Discrete Linear Ensemble Logic

2026-08-11

AI Summary

This work addresses the lack of a unified symbolic logical framework for integrating temporal, spatial, genomic, and multimodal information in biomedical knowledge by proposing an enriched logic over the natural numbers, denoted EL(Nat). EL(Nat) combines shift operators, bounded metric modalities, Boolean connectives, and first-order quantifiers. It strictly extends star-free ω-languages, can define non-ω-regular counting languages, and is incomparable with ω-regular languages; its satisfiability problem is Σ¹₁-complete. By embedding first-order Presburger arithmetic, constructing a Hilbert-style axiom system HEL, and reducing to nondeterministic two-counter automata, the authors establish a forward translation from EL(Nat) into FO(Nat,<,+;P), thereby deriving upper bounds for satisfiability and validity and yielding a relatively complete proof system.

Abstract

We study the discrete point-based fragment of Ensemble Logic $\EL(\Nat)$ over the natural numbers, a logic combining displacement $\varphi_u$, 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 $Σ^1_1$-complete and validity is dually $Π^1_1$-complete. Expressively, $\EL(\Nat)$ strictly extends the star-free $ω$-languages and is incomparable with the $ω$-regular languages: it defines the non-$ω$-regular counting language $\{a^mb^mc^md^m\mid m\geq 1\}\cdotΣ^ω$, 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.

Problem

• Ensemble Logic
• biomedical knowledge representation
• discrete temporal logic
• metric modalities
• Presburger arithmetic

Innovation

• Ensemble Logic
• Monadic Presburger Arithmetic
• Σ¹₁-completeness
• ω-languages
• Hilbert system