Event Calculus

Origin. Kowalski and Sergot introduced Event Calculus (1986). Formalism for reasoning about events and their effects. Alternative to Situation Calculus. Based on time points and fluents. Better handles continuous time and narrative.

Models. Events, times, and fluents. Events: instantaneous occurrences. Fluents: properties that change over time. Events initiate and terminate fluents. Frame axioms implicit via persistence. Time can be discrete or continuous.

Formalism.

Basic predicates:

Happens(e, t): event e occurs at time t
Initiates(e, f, t): event e initiates fluent f at t
Terminates(e, f, t): event e terminates fluent f at t
HoldsAt(f, t): fluent f holds at time t
Clipped(t₁, f, t₂): f terminated between t₁ and t₂

Persistence axiom: HoldsAt(f, t₂) ← Happens(e, t₁) ∧ t₁ < t₂ ∧ Initiates(e, f, t₁) ∧ ¬Clipped(t₁, f, t₂)

Clipped(t₁, f, t₂) ↔ ∃e,t. Happens(e, t) ∧ t₁ < t < t₂ ∧ Terminates(e, f, t)

Circumscription: Minimize Happens, Initiates, Terminates. Commonsense: only stated events occur.

Ramifications: State constraints: ¬(HoldsAt(f, t) ∧ HoldsAt(g, t)) Trigger indirect effects.

Example: Initiates(TurnOn(l), Lit(l), t) Terminates(TurnOff(l), Lit(l), t) Happens(TurnOn(lamp1), 5) ⊢ HoldsAt(Lit(lamp1), 10)

Continuous change: Trajectory(f, t₁, f', d): f changes to f' over duration d. Models gradual processes.

Symbols.

Symbol	Unicode	Name	Meaning
Happens	—	Occurrence	Event happens
HoldsAt	—	Truth at time	Fluent holds
Initiates	—	Start	Event starts fluent
Terminates	—	End	Event ends fluent
Clipped	—	Interrupted	Fluent broken
<	—	Before	Temporal order
f, e, t	—	Fluent, event, time	Basic sorts

Metatheory. Circumscription-based inference. Complexity depends on theory structure. Abductive reasoning supported. Multiple extensions (concurrent events, continuous change). Implementation in logic programming (Prolog, ASP). Sound and complete for various fragments.

Applies to. Narrative understanding. Planning and scheduling. Legal reasoning (contract execution). Databases (temporal queries). Robotics (action effects). Biomedical processes. Business process modeling.

Limitations. Circumscription inference costly. Concurrent events complex. Continuous change approximated. Frame problem addressed but not eliminated. Less efficient than specialized planners. Integration with probabilistic reasoning developing.