Reformulating the Situation Calculus and the Event Calculus in the General Theory of Stable Models and in Answer Set Programming

Circumscription and logic programs under the stable model semantics are two well-known nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical formulas, we reformulate the situation calculus and the event calculus in the general theory of stable models. We also present a translation that turns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus.

💡 Research Summary

The paper establishes a deep connection between two prominent non‑monotonic formalisms—circumscription and the stable‑model semantics of logic programs—and exploits this connection to reformulate two classic action formalisms, the Situation Calculus and the Event Calculus, within the general theory of stable models. The authors first identify a class of “canonical formulas,” essentially quantifier‑free first‑order sentences and Horn‑like definition rules, and prove that for any formula in this class the minimal model produced by circumscription coincides exactly with the stable models defined by the SM operator. This theoretical result provides a solid logical foundation for translating circumscription‑based theories into answer‑set programs without loss of meaning.

Building on this foundation, the paper shows how the axioms of the Situation Calculus—particularly the inertia axiom, action precondition axioms, and successor‑state axioms—can be rewritten as canonical formulas. By applying the canonical‑to‑stable‑model translation, each axiom becomes a set of ASP rules: inertia is encoded via choice rules and integrity constraints that enforce minimal change, action effects are expressed as defeasible rules, and the uniqueness of situations is captured by functional‑style constraints. The resulting ASP encoding preserves the original semantics while making the frame problem amenable to the efficient grounding and solving techniques of modern ASP solvers.

A parallel treatment is given to the Event Calculus. The authors decompose the Event Calculus’s core components—event occurrence, time points, and fluents—into time‑indexed rules. The usual global minimization of event occurrences (the “closed‑world” assumption about events) is replaced by local minimization constraints that are naturally expressed in ASP as negation‑as‑failure and integrity constraints. Initiates and terminates predicates become conditional rules, while the persistence of fluents over intervals is captured by recursive rules that mirror the original axioms but are now grounded in the stable‑model semantics.

The central technical contribution is the SM‑to‑ASP translation algorithm. It proceeds in three stages: (1) normalize circumscribed axioms into a rule‑like form; (2) replace any negative circumscription with ASP’s negation‑as‑failure; (3) encode minimization requirements using choice rules, cardinality constraints, and integrity constraints. The authors prove that the translation is sound and complete with respect to the original circumscribed theory, and that the size of the generated ASP program grows only linearly with the size of the input theory.

Finally, the paper discusses implementation considerations. By feeding the translated ASP program into state‑of‑the‑art solvers such as Clingo or DLV, one can compute models of the Situation Calculus or Event Calculus far more efficiently than with traditional circumscription‑based theorem provers. Although the paper does not present extensive empirical benchmarks, the theoretical analysis suggests substantial performance gains, especially for large dynamic domains like robotic planning or complex event processing. In conclusion, the work bridges the gap between classical logic‑based action formalisms and modern answer‑set programming, offering a principled, scalable pathway to apply high‑performance ASP solvers to reasoning tasks traditionally handled by circumscription.