Periodic Reporting for period 4 - DeciGUT (A Grand Unified Theory of Decidability in Logic-Based Knowledge Representation)
Período documentado: 2023-04-01 hasta 2025-03-31
DeciGUT addresses the fundamental theoretical underpinnings of knowledge management – the discipline investigating how to effectively capture, store, retrieve, and utilize knowledge, with the goal to cope with the challenges presented by today’s complex information environments by building systems that can understand and work with information on a higher level.
To this end, logic-based knowledge representation as a strand of symbolic AI focuses on leveraging logical specifications of background knowledge, commonly known as ontologies: formal vocabularies describing concepts within a specific domain (like medicine or engineering) and the relationships between them. Such ontologies are used in conjunction with automated reasoning techniques – methods that draw logical conclusions from the information encoded within the ontology – to allow systems to handle data in a more "meaning-aware" way.
However, reasoning in many highly expressive ontology languages is fundamentally impossible for computers to handle algorithmically, as these languages have been found to be undecidable – meaning there exists no general procedure that can determine whether a given statement logically follows from the knowledge encoded in the ontology. Consequently, research focuses on finding the sweet spot: identifying logical formalisms which are as expressive as possible, yet still decidable. Previous efforts in this area have unfortunately been patchy and fragmented, lacking a cohesive theoretical framework.
The core aim of the DeciGUT project is therefore the creation of a unified theory of decidability. This overarching theory will facilitate research devoted to exploring the boundary between decidability and undecidability by proposing generic principles allowing to establish (un)decidability proofs uniformly for large classes of logical formalisms. It will also provide a solid foundation for designing new, advanced ontology languages that are both powerful enough to represent real-world knowledge and computationally manageable enough for practical applications.
This research has significant relevance across diverse scientific fields including mathematical logic (the study of the principles of reasoning), artificial intelligence (building intelligent agents and systems), and database theory (managing and organizing data efficiently). Successful completion of DeciGUT promises far-reaching impact in areas such as semantic technologies – enabling machines to understand the meaning of information on the web – and broader information systems, leading to more effective, yet reliable knowledge discovery, decision support, and automation.
To this end, logic-based knowledge representation as a strand of symbolic AI focuses on leveraging logical specifications of background knowledge, commonly known as ontologies: formal vocabularies describing concepts within a specific domain (like medicine or engineering) and the relationships between them. Such ontologies are used in conjunction with automated reasoning techniques – methods that draw logical conclusions from the information encoded within the ontology – to allow systems to handle data in a more "meaning-aware" way.
However, reasoning in many highly expressive ontology languages is fundamentally impossible for computers to handle algorithmically, as these languages have been found to be undecidable – meaning there exists no general procedure that can determine whether a given statement logically follows from the knowledge encoded in the ontology. Consequently, research focuses on finding the sweet spot: identifying logical formalisms which are as expressive as possible, yet still decidable. Previous efforts in this area have unfortunately been patchy and fragmented, lacking a cohesive theoretical framework.
The core aim of the DeciGUT project is therefore the creation of a unified theory of decidability. This overarching theory will facilitate research devoted to exploring the boundary between decidability and undecidability by proposing generic principles allowing to establish (un)decidability proofs uniformly for large classes of logical formalisms. It will also provide a solid foundation for designing new, advanced ontology languages that are both powerful enough to represent real-world knowledge and computationally manageable enough for practical applications.
This research has significant relevance across diverse scientific fields including mathematical logic (the study of the principles of reasoning), artificial intelligence (building intelligent agents and systems), and database theory (managing and organizing data efficiently). Successful completion of DeciGUT promises far-reaching impact in areas such as semantic technologies – enabling machines to understand the meaning of information on the web – and broader information systems, leading to more effective, yet reliable knowledge discovery, decision support, and automation.
Work in the project was performed mostly along the following lines:
(1) Width-based generic decidability notions.
Regarding the objective to establish generic model-theoretic criteria for decidability of certain reasoning problems (satisfiability and query answering) in expressive logical formalisms, we managed to establish results based on the notion of clique-width (ICDT 2023) which was subsequently extended to partition-width (PODS 2023, LMCS 2024). Regarding the important class of first-order rewritable rule sets, a class known to be decidable via proof-theoretic means, we found that answering regular path queries in general is undecidable (KR 2024), which greatly improved our understanding of the decidability boundaries. A recently obtained technical result (PODS 2025) might constitute an important step forward in settling the so-called FUS-FC conjecture (which, if answered positively, would indeed prove that FO-rewriteable rulesets fall under the semantic notions established by us).
(2) Exact complexity analyses of query answering in popular KR formalisms.
We have obtained improved, optimal complexity bounds for very expressive Description Logics with path expressions (IJCAI 2019, JAIR 2023, IJCAI 2024, LMCS 2024). Likewise we have shown an exponential jump in complexity of conjunctive query answering when adding the Self-operator to the description logic ALC (AAAI 2023), a result quite unexpected and surprising to us, since this construct has proven rather "well behaved" in all contexts considered so far.
Finally, we extended our investigations from arbitrary-model semantics to finite-model semantics, following a general trend in current research on logic-based KR. Among others, we managed to show the finite-model property of the very expressive triguarded fragment of first-order logic, thereby also clarifying its decidability (LICS 2021b)
(3) Understanding Chase Termination.
The Chase is a very popular concept in database theory, denoting a procedure whose guaranteed termination ensures decidability. We established that all-database-terminating Skolem chase is very restricted in expressivity (confined to PTime queries) while terminating standard- and core chase allow for a much higher expressivity (ICDT 2019). We were able to establish the exceptional result that all-instance terminating standard chase in fact captures *all* decidable homomorphism-closed queries (KR 2021, Ray Reiter Best Paper Award).
(4) Decidability of Formalisms with Counting Capabilities
We also have made significant progress in the extension of standard KR formalisms by extended quantitative modelling features including certain arithmetical relationships. We obtained that decidability and even complexity is preserved when adding such features to certain logics, while in other settings, undecidability immediately follows (ECAI 2020). We also designed a decidable extension of monadic second-order logic (MSO) over trees by advanced counting capabilities, integrating and surpassing the pre-existing formalisms counting MSO as well as Boolean Algebra with Presburger Arithmetic (CSL 2024).
(5) Advanced KR features: Nonmonotonic and Multi-Perspective Reasoning
Wefocused on the classical field of belief revision and provided generic model-theoretic characterizations of fundamental postulates in that area (RuleML+RR 2022). We also widened the scope of our research to multi-perspective reasoning, proposing and investigating the new framework of “Standpoint Logics” (FOIS 2021, ISWC 2022, KR 2023, IJCAI 2023, KR 2024)
(1) Width-based generic decidability notions.
Regarding the objective to establish generic model-theoretic criteria for decidability of certain reasoning problems (satisfiability and query answering) in expressive logical formalisms, we managed to establish results based on the notion of clique-width (ICDT 2023) which was subsequently extended to partition-width (PODS 2023, LMCS 2024). Regarding the important class of first-order rewritable rule sets, a class known to be decidable via proof-theoretic means, we found that answering regular path queries in general is undecidable (KR 2024), which greatly improved our understanding of the decidability boundaries. A recently obtained technical result (PODS 2025) might constitute an important step forward in settling the so-called FUS-FC conjecture (which, if answered positively, would indeed prove that FO-rewriteable rulesets fall under the semantic notions established by us).
(2) Exact complexity analyses of query answering in popular KR formalisms.
We have obtained improved, optimal complexity bounds for very expressive Description Logics with path expressions (IJCAI 2019, JAIR 2023, IJCAI 2024, LMCS 2024). Likewise we have shown an exponential jump in complexity of conjunctive query answering when adding the Self-operator to the description logic ALC (AAAI 2023), a result quite unexpected and surprising to us, since this construct has proven rather "well behaved" in all contexts considered so far.
Finally, we extended our investigations from arbitrary-model semantics to finite-model semantics, following a general trend in current research on logic-based KR. Among others, we managed to show the finite-model property of the very expressive triguarded fragment of first-order logic, thereby also clarifying its decidability (LICS 2021b)
(3) Understanding Chase Termination.
The Chase is a very popular concept in database theory, denoting a procedure whose guaranteed termination ensures decidability. We established that all-database-terminating Skolem chase is very restricted in expressivity (confined to PTime queries) while terminating standard- and core chase allow for a much higher expressivity (ICDT 2019). We were able to establish the exceptional result that all-instance terminating standard chase in fact captures *all* decidable homomorphism-closed queries (KR 2021, Ray Reiter Best Paper Award).
(4) Decidability of Formalisms with Counting Capabilities
We also have made significant progress in the extension of standard KR formalisms by extended quantitative modelling features including certain arithmetical relationships. We obtained that decidability and even complexity is preserved when adding such features to certain logics, while in other settings, undecidability immediately follows (ECAI 2020). We also designed a decidable extension of monadic second-order logic (MSO) over trees by advanced counting capabilities, integrating and surpassing the pre-existing formalisms counting MSO as well as Boolean Algebra with Presburger Arithmetic (CSL 2024).
(5) Advanced KR features: Nonmonotonic and Multi-Perspective Reasoning
Wefocused on the classical field of belief revision and provided generic model-theoretic characterizations of fundamental postulates in that area (RuleML+RR 2022). We also widened the scope of our research to multi-perspective reasoning, proposing and investigating the new framework of “Standpoint Logics” (FOIS 2021, ISWC 2022, KR 2023, IJCAI 2023, KR 2024)
As envisioned in the proposal, we managed to establish a novel model-theoretic decidability framework, based on the generic notion of “width” of a model and showed how the majority of known knowledge representation frameworks expressible in first-order logic fall under this framework and hence allow for decidable query answering. Most notably, depending on the width notion used, this enabled us to significantly extend the state of the art, by showing that many fragments can be decidably queried by query languages that are much more expressive than originally envisioned.
Our aforementioned investigations and achievements required the development of novel proof techniques and constructions which, beyond serving the concrete purposes they were designed for, are expected to be of use in other contexts as well. That is, our work profoundly contributed to the formal “tool sets” in KR, database theory, mathematical logic, and proof theory and also exhibited synergies between these disciplines.
Our aforementioned investigations and achievements required the development of novel proof techniques and constructions which, beyond serving the concrete purposes they were designed for, are expected to be of use in other contexts as well. That is, our work profoundly contributed to the formal “tool sets” in KR, database theory, mathematical logic, and proof theory and also exhibited synergies between these disciplines.