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Computational Complexity of Probabilistic Reasoning for Neurosymbolic Classification Techniques

Core Concepts
The asymptotic complexity of probabilistic reasoning is crucial to assess the scalability of probabilistic neurosymbolic techniques for large-scale classification tasks, but this topic is rarely tackled in the neurosymbolic literature.
The paper introduces a formalism for informed supervised classification tasks and techniques, and builds upon this formalism to define three abstract neurosymbolic techniques based on probabilistic reasoning: semantic conditioning, semantic regularization, and semantic conditioning at inference. The paper then examines the asymptotic computational complexity of several classes of probabilistic reasoning problems that are often encountered in the neurosymbolic literature. It shows that probabilistic techniques can not scale on certain popular tasks found in the literature, whereas others thought intractable can actually be computed efficiently. Specifically, the paper analyzes the complexity of probabilistic reasoning for the following logics: Hierarchical logic: The fragment of hierarchical logic composed of theories where the graph is a tree is tractable. Hierarchical logic with mutual exclusion assumptions is also tractable. Fixed cardinal constraints: The paper provides a polynomial-time construction to compile fixed cardinal constraints into a concise Sentential Decision Diagram (SDD), making them tractable. Simple path logic: Simple path logic is intractable in general, but the fragment of acyclic simple path theories is tractable. Matching logic: Matching logic is semi-tractable, meaning it is MPE-tractable but not PQE-tractable. The paper concludes by discussing possible future research directions, including a sharper understanding of semi-tractable logics and their practical consequences, exploring approximate methods for intractable prior knowledge, and expanding the formalism beyond supervised classification.

Deeper Inquiries

What are the practical implications of the identified tractable and semi-tractable logics for real-world informed classification tasks

The identification of tractable and semi-tractable logics in the context of informed classification tasks has significant practical implications for real-world applications. Tractable logics, such as the fragment of hierarchical logic composed of tree theories, offer a scalable and efficient way to integrate prior knowledge into neural classification systems. This means that for tasks where hierarchical constraints are prevalent, such as organizing concepts in taxonomies or hierarchical classification tasks, the use of these logics can lead to improved performance and scalability. By understanding which logics are tractable, developers and researchers can choose the most suitable approach for incorporating prior knowledge into their classification systems. Semi-tractable logics, on the other hand, provide a middle ground where certain probabilistic reasoning problems can be solved efficiently while others remain computationally challenging. For instance, matching logic, which is semi-tractable, allows for the efficient solution of MPE problems but retains the complexity of PQE problems. In real-world scenarios where matching problems are common, such as pairing entities or resources optimally, leveraging semi-tractable logics can enable the development of effective probabilistic neurosymbolic techniques. Understanding the boundaries of tractability and semi-tractability in logics helps in selecting the most appropriate techniques for specific informed classification tasks, balancing computational efficiency with accuracy.

How can approximate methods be developed for probabilistic neurosymbolic techniques when the prior knowledge is intractable

When prior knowledge expressed in certain logics is intractable for probabilistic reasoning problems, approximate methods can be developed to address the computational challenges. One approach is to use approximation algorithms or heuristics to find near-optimal solutions to MPE and PQE problems in these logics. By sacrificing optimality for computational efficiency, approximate methods can provide practical solutions for real-world applications where exact solutions are infeasible due to the complexity of the underlying logic. For instance, in cases where matching logic poses challenges for exact probabilistic reasoning, approximate methods can be employed to find good matching solutions that are close to the optimal solution. These methods may involve sampling techniques, approximation algorithms, or iterative optimization approaches to navigate the complexity of intractable logics. By developing and implementing such approximate methods, practitioners can still leverage the benefits of probabilistic neurosymbolic techniques even in the presence of intractable prior knowledge.

How can the formalism introduced in this paper be extended to semi-supervised and weakly-supervised settings

The formalism introduced in the paper can be extended to semi-supervised and weakly-supervised settings by incorporating additional layers of abstraction and complexity. In semi-supervised learning, where only a subset of the data is labeled, the formalism can be adapted to include probabilistic reasoning over both labeled and unlabeled instances. This extension would involve integrating prior knowledge not only in the supervised classification tasks but also in the learning process for the unlabeled data, enhancing the overall model performance. In weakly-supervised settings, where only partial or noisy labels are available, the formalism can be further extended to handle the uncertainty and ambiguity in the labeling process. By incorporating probabilistic reasoning techniques that account for the reliability and quality of weak labels, the formalism can provide a more robust framework for informed classification tasks. Additionally, the formalism can be adapted to incorporate feedback mechanisms and iterative learning processes to improve model performance over time in weakly-supervised scenarios. By extending the formalism to semi-supervised and weakly-supervised settings, practitioners can leverage the power of probabilistic neurosymbolic techniques in a broader range of learning scenarios.