Discrete Linear Ensemble Logic

Abstract

We study the discrete point-based fragment of Ensemble Logic $\EL(\Nat)$ over the natural numbers, a logic combining displacement ${\phi }_u$, bounded metric modalities $\boldBox_t$ and $\mdiamond_t$ with additive bounds, Boolean connectives, and first-order quantification over $\Nat$. Motivated by the need for a unified symbolic layer for biomedical knowledge with temporal, spatial, genomic, and multimodal metric content, we develop the foundational discrete theory of the formalism. We give syntax and semantics, and prove a forward embedding of $\EL(\Nat)$ over a finite proposition set $\mathcal{P}$ into first-order monadic Presburger arithmetic $\FO(\Nat,<,+;\mathcal{P})$. This embedding yields the analytical upper bounds, while a reduction from nondeterministic two-counter machines with recurring control states proves that satisfiability is ${\Sigma }_1^1$-complete and validity is dually ${\Pi }_1^1$-complete. Expressively, $\EL(\Nat)$ strictly extends the star-free $\omega$-languages and is incomparable with the $\omega$-regular languages: it defines the non-$\omega$-regular counting language $\{a^m{b}^m{c}^m{d}^m\mid m\ge 1\}\cdot {\Sigma }^{\omega }$, whereas a delimited parity language remains outside the logic by classical Presburger-arithmetic lower bounds. On the proof-theoretic side, we present a sound Hilbert system $\HEL$ and establish completeness relative to monadic Presburger validity as oracle, noting that completeness relative to plain Presburger arithmetic is impossible.