We clarify the complexity of answering unions of conjunctive queries over knowledge bases formulated in the description logic S, the extension of ALC with transitive roles. Contrary to what existing partial results suggested, we show that the problem is, in fact, 2ExpTime-complete; hardness already holds in the presence of two transitive roles and for Boolean conjunctive queries. We complement this result by showing that the problem remains in coNExpTime when the input query is rooted or is restricted to use at most one transitive role (but may use arbitrarily many non-transitive roles).

Recent research has established complexity results for the problem of deciding the existence of interpolants in logics lacking the Craig interpolation property (CIP). The proof techniques developed so far are non-constructive, and no meaningful bounds on the size of interpolants are known. Hybrid modal logics (or modal logics with nominals) are a particularly interesting class of logics without CIP: in their case, CIP cannot be restored without sacrificing decidability and, in applications, interpolants in these logics can serve as definite descriptions and separators between positive and negative data examples in description logic knowledge bases. In this contribution we show, using a new hypermosaic elimination technique, that in many standard hybrid modal logics Craig interpolants can be computed in fourfold exponential time, if they exist. On the other hand, we show that the existence of uniform interpolants is undecidable, which is in stark contrast to modal or intuitionistic logic where uniform interpolants always exist.

Modal separability for modal fixpoint formulae is the problem to decide for two given modal fixpoint formulae $φ,φ'$ whether there is a modal formula $ψ$ that separates them, in the sense that $φ\modelsψ$ and $ψ\models\negφ'$. We study modal separability and its special case modal definability over various classes of models, such as arbitrary models, finite models, trees, and models of bounded outdegree. Our main results are that modal separability is PSpace-complete over words, that is, models of outdegree $\leq 1$, ExpTime-complete over unrestricted and over binary models, and TwoExpTime-complete over models of outdegree bounded by some $d\geq 3$. Interestingly, this latter case behaves fundamentally different from the other cases also in that modal logic does not enjoy the Craig interpolation property over this class. Motivated by this we study also the induced interpolant existence problem as a special case of modal separability, and show that it is coNExpTime-complete and thus harder than validity in the logic. Besides deciding separability, we also provide algorithms for the effective construction of separators. Finally, we consider in a case study the extension of modal fixpoint formulae by graded modalities and investigate separability by modal formulae and graded modal formulae.

Querying temporal data has recently gained traction in several artificial intelligence applications. As operational domains of intelligent agents are constantly being expanded, there is a strong need for representing domain knowledge. This comes in the form of ontologies, which are predominantly expressed in description logics and enrich time-stamped data to temporal knowledge bases. For modeling highly complex system environments, expressive description logics are often the formalism of choice. Querying such temporal knowledge bases is a challenging task, but recently a first practical solution has been put forward. We propose a novel approach to the query answering problem based on two well-known rewriting rules from temporal logic. After a careful theoretical analysis of our algorithm, we show in a practical evaluation on several benchmarks that it outperforms state of the art, sometimes by orders of magnitude. Based on our findings, we also propose a fragment of temporal conjunctive queries which guides users towards well-performing queries.

We study the problem of fitting ontologies and constraints to positive and negative examples that take the form of a finite relational structure. As ontology and constraint languages, we consider the description logics EL and ELI as well as several classes of tuple-generating dependencies (TGDs): full, guarded, frontier-guarded, frontier-one, and unrestricted TGDs as well as inclusion dependencies. We pinpoint the exact computational complexity, design algorithms, and analyze the size of fitting ontologies and TGDs. We also investigate the related problem of constructing a finite basis of concept inclusions / TGDs for a given set of finite structures. While finite bases exist for EL, ELI, guarded TGDs, and inclusion dependencies, they in general do not exist for full, frontier-guarded and frontier-one TGDs.

Modal separability for modal fixpoint formulae is the problem to decide for two given modal fixpoint formulae φ,φ' whether there is a modal formula ψ that separates them, in the sense that φ ⊧ ψ and ψ ⊧ ¬φ'. We study modal separability and its special case modal definability over various classes of models, such as arbitrary models, finite models, trees, and models of bounded outdegree. Our main results are that modal separability is PSpace-complete over words, that is, models of outdegree ≤ 1, ExpTime-complete over unrestricted and over binary models, and 2-ExpTime-complete over models of outdegree bounded by some d ≥ 3. Interestingly, this latter case behaves fundamentally different from the other cases also in that modal logic does not enjoy the Craig interpolation property over this class. Motivated by this we study also the induced interpolant existence problem as a special case of modal separability, and show that it is coNExpTime-complete and thus harder than validity in the logic. Besides deciding separability, we also investigate the problem of efficient construction of separators. Finally, we consider in a case study the extension of modal fixpoint formulae by graded modalities and investigate separability by modal formulae and graded modal formulae.

Algorithms for learning database queries from examples and unique characterisations of queries by examples are prominent starting points for developing automated support for query explanation and construction. We investigate how far recent results and techniques on learning and unique characterisations of atemporal queries mediated by an ontology can be extended to temporal data and queries. Based on a systematic review of the relevant approaches in the atemporal case, we obtain general transfer results identifying conditions under which temporal queries composed of atemporal ones are (polynomially) learnable and uniquely characterisable.

Abstract For developing safe automated systems, recognizing safety-critical situations in data from their complex operational domain is imperative. This capability is, for example, essential when evaluating the system’s conformance to specified requirements in test run data. The requirements involve a temporal dimension, as the system operates over time. Moreover, the generated data are usually relational and require additional background knowledge about the domain for correctly recognizing the situation. This fact makes propositional temporal logics, an established tool, unsuitable for the task. We address this issue by developing a tailored temporal logic to query for situations in relational data over complex domains. Our language combines mission-time linear temporal logic with conjunctive queries to access time-stamped data with background knowledge formulated in an expressive description logic. Currently, however, no tools exist for answering queries in such settings. We hence also contribute an implementation in the logic reasoner Openllet, leveraging the efficacy of well-established conjunctive query answering. Moreover, we present a benchmark generator in the setting of automated driving and demonstrate that our tool performs well when tasked with recognizing safety-critical situations in road traffic.

This note serves three purposes: (i) we provide a self-contained exposition of the fact that conjunctive queries are not efficiently learnable in the Probably-Approximately-Correct (PAC) model, paying clear attention to the complicating fact that this concept class lacks the polynomial-size fitting property, a property that is tacitly assumed in much of the computational learning theory literature; (ii) we establish a strong negative PAC learnability result that applies to many restricted classes of conjunctive queries (CQs), including acyclic CQs for a wide range of notions of acyclicity; (iii) we show that CQs (and UCQs) are efficiently PAC learnable with membership queries.

We propose bounded fitting as a scheme for learning description logic concepts in the presence of ontologies. A main advantage is that the resulting learning algorithms come with theoretical guarantees regarding their generalization to unseen examples in the sense of PAC learning. We prove that, in contrast, several other natural learning algorithms fail to provide such guarantees. As a further contribution, we present the system SPELL which efficiently implements bounded fitting for the description logic ELHr based on a SAT solver, and compare its performance to a state-of-the-art learner.

A prime application of description logics is ontology-mediated query answering, with the query language often reaching far beyond instance queries. Here, we investigate this task for positive existential two-way regular path queries and ontologies formulated in the expressive description logic SQu, where SQu denotes the extension of the basic description logic ALC with transitive roles (S) and qualified number restrictions (Q) which can be unrestrictedly applied to both non-transitive and transitive roles (⋅u). Notably, the latter is usually forbidden in expressive description logics. As the main contribution, we show decidability of ontology-mediated query answering in that setting and establish tight complexity bounds, namely 2ExpTime-completeness in combined complexity and coNP-completeness in data complexity. Since the lower bounds are inherited from the fragment ALC, we concentrate on providing upper bounds. As main technical tools we establish a tree-like countermodel property and a characterization of when a query is not satisfied in a tree-like interpretation. Together, these results allow us to use an automata-based approach to query answering.

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article, we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as 𝒜ℒ𝒞𝒪, 𝒜ℒ𝒞ℋ, and 𝒜ℒ𝒞ℋ𝒪ℐ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime -complete in the presence of an ontology or the universal modality, and coNExpTime -complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case, the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.

We study ELI queries (ELIQs) in the presence of ontologies formulated in the description logic DL-Lite. For the dialect DL-LiteH, we show that ELIQs have a frontier (set of least general generalizations) that is of polynomial size and can be computed in polynomial time. In the dialect DL-LiteF, in contrast, frontiers may be infinite. We identify a natural syntactic restriction that enables the same positive results as for DL-LiteH. We use our results on frontiers to show that ELIQs are learnable in polynomial time in the presence of a DL-LiteH / restricted DL-LiteF ontology in Angluin's framework of exact learning with only membership queries.

We introduce Bule, a modeling language for problems from the complexity class PSPACE via quantified Boolean formulas (QBF) - that is, propositional formulas in which the variables are existentially or universally quantified. Bule allows the user to write a high-level representation of the problem in a natural, rule-based language, that is inspired by stratified Datalog. We implemented a tool of the same name that converts the high-level representation into DIMACS format and thus provides an interface to aribtrary QBF solvers, so that the modeled problems can also be solved. We analyze the complexity-theoretic properties of our modeling language, provide a library for common modeling patterns, and evaluate our language and tool on several examples.

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP transform potentially hard existence problems into deduction problems in the underlying logic. Description Logics with nominals and/or role inclusions do not enjoy the CIP nor PBDP, but interpolants and explicit definitions have many potential applications in ontology engineering and ontology-based data management. In this article we show the following: even without Craig and Beth, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as ALCO, ALCH and ALCHIO. However, living without Craig and Beth makes this problem harder than deduction: we prove that the existence problems become 2EXPTIME-complete, thus one exponential harder than validity. The existence of explicit definitions is 2EXPTIME-hard even if one asks for a definition of a nominal using any symbol distinct from that nominal, but it becomes EXPTIME-complete if one asks for a definition of a concept name using any symbol distinct from that concept name.

We consider the problem to learn a concept or a query in the presence of an ontology formulated in the description logic ELr, in Angluin's framework of active learning that allows the learning algorithm to interactively query an oracle (such as a domain expert). We show that the following can be learned in polynomial time: (1) EL-concepts, (2) symmetry-free ELI-concepts, and (3) conjunctive queries (CQs) that are chordal, symmetry-free, and of bounded arity. In all cases, the learner can pose to the oracle membership queries based on ABoxes and equivalence queries that ask whether a given concept/query from the considered class is equivalent to the target. The restriction to bounded arity in (3) can be removed when we admit unrestricted CQs in equivalence queries. We also show that EL-concepts are not polynomial query learnable in the presence of ELI-ontologies.

We study the separation of positive and negative data examples in terms of description logic concepts in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols that can be used for separation, and we admit individual names in that signature. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated and we distinguish between separation with and without helper symbols. Within this framework, we compare the separating power of different languages and investigate the complexity of deciding separability. While weak separability is shown to be closely related to conservative extensions, strongly separating concepts coincide with Craig interpolants, for suitably defined encodings of the data and ontology. This enables us to transfer known results from those fields to separability. Conversely, we obtain original results on separability that can be transferred backward. For example, rather surprisingly, conservative extensions and weak separability in ALCO are both 3ExpTime-complete.

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3EXPTIME-complete in general, and 2EXPTIME-complete if the arity of relation symbols is bounded by a constant c ≥ 3. In FO2, we prove a CON2EXPTIME upper bound and a 2EXPTIME lower bound for both problems. Thus, both for GF and FO2existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2under standard complexity assumptions).

We study two forms of least general generalizations in description logic, the least common subsumer (LCS) and most specific concept (MSC). While the LCS generalizes from examples that take the form of concepts, the MSC generalizes from individuals in data. Our focus is on the complexity of existence and verification, the latter meaning to decide whether a candidate concept is the LCS or MSC. We consider cases with and without a background TBox and a target signature. Our results range from coNP-complete for LCS and MSC verification in the description logic εℒ without TBoxes to undecidability of LCS and MSC verification and existence in εℒI with TBoxes. To obtain results in the presence of a TBox, we establish a close link between the problems studied in this paper and concept learning from positive and negative examples. We also give a way to regain decidability in εℒI with TBoxes and study single example MSC as a special case.

Finding a logical formula that separates positive and negative examples given in the form of labeled data items is fundamental in applications such as concept learning, reverse engineering of database queries, and generating referring expressions. In this paper, we investigate the existence of a separating formula for incomplete data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and three important fragments thereof: the description logic ALCI, the guarded fragment, and the two-variable fragment. We consider several forms of separability that differ in the treatment of negative examples and in whether or not they admit the use of additional helper symbols to achieve separation. We characterize separability in a model-theoretic way, compare the separating power of the different languages, and determine the computational complexity of separability as a decision problem.

We consider fragments of the description logic SHOIF extended with regular expressions on roles. Our main result is that satisfiability and finite satisfiability are decidable in two fragments SHOIF^1 and SHOIF^2, NExpTime-complete for the former and in 2NExpTime for the more expressive latter fragment. Both fragments impose restrictions on regular role expressions of the form r*. SHOIF^1 encompasses the extension of SHOIF with transitive closure of roles (when functional roles have no subroles) and the modal logic of linear orders and successor, with converse. Consequently, these logics are also decidable and NExpTime-complete.

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF_bot. The same results hold for inseparability and entailment.

We study ontology-mediated querying over probabilistic data for the case when the ontology is formulated in EL(hdr), an expressive member of the EL family of description logics. We leverage techniques that have been developed (i) for classical ontology-mediated querying and (ii) for probabilistic logic programming and provide an implementation based on our findings. We include both theoretical considerations and an experimental evaluation of our approach.

Learning description logic (DL) concepts from positive and negative examples given in the form of labeled data items in a KB has received significant attention in the literature. We study the fundamental question of when a separating DL concept exists and provide useful model-theoretic characterizations as well as complexity results for the associated decision problem. For expressive DLs such as ALC and ALCQI, our characterizations show a surprising link to the evaluation of ontology-mediated conjunctive queries. We exploit this to determine the combined complexity (between ExpTime and NExpTime) and data complexity (second level of the polynomial hierarchy) of separability. For the Horn DL EL, separability is ExpTime-complete both in combined and in data complexity while for its modest extension ELI it is even undecidable. Separability is also undecidable when the KB is formulated in ALC and the separating concept is required to be in EL or ELI.

We study the description logic SQ with number restrictions applicable to transitive roles, extended with either nominals or inverse roles. We show tight 2EXPTIME upper bounds for unrestricted entailment of regular path queries for both extensions and finite entailment of positive existential queries for nominals. For inverses, we establish 2EXPTIME-completeness for unrestricted and finite entailment of instance queries (the latter under restriction to a single, transitive role).

Horn description logics are syntactically defined fragments of standard description logics that fall within the Horn fragment of first-order logic and for which ontology-mediated query answering is in PTime for data complexity. They were independently introduced in modal logic to capture the intersection of Horn first-order logic with modal logic. In this paper, we introduce model comparison games for the basic Horn description logic hornALC (corresponding to the basic Horn modal logic) and use them to obtain an Ehrenfeucht-Fraïssé type definability result and a van Benthem style expressive completeness result for hornALC. We also establish a finite model theory version of the latter. The Ehrenfeucht-Fraïssé type definability result is used to show that checking hornALC indistinguishability of models is ExpTime-complete, which is in sharp contrast to ALC indistinguishability (i.e., bisimulation equivalence) checkable in PTime. In addition, we explore the behavior of Horn fragments of more expressive description and modal logics by defining a Horn guarded fragment of first-order logic and introducing model comparison games for it.

The unary negation fragment of first-order logic (UNFO) has recently been proposed as a generalization of modal logic that shares many of its good computational and model-theoretic properties. It is attractive from the perspective of database theory because it can express conjunctive queries (CQs) and ontologies formulated in many description logics (DLs). Both are relevant for ontology-mediated querying and, in fact, CQ evaluation under UNFO ontologies (and thus also under DL ontologies) can be `expressed' in UNFO as a satisfiability problem. In this paper, we consider the natural extension of UNFO with regular expressions on binary relations. The resulting logic UNFOreg can express (unions of) conjunctive two-way regular path queries (C2RPQs) and ontologies formulated in DLs that include transitive roles and regular expressions on roles. Our main results are that evaluating C2RPQs under UNFOreg ontologies is decidable, 2ExpTime-complete in combined complexity, and coNP-complete in data complexity, and that satisfiability in UNFOreg is 2ExpTime-complete, thus not harder than in UNFO.

We introduce the query-by-example (QBE) paradigm for query answering in the presence of ontologies. Intuitively, QBE permits non-expert users to explore the data by providing examples of the information they (do not) want, which the system then generalizes into a query. Formally, we study the following question: given a knowledge base and sets of positive and negative examples, is there a query that returns all positive but none of the negative examples? We focus on description logic knowledge bases with ontologies formulated in Horn-ALCI and (unions of) conjunctive queries. Our main contributions are characterizations, algorithms and tight complexity bounds for QBE.

We study description logics (DLs) supporting number restrictions on transitive roles. We first take a look at SOQ and SON with binary and unary coding of numbers, and provide algorithms for the satisfiability problem and tight complexity bounds ranging from EXPTIME to NEXPTIME. We then show that by allowing for counting only up to one (functionality), inverse roles and role inclusions can be added without losing decidability. We finally investigate DLs of the DL-Lite-family, and show that, in the presence of role inclusions, the core fragment becomes undecidable.

We investigate the decidability and computational complexity of (deductive) conservative extensions in fragments of first-order logic (FO), with a focus on the two-variable fragment FO$^2$ and the guarded fragment GF. We prove that conservative extensions are undecidable in any FO fragment that contains FO$^2$ or GF (even the three-variable fragment thereof), and that they are decidable and 2\ExpTime-complete in the intersection GF$^2$ of FO$^2$ and GF.

We investigate the decidability and computational complexity of query conservative extensions in Horn description logics (DLs) with inverse roles. This is more challenging than without inverse roles because characterizations in terms of unbounded homomorphisms between universal models fail, blocking the standard approach to establishing decidability. We resort to a combination of automata and mosaic techniques, proving that the problem is 2EXPTIME-complete in Horn-ALCHIF (and also in Horn-ALC and in ELI). We obtain the same upper bound for deductive conservative extensions, for which we also prove a coNEXPTIME lower bound.

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable.

We introduce metric temporal description logics (mTDLs) as combinations of the classical description logic ALC with (a) LTLbin, an extension of the temporal logic LTL with succinctly represented intervals, and (b) metric temporal logic MTL, extending LTLbinwith capabilities to quantitatively reason about time delays. Our main contributions are algorithms and tight complexity bounds for the satisfiability problem in these mTDLs: For mTDLs based on (fragments of) LTLbin, we establish complexity bounds ranging from EXPTIME to 2EXPSPACE. For mTDLs based on (fragments of) MTL interpreted over the naturals, we establish complexity bounds ranging from EXPSPACE to 2EXPSPACE.

We study the satisfiability problem of the logic K 2 = K × K—the two-dimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far, the best-known lower bound is NEXP-hardness shown by Marx and Mikulás in 2001. Our first main result closes this complexity gap. We show that satisfiability in K 2 is nonelementary. More precisely, we prove that it is k -NEXP-complete, where k is the switching depth (the minimal modal rank among the two dimensions) of the input formula, hereby solving a conjecture of Marx and Mikulás. Using our lower-bound technique also allows us to derive nonelementary lower bounds for the two-dimensional modal logics K4 × K and S5 2 × K, for which only elementary lower bounds were previously known. Moreover, we apply our technique to prove nonelementary lower bounds for the sizes of Feferman-Vaught decompositions with respect to product for any decomposable logic that is at least as expressive as unimodal K, generalizing a recent result by the first author and Lin. For the three-variable fragment FO 3 of first-order logic, we obtain the following two immediate corollaries: the size of Feferman-Vaught decompositions with respect to disjoint sum are inherently nonelementary, and equivalent formulas in Gaifman normal form are inherently nonelementary. Our second main result consists in providing effective elementary (more precisely, doubly exponential) upper bounds for the two-variable fragment FO 2 of first-order logic both for Feferman-Vaught decompositions and for equivalent formulas in Gaifman normal form.

We show that the satisfiability problem for the two-dimensional extension KxK of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulas from 2001. Our lower bound technique allows us to derive further lower bounds for many-dimensional modal logics for which only elementary lower bounds were previously known. We also derive nonelementary lower bounds on the sizes of Feferman-Vaught decompositions w.r.t. product for any decomposable logic that is at least as expressive as unimodal K. Finally, we study the sizes of Feferman-Vaught decompositions and formulas in Gaifman normal form for fixed-variable fragments of first-order logic.

We study probabilistic variants of the description logic EL. For the case where probabilities apply only to concepts, we provide a careful analysis of the borderline between tractability and ExpTime-completeness. One outcome is that any probability value except zero and one leads to intractability in the presence of general TBoxes, while this is not the case for classical TBoxes. For the case where probabilities can also be applied to roles, we show PSpace-completeness. This result is (positively) surprising as the best previously known upper bound was 2-ExpTime and there were reasons to believe in completeness for this class.

Researchers are looking for alternatives to overcome the upcoming limits of conventional hardware technologies. Reversible logic thereby established itself as a promising direction so that several methods for synthesis, verification, and testing of reversible circuits have already been proposed. However, also methods for debugging, i.e., to determine error candidates in case of a failed verification, are required to complete the design flow. Even if first approaches have already been proposed, debugging of reversible circuits still is in the beginning. In this paper, we present an alternative method to automatically debug reversible circuits. We thereby focus on missing control errors -- an established error model in the design of reversible circuits. A new notion of an error candidate is proposed that relies on the observation of a necessary condition for error locations in reversible circuits. Using this notion, a set of error candidates is obtained that differs from the error candidates returned by previous methods. Thus, combining the approaches enhances the overall debugging flow. Experimental results demonstrate that a higher accuracy is obtained in significantly shorter run-time.