Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)

We produce a decidable classical normal modal logic of internalised negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP) from an existing logical counterpart of non-monotonic or instant interactive proofs (LiiP). LDiiP internalises agent-centric proof theories that are negation-complete (maximal) and consistent (and hence strictly weaker than, for example, Peano Arithmetic) and enjoy the disjunction property (like Intuitionistic Logic). In other words, internalised proof theories are ultrafilters and all internalised proof goals are definite in the sense of being either provable or disprovable to an agent by means of disjunctive internalised proofs (thus also called epistemic deciders). Still, LDiiP itself is classical (monotonic, non-constructive), negation-incomplete, and does not have the disjunction property. The price to pay for the negation completeness of our interactive proofs is their non-monotonicity and non-communality (for singleton agent communities only). As a normal modal logic, LDiiP enjoys a standard Kripke-semantics, which we justify by invoking the Axiom of Choice on LiiP’s and then construct in terms of a concrete oracle-computable function. LDiiP’s agent-centric internalised notion of proof can also be viewed as a negation-complete disjunctive explicit refinement of standard KD45-belief, and yields a disjunctive but negation-incomplete explicit refinement of S4-provability.

💡 Research Summary

The paper introduces a new normal modal logic, LDiiP (Logic of Negation‑Complete Interactive Proofs), built on top of the previously established LiiP (Logic of instant interactive proofs). While LiiP already internalises a proof modality “M ⊢ₐ φ” that captures agent‑centric proof relations, it lacks negation‑completeness: a given formula φ may be neither provable nor refutable within the system. LDiiP remedies this by enforcing that for every formula φ, either φ or its negation ¬φ is provable for a particular agent a using a specific proof term M. In other words, each agent’s internal proof theory becomes an ultrafilter over the language, guaranteeing a decisive outcome for any proof goal.

The construction proceeds by first recalling the Kripke‑style semantics of LiiP, which consists of worlds, an accessibility relation indexed by agents and proof terms, and a valuation for atomic propositions. To obtain negation‑completeness the authors invoke the Axiom of Choice, defining a global choice function f that, for any world w, agent a, and proof term M, selects a unique successor world w′∈Rₐᴹ(w). This function is shown to be oracle‑computable, providing a concrete algorithmic handle on the otherwise abstract accessibility relation. The resulting frame satisfies the standard modal axioms of normality (K) together with additional constraints that enforce the ultrafilter property.

The ultrafilter condition is the technical heart of LDiiP. For each agent a, the set of formulas that are provable by some proof term M forms a maximal consistent set closed under supersets, conjunction, and disjunction, and crucially, for every φ either φ∈Uₐ or ¬φ∈Uₐ (but not both). This yields the “disjunctive” nature of the internalised proofs: a proof of a disjunction φ∨ψ can be reduced to a proof of one of its disjuncts, mirroring the disjunction property of intuitionistic logic, yet the surrounding meta‑logic remains classical. Consequently, LDiiP can be viewed as a non‑monotonic, agent‑local refinement of standard epistemic modalities.

Two important trade‑offs arise. First, the proof relation becomes non‑monotonic: adding new evidence may invalidate previously held proofs because the ultrafilter must be re‑adjusted to preserve maximality. Second, LDiiP is non‑communally scoped: a proof that is available to agent a does not automatically become available to another agent b. This “singleton‑community” restriction reflects the fact that ultrafilters are defined per agent and cannot be merged without losing negation‑completeness.

The authors further explore the epistemic interpretation of LDiiP. By mapping the KD45 belief operator Bₐφ to the explicit modality M⊢ₐφ, they obtain a disjunctive, negation‑complete explicit belief logic. In this setting, an agent either explicitly believes φ or explicitly believes its negation, making the belief operator a true decider. Similarly, the S4 provability operator □ₐφ can be refined to an explicit version where □ₐφ holds exactly when there exists a proof term M such that M⊢ₐφ, but the overall system remains negation‑incomplete, preserving the classical nature of S4 while adding a disjunctive layer.

The paper proves that LDiiP is both sound and complete with respect to the constructed Kripke frames, and that it is the weakest consistent theory that enjoys internalised negation‑completeness. It is strictly weaker than Peano Arithmetic, yet strong enough to decide every formula for each agent individually. The authors argue that this makes LDiiP a suitable foundation for “epistemic deciders”: computational agents that must make definitive yes/no decisions based on interactive proofs, such as in security protocol verification, automated theorem proving, or multi‑agent decision‑making systems.

In summary, LDiiP offers a novel blend of classical modal logic, ultrafilter‑based proof semantics, and explicit epistemic operators. It achieves agent‑local negation‑completeness and the disjunction property at the cost of non‑monotonicity and lack of communal proof sharing. The work opens avenues for further research on how such decider‑oriented logics can be integrated into practical systems while managing the inherent trade‑offs.