Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: SatisfiabilitySearch and Optimization: Non-convex Optimization
Complexity of Contextuality
🤖 AI Summary
This paper systematically investigates three fundamental computational problems concerning generalized contextuality: (1) deciding whether a given theory admits a noncontextual ontological model; (2) for a fixed dimension $k$, determining whether a $k$-dimensional noncontextual model exists; and (3) efficiently constructing a minimal-size such model, if one exists. Methodologically, it introduces the first reduction of dimension-constrained noncontextuality decision to the intermediate simplex problem in computational geometry. This yields a tight complexity characterization: the problem is $exp(Omega(k))$-hard (lower bound) and solvable in $exp(O(k))$ time (upper bound). Furthermore, the work reveals an intrinsic gap between minimal noncontextual models and general minimal ontological models—e.g., their sizes differ strictly (5 vs. 4 in a canonical counterexample)—demonstrating that dimensional constraints impose an inherent modeling cost.
Technology Category
Application Category
📝 Abstract
Generalized contextuality is a hallmark of nonclassical theories like quantum mechanics. Yet, three fundamental computational problems concerning its decidability and complexity remain open. First, determining the complexity of deciding if a theory admits a noncontextual ontological model; Second, determining the complexity of deciding if such a model is possible for a specific dimension $k$; Third, efficiently computing the smallest such model when it exists, given that finding the smallest ontological model is NP-hard. We address the second problem by presenting an algorithm derived from a geometric formulation and its reduction to the intermediate simplex problem in computational geometry. We find that the complexity of deciding the existence of a noncontextual ontological model of dimension $k$ is at least exponential in the dimension of the theory and at most exponential in $k$. This, in turn, implies that computing the smallest noncontextual ontological model is inefficient in general. Finally, we demonstrate the fundamental difference between finding the smallest noncontextual ontological model and the smallest ontological model using an explicit example wherein the respective minimum ontic sizes are five and four.
Problem
Research questions and friction points this paper is trying to address.
Determine complexity of deciding noncontextual ontological model existence
Assess complexity of dimension-specific noncontextual model decidability
Compute smallest noncontextual model efficiently despite NP-hardness
Innovation
Methods, ideas, or system contributions that make the work stand out.
Geometric formulation reduces to simplex problem
Exponential complexity in theory dimension
Demonstrates difference in minimum ontic sizes
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