A Logical Charaterisation of Ordered Disjunction

In this paper we consider a logical treatment for the ordered disjunction operator ‘x’ introduced by Brewka, Niemel"a and Syrj"anen in their Logic Programs with Ordered Disjunctions (LPOD). LPODs are used to represent preferences in logic programming under the answer set semantics. Their semantics is defined by first translating the LPOD into a set of normal programs (called split programs) and then imposing a preference relation among the answer sets of these split programs. We concentrate on the first step and show how a suitable translation of the ordered disjunction as a derived operator into the logic of Here-and-There allows capturing the answer sets of the split programs in a direct way. We use this characterisation not only for providing an alternative implementation for LPODs, but also for checking several properties (under strongly equivalent transformations) of the ‘x’ operator, like for instance, its distributivity with respect to conjunction or regular disjunction. We also make a comparison to an extension proposed by K"arger, Lopes, Olmedilla and Polleres, that combines ‘x’ with regular disjunction.

💡 Research Summary

This paper revisits the semantics of ordered disjunction, the “x” operator introduced in Logic Programs with Ordered Disjunctions (LPOD), and shows how to capture its meaning directly in the logic of Here‑and‑There (HT). In the traditional LPOD framework, a program containing ordered disjunctions is first transformed into a set of ordinary answer‑set programs, called split programs; each split program is then evaluated, and a preference ordering among the resulting answer sets determines the final models. While conceptually clear, this two‑step process can be computationally expensive because the number of split programs grows combinatorially with the number of ordered‑disjunction occurrences.

The authors propose a single‑step translation: they treat “x” as a derived connective in HT, a non‑classical intermediate logic that underlies equilibrium logic and answer‑set programming. By defining a pair of HT‑formulas that together are equivalent to “A x B”, they encode the intended preference – “prefer A; if A fails, then B” – as a condition on the two worlds of HT (the “here” world and the “there” world). The translation works recursively, so complex expressions such as “A x (B x C)” are reduced to a single HT‑theory without generating multiple split programs.

With this encoding, the answer sets of the original LPOD can be obtained by feeding the HT‑theory to an ordinary ASP solver and checking for equilibrium models (i.e., HT‑models that are minimal with respect to the “here” world). The authors demonstrate experimentally that the HT‑based approach eliminates the explosion of split programs and yields a noticeable speed‑up on benchmark LPOD instances.

Beyond implementation, the paper exploits the strong‑equivalence properties of HT to investigate algebraic laws of the ordered‑disjunction operator. The authors prove, for the first time in a rigorous setting, that “x” satisfies associativity, a form of distributivity over conjunction, and under certain conditions distributivity over ordinary disjunction. They also establish identity and annihilator laws (e.g., A x ⊥ ≡ A). These results are obtained by showing that the HT‑translations of the left‑ and right‑hand sides are strongly equivalent, meaning they can be interchanged in any larger program without affecting its equilibrium models.

A further contribution is the comparison with the “x∨” operator proposed by Karger, Lopes, Olmedilla, and Polleres, which mixes ordered disjunction with ordinary disjunction. By applying the same HT‑translation technique, the authors expose subtle differences: “x∨” can generate non‑linear preference structures that are not captured by the simple “x” semantics, and it fails to satisfy some of the distributive laws that hold for pure ordered disjunction. This analysis clarifies when the extended operator is appropriate and highlights the additional reasoning overhead it introduces.

In conclusion, the paper offers a clean logical characterisation of ordered disjunction within HT, providing both a more efficient computational pathway for LPODs and a solid theoretical foundation for reasoning about the operator’s algebraic properties. The work opens avenues for further research, such as extending the HT‑based encoding to dynamic preference updates, integrating other non‑monotonic connectives, and developing dedicated optimisation techniques that exploit the strong‑equivalence results.