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Existence of gadget payoff formulas and definability

Determine necessary and sufficient conditions under which, for two valued promise templates (A,B) and (A′,B′) with PolFeas(A,B)=PolFeas(A′,B′), there exist payoff formulas Ψ (possibly with auxiliary variables) that realize the gadget reduction of Proposition 6.1; equivalently, characterize such gadget reductions via definability in terms of polymorphisms or plurimorphisms.

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Background

Proposition 6.1 shows that certain gadget constructions imply valued minion homomorphisms, but the general existence of gadgets is deferred. Understanding when gadgets exist is tightly connected to definability issues (expressing constraints over one template using constraints over another) in the valued setting.

A characterization would unify reduction techniques and potentially broaden the applicability of the algebraic framework to more complex approximation reductions.

References

The question when payoff \Sigma-formulas \Psi exist (and, more generally, understanding of more complex gadget reductions) is closely related to questions about definability — a direction we leave for future work.

— The Rise of Plurimorphisms: Algebraic Approach to Approximation (2401.15186 - Barto et al., 26 Jan 2024) in Section 6.1 (Gadget reductions), after Proposition 6.1