Decidability of Querying First-Order Theories via Countermodels of Finite Width

Основные понятия

The decidability of logical entailment problems can be established by exploiting the existence of countermodels that are structurally simple, as measured by certain types of width measures.

Аннотация

The paper proposes a generic framework for establishing the decidability of a wide range of logical entailment problems, based on the existence of countermodels that are structurally simple, as measured by certain types of width measures.

The key insights are:

1. Entailment problems of the form Φ |= Ψ, where Φ is a first-order (FO) sentence and Ψ is a universal second-order (∀SO) sentence, can be reduced to FO entailment Φ |= Ψ*, which ensures semidecidability.

2. To achieve full decidability, the authors introduce the notion of "width-based controllability", where the existence of countermodels of finite width is guaranteed. This is formalized in Theorem 7, which provides a generic decidability result.

3. An important special case is when the specification language F exhibits finitely universal model sets of finite width, which ensures decidability of entailment for a wide range of homomorphism-closed queries in ∀SO (Corollary 15). This covers many expressive query languages like nested monadically defined queries and monadic disjunctive datalog.

4. The authors propose partitionwidth as a powerful width measure that subsumes various other notions and exhibits favorable computational and structural properties.

5. The framework is applied to the setting of existential rules, where the authors define finite-partitionwidth sets of rules and show how they can incorporate various types of datalog layers, yielding a flexible toolset for analyzing and creating rulesets with favorable querying properties.

Статистика

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Ключевые выводы из

Decidability of Querying First-Order Theories via Countermodels of Finite Width

by Thomas Felle... в arxiv.org 09-17-2024

https://arxiv.org/pdf/2304.06348.pdf

Дополнительные вопросы

Can the proposed framework be extended to cover more expressive specification languages beyond first-order logic, such as fragments of monadic second-order logic?

The proposed framework primarily focuses on first-order logic (FO) and its interaction with universal second-order logic (∀SO). However, there is potential for extending this framework to encompass more expressive specification languages, including fragments of monadic second-order logic (MSO). The key to such an extension lies in the properties of the width measures employed and the effective expressibility of the fragments involved.
For instance, if a width measure can be shown to be MSO-friendly, it may allow for the decidability of entailment problems involving MSO fragments. The challenge, however, is that many width measures, including those discussed in the context of this framework, do not generalize well to MSO. This is primarily due to the complexity of MSO, which allows for quantification over sets of elements, thus increasing the expressivity and potentially complicating the decidability landscape.
To successfully extend the framework, one would need to establish a computable function that bounds the width of countermodels for the MSO fragments in question. This would require a careful analysis of the specific properties of the MSO fragments and their interaction with the chosen width measures. If such a function can be identified, it would facilitate the application of the existing framework to MSO, thereby broadening the scope of decidable entailment problems.

Are there other width measures, beyond partitionwidth, that could be leveraged to obtain decidability results for even more expressive query languages?

Yes, there are several other width measures that could potentially be leveraged to obtain decidability results for more expressive query languages. Some notable examples include:

Treewidth: This is a well-established width measure that has been successfully applied in various contexts, particularly in graph theory and database theory. It is known to be GSO-friendly and can be used to analyze the decidability of entailment problems involving homomorphism-closed queries.

Cliquewidth: Similar to treewidth, cliquewidth is another width measure that can be applied to structures and has been shown to be MSO-friendly. It can be particularly useful in contexts where the structure of the data can be represented as graphs, allowing for the analysis of more complex query languages.

Rank-width: This is a relatively newer width measure that has gained attention in the context of graph theory. It can be applied to various logical frameworks and may provide new avenues for establishing decidability results, especially in cases where traditional measures like treewidth and cliquewidth fall short.

Pathwidth: This measure is particularly relevant for structures that can be represented as paths or trees. It has applications in database theory and can be useful for analyzing the complexity of certain query languages.

By exploring these and potentially other width measures, researchers can identify new decidability results for a broader range of expressive query languages, thereby enhancing the applicability of the framework in practical scenarios.

How can the insights from this work be applied to practical problems in areas like databases, knowledge representation, and formal verification, where logical entailment plays a central role?

The insights from this work have significant implications for practical problems in databases, knowledge representation, and formal verification, particularly in the context of logical entailment. Here are several ways these insights can be applied:

Database Query Optimization: The framework's focus on decidability and width measures can inform the design of query languages and optimization strategies in databases. By identifying which queries can be efficiently evaluated based on their structural properties, database systems can optimize query execution plans, leading to improved performance.

Knowledge Representation and Reasoning: In knowledge representation, the ability to determine entailment between knowledge bases and queries is crucial. The framework can help in developing reasoning systems that can efficiently handle complex queries, particularly in scenarios involving incomplete or uncertain information. This is especially relevant in ontology-based query answering, where the expressivity of the query language can significantly impact the reasoning capabilities.

Formal Verification: In formal verification, the framework can assist in establishing the decidability of various verification problems, such as model checking and property verification. By leveraging width measures, verification tools can be designed to handle more expressive specifications, allowing for the analysis of complex systems while ensuring that the verification process remains decidable.

Interdisciplinary Applications: The insights can also be applied across disciplines, such as artificial intelligence, where logical reasoning is fundamental. By understanding the decidability of entailment in various logical frameworks, AI systems can be designed to reason more effectively about knowledge and make informed decisions based on logical constraints.

In summary, the framework's contributions to understanding the decidability of logical entailment problems can lead to practical advancements in various fields, enhancing the efficiency and effectiveness of systems that rely on logical reasoning.