Every Formula-Based Logic Program Has a Least Infinite-Valued Model

Every definite logic program has as its meaning a least Herbrand model with respect to the program-independent ordering “set-inclusion”. In the case of normal logic programs there do not exist least models in general. However, according to a recent approach by Rondogiannis and Wadge, who consider infinite-valued models, every normal logic program does have a least model with respect to a program-independent ordering. We show that this approach can be extended to formula-based logic programs (i.e., finite sets of rules of the form A\leftarrowF where A is an atom and F an arbitrary first-order formula). We construct for a given program P an interpretation M_P and show that it is the least of all models of P. Keywords: Logic programming, semantics of programs, negation-as-failure, infinite-valued logics, set theory

💡 Research Summary

The paper extends the recent infinite‑valued semantics for normal logic programs, originally introduced by Rondogiannis and Wadge, to a much broader class called formula‑based logic programs. A formula‑based program consists of a finite set of rules of the form A ← φ where A is an atom and φ is an arbitrary first‑order formula (not limited to conjunctions of literals). The authors first recall that definite programs have a unique least Herbrand model under set‑inclusion, while normal programs generally lack a least model in the classical two‑valued setting. The infinite‑valued approach solves this by introducing a linearly ordered set of truth values

W = {F₀ < F₁ < … < F_α < … < 0 < … < T_α < … < T₁ < T₀}

where each F_α (resp. T_α) represents a “degree‑α false” (resp. “degree‑α true”) and 0 denotes undefined. This ordering generalises the classical ⊆ ordering on Herbrand interpretations.

The paper defines an interpretation I as a mapping from the Herbrand base HB to W, together with variable assignments h. The semantics of formulas is given by a recursive evaluation that uses min/max for ∧/∨, sup/inf for ∃/∀, and a special negation rule that maps F_α to T_{α+1}, T_α to F_{α+1}, and leaves 0 unchanged. A rule A ← φ is satisfied by I iff for every assignment h, the truth value of A is at least that of φ. A model of a program P satisfies all its ground instances.

The central technical contribution is the construction of a least model M_P for any formula‑based program P. The authors introduce the immediate consequence operator T_P, which maps an interpretation I to a new interpretation J where for each atom A, J(A) is the supremum of the truth values of all bodies φ of ground instances A ← φ in P. They prove that T_P is α‑monotonic for every countable ordinal α: if I ⊑_α J then T_P(I) ⊑_α T_P(J). This relies on two “extension theorems”. The first shows that if I ⊑_α J then the evaluation of any formula φ agrees on all truth values whose degree is ≤ α, and on values of degree < α the evaluations are identical. The second theorem handles infimum and supremum of families of formulas, ensuring that the monotonicity carries over to complex formulas built with ∧, ∨, ∃, ∀ and ¬.

Because formula bodies may refer to countably infinite sets of ground atoms, ω‑many iterations of T_P (as sufficient for normal programs) are not enough. The authors exploit the regularity of the first uncountable cardinal ℵ₁: any countable increasing sequence of ordinals has its limit in ℵ₁. They therefore iterate T_P through all countable ordinals up to ℵ₁, defining a transfinite sequence I_0, I_1, …, I_α, …, I_ℵ₁ where I_{α+1}=T_P(I_α) and at limit stages I_λ = sup_{β<λ} I_β. By regularity, this process stabilises at some countable ordinal δ_max < ℵ₁, yielding a fixed point M_P = I_{δ_max}. The authors prove that M_P satisfies all rules of P and that for any other model J of P we have M_P ⊑_∞ J; thus M_P is the least infinite‑valued model.

Finally, the paper shows that collapsing the infinite hierarchy of truth values to a three‑valued interpretation (mapping all T_α to True, all F_α to False, and keeping 0 as Undefined) yields a 3‑valued model that coincides with the well‑founded model of the program, as defined by Van Gelder, Ross, and Schlipf. However, unlike the normal‑program case, this collapsed model is not necessarily minimal among all 3‑valued models; a simple syntactic restriction on the program (e.g., bodies that are “regular” in a specific sense) guarantees minimality.

In summary, the paper provides a rigorous extension of infinite‑valued semantics from normal to arbitrary formula‑based logic programs, establishes the existence of a unique least model via transfinite iteration up to ℵ₁, connects the construction to well‑founded semantics, and discusses conditions under which the three‑valued collapse remains minimal. This work deepens the theoretical foundations of logic programming by showing that a purely model‑theoretic, program‑independent ordering can be used to assign a canonical meaning to a far broader class of logic programs than previously possible.