Periodic Reporting for period 3 - DeciGUT (A Grand Unified Theory of Decidability in Logic-Based Knowledge Representation)
Période du rapport: 2021-10-01 au 2023-03-31
DeciGUT deals with the formal foundations of knowledge management as well as their application in today's information society. Among the biggest challenges in this area are the intelligent access to digital information as well as the automated composition of information from diverse sources. For these purposes, logical specifications of background knowledge - so-called ontologies - can be used together with automated reasoning techniques in order to enable a "meaning-aware" handling of data.
Unfortunately, reasoning in ontology languages of high expressivity is impossible to capture algorithmically – they are undecidable. Therefore, the quest for "good" ontology languages consists in identifying logical formalisms which are as expressive as possible, yet still decidable. Hitherto, the obtained results in this area have, however, been patchy and fragmented.
The goal of the DeciGUT project is the creation of a unified theory of decidability, which in turn will enable the definition of new, advanced ontology languages.
The project is of high relevance to diverse scientific fields like mathematical logic, artificial intelligence, and database theory with potentially far-reaching impact in areas such as semantic technologies and information systems.
Unfortunately, reasoning in ontology languages of high expressivity is impossible to capture algorithmically – they are undecidable. Therefore, the quest for "good" ontology languages consists in identifying logical formalisms which are as expressive as possible, yet still decidable. Hitherto, the obtained results in this area have, however, been patchy and fragmented.
The goal of the DeciGUT project is the creation of a unified theory of decidability, which in turn will enable the definition of new, advanced ontology languages.
The project is of high relevance to diverse scientific fields like mathematical logic, artificial intelligence, and database theory with potentially far-reaching impact in areas such as semantic technologies and information systems.
Work in the project was performed along the following lines:
(1) Exact complexity analyses of query answering in popular KR formalisms.
As answering queries over knowledge sources has become the central task in knowledge representation and information systems, it is of utmost importance to clarify the computational effort required for this task (characterized in terms of computational complexity), depending on the used knowledge representation formalism. Notably, logical features that cause high complexities in some settings have shown to cause undecidability in others, which directly links these investigations to the core questions of the DeciGUT project.
We have obtained improved, optimal complexity bounds for very expressive Description Logics with regular path expressions (IJCAI 2019), as well as a restricted version of the guarded fragment of first-order logic, called forward-guarded fragment, inspired by a notion of "directionality" (JELIA 2021). Likewise we have shown an exponential jump in complexity of conjunctive query answering when adding the Self-operator to the description logic ALC (submitted to KR 2021), a result quite unexpected and surprising to us, since this construct has proven rather "well behaved" in all contexts considered so far.
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).
Finally, we extended our investigations from arbitrary-model semantics to finite-model semantics, following a general trend in current resear5ch 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 (accepted at LICS 2021)
(2) Understanding Chase Termination.
The Chase is a very popular concept in database theory and denotes a (potentially infinite) procedure of repeatedly applying rules (also referred to as "tuple-generating dependencies", short TGDs) to a given database (a method also referred to as "forward chaining"). Guaranteed termination of this procedure is one of the classical central ways of establishing decidability. However, several variants of the Chase exist and their exact relationship in terms of expressivity had been hitherto unknown.
We sharpened the existing result that termination of the oblivious chase over of a given rule set over all databases is undecidable, by showing that this even holds in the very restricted setting of single-head TGDs with binary predicates only (IJCAI 2020, note that such a restriction of the signature often turns undeciudble problems decidable - not so in this case). We established that all-database-terminating Skolem chase is very restricted in expressivity (confined to PTime queries) while terminating standard- and Skolem chase allow for a much higher expressivity, including queries of non-elementary complexity (ICDT 2019). Very recently, we were able to establish the exceptional result that all-instance terminating standard chase in fact captures *all* decidable homomorphism-closed queries (submitted to KR 2021). As an effect, it was shown that standar5d chase, core chase and some known intermediate variants have the same expressivity.
(3) Investigation of computational problems related to the homomorphism-closure of model sets.
For a big share of query laguages, the set of models exhibiting a match of a given query is closed under homomorphisms. A better understanding of the decidability and complexity status of problems related to the homomorphism-closure of the model set of a given sentence therefore helps the investigation of decidability of query answering as a whole. We clarified the computational properties of 4 key tasks for a comprehensive selection of fragments of first- and second order logic (accepted at LICS 2021). Moreover, along the same investigative lines, we investigated the expressive relationship of Datalog (a querying formalism having homomorphism-closed model sets), fragments of Second-Order Logic, and certain Constraint Satisfaction Problems (accepted at ICALP 2021).
(1) Exact complexity analyses of query answering in popular KR formalisms.
As answering queries over knowledge sources has become the central task in knowledge representation and information systems, it is of utmost importance to clarify the computational effort required for this task (characterized in terms of computational complexity), depending on the used knowledge representation formalism. Notably, logical features that cause high complexities in some settings have shown to cause undecidability in others, which directly links these investigations to the core questions of the DeciGUT project.
We have obtained improved, optimal complexity bounds for very expressive Description Logics with regular path expressions (IJCAI 2019), as well as a restricted version of the guarded fragment of first-order logic, called forward-guarded fragment, inspired by a notion of "directionality" (JELIA 2021). Likewise we have shown an exponential jump in complexity of conjunctive query answering when adding the Self-operator to the description logic ALC (submitted to KR 2021), a result quite unexpected and surprising to us, since this construct has proven rather "well behaved" in all contexts considered so far.
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).
Finally, we extended our investigations from arbitrary-model semantics to finite-model semantics, following a general trend in current resear5ch 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 (accepted at LICS 2021)
(2) Understanding Chase Termination.
The Chase is a very popular concept in database theory and denotes a (potentially infinite) procedure of repeatedly applying rules (also referred to as "tuple-generating dependencies", short TGDs) to a given database (a method also referred to as "forward chaining"). Guaranteed termination of this procedure is one of the classical central ways of establishing decidability. However, several variants of the Chase exist and their exact relationship in terms of expressivity had been hitherto unknown.
We sharpened the existing result that termination of the oblivious chase over of a given rule set over all databases is undecidable, by showing that this even holds in the very restricted setting of single-head TGDs with binary predicates only (IJCAI 2020, note that such a restriction of the signature often turns undeciudble problems decidable - not so in this case). We established that all-database-terminating Skolem chase is very restricted in expressivity (confined to PTime queries) while terminating standard- and Skolem chase allow for a much higher expressivity, including queries of non-elementary complexity (ICDT 2019). Very recently, we were able to establish the exceptional result that all-instance terminating standard chase in fact captures *all* decidable homomorphism-closed queries (submitted to KR 2021). As an effect, it was shown that standar5d chase, core chase and some known intermediate variants have the same expressivity.
(3) Investigation of computational problems related to the homomorphism-closure of model sets.
For a big share of query laguages, the set of models exhibiting a match of a given query is closed under homomorphisms. A better understanding of the decidability and complexity status of problems related to the homomorphism-closure of the model set of a given sentence therefore helps the investigation of decidability of query answering as a whole. We clarified the computational properties of 4 key tasks for a comprehensive selection of fragments of first- and second order logic (accepted at LICS 2021). Moreover, along the same investigative lines, we investigated the expressive relationship of Datalog (a querying formalism having homomorphism-closed model sets), fragments of Second-Order Logic, and certain Constraint Satisfaction Problems (accepted at ICALP 2021).
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.
In collaboration with Manuel Bodirsky (another ERC Grantee located in Dresden), we uncovered interesting relationships between query answering and the area of mathematical Constraint Satisfaction Problem, which we are going to jointly explore.
Our results on expressivity of formalisms falling under diverse chase-termination-criteria clarified open questions that had for a significant time bothered researchers in database theory.
In collaboration with Manuel Bodirsky (another ERC Grantee located in Dresden), we uncovered interesting relationships between query answering and the area of mathematical Constraint Satisfaction Problem, which we are going to jointly explore.
Our results on expressivity of formalisms falling under diverse chase-termination-criteria clarified open questions that had for a significant time bothered researchers in database theory.