We study the problem of learning linear temporal logic (LTL) formulas from
examples, as a first step towards expressing a property separating positive and
negative instances in a way that is comprehensible for humans. In this paper we
initiate the study of the computational complexity of the problem. Our main
results are hardness results: we show that the LTL learning problem is
NP-complete, both for the full logic and for almost all of its fragments. This
motivates the search for efficient heuristics, and highlights the complexity of
expressing separating properties in concise natural language.

The study of learning linear temporal logic (LTL) formulas from examples is an important step for humans to be able to comprehend and express properties that separate positive and negative instances. This interdisciplinary concept involves elements of computer science, logic, and linguistics. In this paper, the focus is on understanding the computational complexity of the problem.

One of the key findings of this research is that the LTL learning problem is NP-complete. This means that finding an efficient solution to the problem is challenging, as it falls into the category of computationally difficult problems that often require exponential time to solve. This finding highlights the need for developing efficient heuristics to tackle this complexity.

The study also considers the computational complexity of various fragments of LTL. The results show that the problem remains NP-complete for almost all of its fragments. This implies that even when the problem is simplified by restricting the complexity of the logic, it still remains computationally challenging.

The Multi-disciplinary Nature of the Problem

The problem of learning LTL formulas from examples requires a multi-disciplinary approach. Firstly, it requires a deep understanding of computer science, particularly in the field of computational complexity theory. The NP-completeness result signifies that the problem may not have a polynomial-time algorithm, which adds significance to the research in developing heuristics.

Secondly, the problem delves into the realm of logic, specifically temporal logic. Temporal logic allows for reasoning about events and time, and it plays a crucial role in modeling and specifying properties of reactive systems. Understanding temporal logic concepts and their application is essential for comprehending and solving the LTL learning problem.

Finally, the study brings attention to the complexity of expressing separating properties in concise natural language. This aspect incorporates insights from linguistics and human cognition. Finding an expressive yet understandable way to represent these properties adds another layer of complexity to the overall problem.

Future Directions

Given the computational complexity highlighted in this research, future directions in the field could focus on developing efficient heuristics and approximation algorithms to solve the LTL learning problem. These approaches may not guarantee an optimal solution, but they could provide practical and feasible solutions within a reasonable amount of time.

Furthermore, there is room for exploring the integration of machine learning techniques to aid in learning LTL formulas from examples. By leveraging large datasets and advanced algorithms, it may be possible to train models that can effectively learn the underlying patterns of positive and negative instances.

Additionally, collaborations between computer scientists, logicians, linguists, and cognitive scientists could bring valuable insights into finding a more intuitive representation of separating properties in natural language. By drawing on expertise from different disciplines, a more comprehensive and effective approach may evolve.

In conclusion, the study of learning LTL formulas from examples is a multi-disciplinary endeavor with challenges inherent in both the computational complexity and the representation of properties in natural language. The results presented in this paper serve as a starting point for further exploration and collaboration in finding efficient solutions and expanding our understanding of this complex problem.

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