority could refine it?
Must the system decide before the uncertainty is resolved?
Which mistakes are safe, reversible, compensatable, or forbidden?
Does a probability describe system behavior or the observer's belief?
Which differences are intentionally hidden by equivalence or confluence?

External References

Joseph Y. Halpern, Reasoning about Uncertainty, second edition, MIT Press, 2017.