Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curved Boolean Logic: A Contextual Generalization of Propositional Logic with Algorithmic Consequences

Maximilian R. P. von Liechtenstein

TL;DR

Curved Boolean Logic (CBL) generalizes propositional logic by allowing local truth assignments that do not always extend to a global valuation, with curvature $ ext{κ}$ quantifying the obstruction. The work develops equivalent semantic frameworks (sheaf and exclusivity-graph), a context-aware proof calculus, and formalizes CBL-SAT (NP-complete) alongside curvature-aware solvers (CBL-AC, CBL-CONS). It introduces robustness via $oldsymbol{ ext{epsilon}}$-stability models, analyzes noise (i.i.d., AR(1), adversarial) and provides a Colab-ready reproducibility notebook. Canonical curved cores (KCBS, Mermin square) illustrate how curvature enables early pruning and certificates, supported by a solver prototype and two drop-in operators for SAT/CSP and 3-colorability. The paper connects CBL to large language models, clustering contextual stability with compression and adapter stability, and outlines open problems, empirical validation plans, and broader implications for AI, verification, and complexity theory.

Abstract

Curved Boolean Logic (CBL) generalizes propositional logic by allowing local truth assignments that do not extend to a single global valuation, analogous to curvature in geometry. We give equivalent sheaf and exclusivity-graph semantics and a context-aware proof calculus that is conservative in the flat limit. We formalize CBL-SAT and basic complexity (NP-complete in general) and present operational operators (CBL-AC and CBL-CONS) that prune contradictions earlier on classical hardware. We model noise with iid, AR(1)-correlated, and adversarial bounded perturbations and provide permutation-based significance with Benjamini-Hochberg FDR control. A Colab-ready notebook (ancillary files) regenerates all figures and statistics. We position CBL relative to KCBS, CSW, and sheaf frameworks and outline links to SAT/CSP and robustness/adapter stability in large language models.

Curved Boolean Logic: A Contextual Generalization of Propositional Logic with Algorithmic Consequences

TL;DR

Curved Boolean Logic (CBL) generalizes propositional logic by allowing local truth assignments that do not always extend to a global valuation, with curvature quantifying the obstruction. The work develops equivalent semantic frameworks (sheaf and exclusivity-graph), a context-aware proof calculus, and formalizes CBL-SAT (NP-complete) alongside curvature-aware solvers (CBL-AC, CBL-CONS). It introduces robustness via -stability models, analyzes noise (i.i.d., AR(1), adversarial) and provides a Colab-ready reproducibility notebook. Canonical curved cores (KCBS, Mermin square) illustrate how curvature enables early pruning and certificates, supported by a solver prototype and two drop-in operators for SAT/CSP and 3-colorability. The paper connects CBL to large language models, clustering contextual stability with compression and adapter stability, and outlines open problems, empirical validation plans, and broader implications for AI, verification, and complexity theory.

Abstract

Curved Boolean Logic (CBL) generalizes propositional logic by allowing local truth assignments that do not extend to a single global valuation, analogous to curvature in geometry. We give equivalent sheaf and exclusivity-graph semantics and a context-aware proof calculus that is conservative in the flat limit. We formalize CBL-SAT and basic complexity (NP-complete in general) and present operational operators (CBL-AC and CBL-CONS) that prune contradictions earlier on classical hardware. We model noise with iid, AR(1)-correlated, and adversarial bounded perturbations and provide permutation-based significance with Benjamini-Hochberg FDR control. A Colab-ready notebook (ancillary files) regenerates all figures and statistics. We position CBL relative to KCBS, CSW, and sheaf frameworks and outline links to SAT/CSP and robustness/adapter stability in large language models.

Paper Structure

This paper contains 180 sections, 17 theorems, 21 equations, 11 figures, 14 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Reader guide.
  3. Historical context.
  4. Outlook.
  5. Referee-preemption.
  6. Background and Related Work
  7. Position within Existing Frameworks
  8. Goal.
  9. Robustness and $\varepsilon$-stability.
  10. Temporal and causal extensions.
  11. Complexity and algorithmic parallels.
  12. Provenance and auditability.
  13. Assumptions and Dependency Matrix
  14. Goal.
  15. Dependency matrix.
  16. ...and 165 more sections

Key Result

Theorem 1

For finite context systems, sheaf semantics and exclusivity-hypergraph semantics are equivalent: a compatible family admits a global section iff the exclusivity formulation admits an assignment consistent on overlaps.

Figures (11)

  • Figure 1: Timeline: from foundational results to Curved Boolean Logic.
  • Figure 2: Context lattice for two overlapping contexts. Each arrow denotes set inclusion.
  • Figure 3: Flat vs. curved proof trees. In flat cases, a global valuation ensures consistency; in curved cases, local branches exist but no global valuation exists, producing a structural contradiction.
  • Figure 4: Synthetic placeholder: runtime vs curvature rank $\kappa$.
  • Figure 5: Synthetic runtime scaling. Placeholder; to be replaced with data.
  • ...and 6 more figures

Theorems & Definitions (42)

  • Definition 1: Axiom A1: Measurement independence
  • Definition 2: Axiom A2: $\varepsilon$-bounded perturbations
  • Definition 3: Axiom A3: Mixing for folds
  • Definition 4: Axiom A4: Context-preserving instrumentation
  • Definition 5: Axiom A5: Finite-moment regularity
  • Definition 6: Local and global valuations
  • Definition 7: Curvature functional
  • Theorem 1: Equivalence of semantics
  • proof : Sketch
  • Definition 8: Curved core
  • ...and 32 more