arXiv subjects

Franz Baader

Publications and source records attributed to Franz Baader.

13 recordsLinked to original sources

arXiv2026

The Unification Type of an Equational Theory May Depend on the Instantiation Preorder: From Results for Single Theories to Results for Classes of Theories

The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It has been known for more than three decades that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More precisely, it was shown in 1991 that the theory ACUI of an associative, commutative, and idempotent binary function symbol with a unit is unitary w.r.t. the restricted instantiation preorder, but not unitary w.r.t. the unrestricted one. In 2016 this result was strengthened by showing that the unrestricted type of this theory also cannot be finitary. In the conference version of this article, we considerably improved on this result by proving that ACUI is infinitary w.r.t. the unrestricted instantiation preorder, thus precluding type zero. We also showed that, w.r.t. this preorder, the unification type of ACU (where idempotency is removed from the axioms) and of AC (where additionally the unit is removed) is infinitary, though it is respectively unitary and finitary in the restricted case. In the other direction, we proved (using the example of unification in the description logic EL) that the unification type may actually improve from type zero to infinitary when switching from the restricted instantiation preorder to the unrestricted one. In the present article, we not only determine the unrestricted unification type of considerably more equational theories, but we also prove general results for whole classes of theories. In particular, we show that theories that are regular and finite, regular and locally finite, or regular, monoidal, and satisfy an additional condition are Noetherian, and thus cannot have unrestricted unification type zero.

cs.LO↗

arXiv2025

Concrete Domains Meet Expressive Cardinality Restrictions in Description Logics (Extended Version)

Standard Description Logics (DLs) can encode quantitative aspects of an application domain through either number restrictions, which constrain the number of individuals that are in a certain relationship with an individual, or concrete domains, which can be used to assign concrete values to individuals using so-called features. These two mechanisms have been extended towards very expressive DLs, for which reasoning nevertheless remains decidable. Number restrictions have been generalized to more powerful comparisons of sets of role successors in $\mathcal{ALCSCC}$, while the comparison of feature values of different individuals in $\mathcal{ALC}(\mathfrak{D})$ has been studied in the context of $\omega$-admissible concrete domains $\mathfrak{D}$. In this paper, we combine both formalisms and investigate the complexity of reasoning in the thus obtained DL $\mathcal{ALCOSCC}(\mathfrak{D})$, which additionally includes the ability to refer to specific individuals by name. We show that, in spite of its high expressivity, the consistency problem for this DL is ExpTime-complete, assuming that the constraint satisfaction problem of $\mathfrak{D}$ is also decidable in exponential time. It is thus not higher than the complexity of the basic DL $\mathcal{ALC}$. At the same time, we show that many natural extensions to this DL, including a tighter integration of the concrete domain and number restrictions, lead to undecidability.

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arXiv2024

Small Term Reachability and Related Problems for Terminating Term Rewriting Systems

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system $R$, a term $s$, and a natural number $n$, decide whether there is a term $t$ of size $\leq n$ reachable from $s$ using the rules of $R$. We investigate the complexity of this problem depending on how termination of $R$ can be established. We show that the problem is in general NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases. Finally, we consider the large term reachability problem, a variant of the problem where we are interested in reachability of a term of size $\geq n$. It turns out that this seemingly innocuous modification in some cases changes the complexity of the problem, which may also become dependent on whether the number $n$ is is represented in unary or binary encoding, whereas this makes no difference for the complexity of the small term reachability problem.

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arXiv2023

Combining Proofs for Description Logic and Concrete Domain Reasoning (Technical Report)

Logic-based approaches to AI have the advantage that their behavior can in principle be explained with the help of proofs of the computed consequences. For ontologies based on Description Logic (DL), we have put this advantage into practice by showing how proofs for consequences derived by DL reasoners can be computed and displayed in a user-friendly way. However, these methods are insufficient in applications where also numerical reasoning is relevant. The present paper considers proofs for DLs extended with concrete domains (CDs) based on the rational numbers, which leave reasoning tractable if integrated into the lightweight DL $\mathcal{E}\hspace{-0.1em}\mathcal{L}_\bot$. Since no implemented DL reasoner supports these CDs, we first develop reasoning procedures for them, and show how they can be combined with reasoning approaches for pure DLs, both for $\mathcal{E}\hspace{-0.1em}\mathcal{L}_\bot$ and the more expressive DL $\mathcal{ALC}$. These procedures are designed such that it is easy to extract proofs from them. We show how the extracted CD proofs can be combined with proofs on the DL side into integrated proofs that explain both the DL and the CD reasoning.

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arXiv2022

Finding Good Proofs for Description Logic Entailments Using Recursive Quality Measures (Extended Technical Report)

Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can explain such an entailment by presenting a proof of the consequence in an appropriate calculus. How comprehensible such a proof is depends not only on the employed calculus, but also on the properties of the particular proof, such as its overall size, its depth, the complexity of the employed sentences and proof steps, etc. For this reason, we want to determine the complexity of generating proofs that are below a certain threshold w.r.t. a given measure of proof quality. Rather than investigating this problem for a fixed proof calculus and a fixed measure, we aim for general results that hold for wide classes of calculi and measures. In previous work, we first restricted the attention to a setting where proof size is used to measure the quality of a proof. We then extended the approach to a more general setting, but important measures such as proof depth were not covered. In the present paper, we provide results for a class of measures called recursive, which yields lower complexities and also encompasses proof depth. In addition, we close some gaps left open in our previous work, thus providing a comprehensive picture of the complexity landscape.

cs.AI↗

arXiv2022

Evonne: Interactive Proof Visualization for Description Logics (System Description) -- Extended Version

Explanations for description logic (DL) entailments provide important support for the maintenance of large ontologies. The "justifications" usually employed for this purpose in ontology editors pinpoint the parts of the ontology responsible for a given entailment. Proofs for entailments make the intermediate reasoning steps explicit, and thus explain how a consequence can actually be derived. We present an interactive system for exploring description logic proofs, called Evonne, which visualizes proofs of consequences for ontologies written in expressive DLs. We describe the methods used for computing those proofs, together with a feature called signature-based proof condensation. Moreover, we evaluate the quality of generated proofs using real ontologies.

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arXiv2021

Efficient TBox Reasoning with Value Restrictions using the $\mathcal{FL}_{o}$wer reasoner

The inexpressive Description Logic (DL) $\mathcal{FL}_0$, which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in $\mathcal{FL}_0$ w.r.t. general TBoxes is ExpTime-complete, i.e., as hard as in the considerably more expressive logic $\mathcal{ALC}$. In this paper, we rehabilitate $\mathcal{FL}_0$ by presenting a dedicated subsumption algorithm for $\mathcal{FL}_0$, which is much simpler than the tableau-based algorithms employed by highly optimized DL reasoners. Our experiments show that the performance of our novel algorithm, as prototypically implemented in our $\mathcal{FL}_o$wer reasoner, compares very well with that of the highly optimized reasoners. $\mathcal{FL}_o$wer can also deal with ontologies written in the extension $\mathcal{FL}_{\bot}$ of $\mathcal{FL}_0$ with the top and the bottom concept by employing a polynomial-time reduction, shown in this paper, which eliminates top and bottom. We also investigate the complexity of reasoning in DLs related to the Horn-fragments of $\mathcal{FL}_0$ and $\mathcal{FL}_{\bot}$.

cs.AI↗

arXiv2020

Finding Small Proofs for Description Logic Entailments: Theory and Practice (Extended Technical Report)

Logic-based approaches to AI have the advantage that their behaviour can in principle be explained by providing their users with proofs for the derived consequences. However, if such proofs get very large, then it may be hard to understand a consequence even if the individual derivation steps are easy to comprehend. This motivates our interest in finding small proofs for Description Logic (DL) entailments. Instead of concentrating on a specific DL and proof calculus for this DL, we introduce a general framework in which proofs are represented as labeled, directed hypergraphs, where each hyperedge corresponds to a single sound derivation step. On the theoretical side, we investigate the complexity of deciding whether a certain consequence has a proof of size at most $n$ along the following orthogonal dimensions: (i) the underlying proof system is polynomial or exponential; (ii) proofs may or may not reuse already derived consequences; and (iii) the number $n$ is represented in unary or binary. We have determined the exact worst-case complexity of this decision problem for all but one of the possible combinations of these options. On the practical side, we have developed and implemented an approach for generating proofs for expressive DLs based on a non-standard reasoning task called forgetting. We have evaluated this approach on a set of realistic ontologies and compared the obtained proofs with proofs generated by the DL reasoner ELK, finding that forgetting-based proofs are often better w.r.t. different measures of proof complexity.

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arXiv2020

Satisfiability and Query Answering in Description Logics with Global and Local Cardinality Constraints

We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. The latter result is based on a locally-acyclic model construction, and it reduces query entailment to ABox consistency in the restricted setting, i.e., to ABox consistency w.r.t. restricted cardinality constraints in ALCSCC, for which we can show an ExpTime upper bound.

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arXiv2018

Repairing Description Logic Ontologies by Weakening Axioms

The classical approach for repairing a Description Logic ontology O in the sense of removing an unwanted consequence $α$ is to delete a minimal number of axioms from O such that the resulting ontology O' does not have the consequence $α$. However, the complete deletion of axioms may be too rough, in the sense that it may also remove consequences that are actually wanted. To alleviate this problem, we propose a more gentle way of repair in which axioms are not necessarily deleted, but only weakened. On the one hand, we investigate general properties of this gentle repair method. On the other hand, we propose and analyze concrete approaches for weakening axioms expressed in the Description Logic EL.

cs.LO↗

arXiv2016

Extending Unification in $\mathcal{EL}$ to Disunification: The Case of Dismatching and Local Disunification

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic $\mathcal{EL}$ to disunification since negative constraints can be used to avoid unwanted unifiers. While decidability of the solvability of general $\mathcal{EL}$-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we consider only solutions that are constructed from terms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of disunification problems.

cs.LO↗

arXiv2010

Unification in the Description Logic EL

The Description Logic EL has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, EL is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in EL is decidable. More precisely, EL-unification is NP-complete, and thus has the same complexity as EL-matching. We also show that, w.r.t. the unification type, EL is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.

cs.AI↗

arXiv2004

The Inverse Method Implements the Automata Approach for Modal Satisfiability

Tableaux-based decision procedures for satisfiability of modal and description logics behave quite well in practice, but it is sometimes hard to obtain exact worst-case complexity results using these approaches, especially for EXPTIME-complete logics. In contrast, automata-based approaches often yield algorithms for which optimal worst-case complexity can easily be proved. However, the algorithms obtained this way are usually not only worst-case, but also best-case exponential: they first construct an automaton that is always exponential in the size of the input, and then apply the (polynomial) emptiness test to this large automaton. To overcome this problem, one must try to construct the automaton "on-the-fly" while performing the emptiness test. In this paper we will show that Voronkov's inverse method for the modal logic K can be seen as an on-the-fly realization of the emptiness test done by the automata approach for K. The benefits of this result are two-fold. First, it shows that Voronkov's implementation of the inverse method, which behaves quite well in practice, is an optimized on-the-fly implementation of the automata-based satisfiability procedure for K. Second, it can be used to give a simpler proof of the fact that Voronkov's optimizations do not destroy completeness of the procedure. We will also show that the inverse method can easily be extended to handle global axioms, and that the correspondence to the automata approach still holds in this setting. In particular, the inverse method yields an EXPTIME-algorithm for satisfiability in K w.r.t. global axioms.

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