arXiv subjects
Publications and source records attributed to Frank Wolter.
We introduce Craig interpolation and related notions such as uniform interpolation, Beth definability, and theory decomposition in classical propositional logic. We present four approaches to computing interpolants: via quantifier elimination, from formulas in disjunctive normal form, and by extraction from resolution or tableau refutations. We close with a discussion of the size of interpolants and links to circuit complexity.
We investigate preservation results for the independent fusion of one-variable first-order modal logics. We show that, without equality, Kripke completeness and decidability of global and local consequence relations are preserved, under both expanding and constant domain semantics. By contrast, Kripke completeness and decidability are not preserved for fusions with equality and non-rigid constants (or, equivalently, counting up to one), again for the global and local consequence and under both expanding and constant domain semantics. This result is shown by encoding Diophantine equations. Even without equality, the finite model property is preserved only in the local case. Finally, we view fusions of one-variable modal logics as fusions of propositional modal logics sharing an S5 modality and provide a general sufficient condition for transfer of Kripke completeness and decidability (but not of finite model property).
Normal modal logics extending the logic K4.3 of linear transitive frames are known to lack the Craig interpolation property, except some logics of bounded depth such as S5. We turn this `negative' fact into a research question and pursue a non-uniform approach to Craig interpolation by investigating the following interpolant existence problem: decide whether there exists a Craig interpolant between two given formulas in any fixed logic above K4.3. Using a bisimulation-based characterisation of interpolant existence for descriptive frames, we show that this problem is decidable and coNP-complete for all finitely axiomatisable normal modal logics containing K4.3. It is thus not harder than entailment in these logics, which is in sharp contrast to other recent non-uniform interpolation results. We also extend our approach to Priorean temporal logics (with both past and future modalities) over the standard time flows-the integers, rationals, reals, and finite strict linear orders-none of which is blessed with the Craig interpolation property.
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.
We start a systematic investigation of the size of Craig interpolants, uniform interpolants, and strongest implicates for (quasi-)normal modal logics. Our main upper bound states that for tabular modal logics, the computation of strongest implicates can be reduced in polynomial time to uniform interpolant computation in classical propositional logic. Hence they are of polynomial dag-size iff NP is included in P/poly. The reduction also holds for Craig interpolants and uniform interpolants if the tabular modal logic has the Craig interpolation property. Our main lower bound shows an unconditional exponential lower bound on the size of Craig interpolants and strongest implicates covering almost all non-tabular standard normal modal logics. For normal modal logics contained in or containing S4 or GL we obtain the following dichotomy: tabular logics have ``propositionally sized'' interpolants while for non-tabular logics an unconditional exponential lower bound holds.
Traditionally, research on Craig interpolation is concerned with (a) establishing the Craig interpolation property (CIP) of a logic saying that every valid implication in the logic has a Craig interpolant and (b) designing algorithms that extract Craig interpolants from proofs. Logics that lack the CIP are regarded as `pathological' and excluded from consideration. In this chapter, we survey variations and generalisations of traditional Craig interpolation. First, we consider Craig interpolants for implications in logics without the CIP, focusing on the decidability and complexity of deciding their existence. We then generalise interpolation by looking for Craig interpolants in languages L' that can be weaker than the language L of the given implication. Thus, do not only we restrict the non-logical symbols of Craig interpolants but also the logical ones. The resulting L/L'-interpolation problem generalises L/L'-definability, the question whether an L-formula is equivalent to some L'-formula. After that, we move from logical languages to formal languages where interpolation disguises itself as separation: given two disjoint languages in a class C, does there exist a separating language in a smaller class C'? This question is particularly well-studied in the case when the input languages are regular and the separating language is first-order definable. Finally, we connect the different research strands by showing how the decidability of the separation problem for regular languages can be used to prove the decidability of Craig interpolant existence for linear temporal logic LTL.
None of the first-order modal logics between $\mathsf{K}$ and $\mathsf{S5}$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $\mathsf{S5}$: the one-variable fragment $\mathsf{Q^1S5}$ and its extension $\mathsf{S5}_{\mathcal{ALC}^u}$ that combines $\mathsf{S5}$ and the description logic$\mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $\mathsf{Q^1S5}$ and $\mathsf{S5}_{\mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $\mathsf{Q^1K}$ of first-order modal logic $\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable.
While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only holds, however, under the provision that non-rigid constants, definite descriptions and non-trivial counting are not admitted. Indeed, several monodic fragments having at least one of these features are known to be undecidable. We investigate these features systematically and show that fundamental monodic fragments such as the two-variable fragment with counting and the guarded fragment of standard first-order modal logics $\mathbf{K}_{n}$ and $\mathbf{S5}_{n}$ are decidable. Tight complexity bounds are established as well. Under the expanding-domain semantics, we show decidability of the basic modal logic extended with the transitive closure operator on finite acyclic frames; this logic, however, is Ackermann-hard.
For fragments L of first-order logic (FO) with counting quantifiers, we consider the definability problem, which asks whether a given L-formula can be equivalently expressed by a formula in some fragment of L without counting, and the more general separation problem asking whether two mutually exclusive L-formulas can be separated in some counting-free fragment of L. We show that separation is undecidable for the two-variable fragment of FO extended with counting quantifiers and for the graded modal logic with inverse, nominals and universal modality. On the other hand, if inverse or nominals are dropped, separation becomes coNExpTime- or 2ExpTime-complete, depending on whether the universal modality is present. In contrast, definability can often be reduced in polynomial time to validity in L. We also consider uniform separation and show that it often behaves similarly to definability.
While the computation of Craig interpolants for description logics (DLs) with the Craig Interpolation Property (CIP) is well understood, very little is known about the computation and size of interpolants for DLs without CIP or if one aims at interpolating concepts in a weaker DL than the DL of the input ontology and concepts. In this paper, we provide the first elementary algorithms computing (i) ALC-interpolants between ALC-concepts under ALCH-ontologies and (ii) ALC-interpolants between ALCQ-concepts under ALCQ-ontologies. The algorithms are based on recent decision procedures for interpolant existence. We also observe that, in contrast, uniform (possibly depth restricted) interpolants might be of non-elementary size.
Definite descriptions, such as 'the General Chair of KR 2024', are a semantically transparent device for object identification in knowledge representation. In first-order modal logic, definite descriptions have been widely investigated for their non-rigidity, which allows them to designate different objects (or none at all) at different states. We propose expressive modal description logics with non-rigid definite descriptions and names, and investigate decidability and complexity of the satisfaction problem. We first systematically link satisfiability for the one-variable fragment of first-order modal logic with counting to our modal description logics. Then, we prove a promising NEXPTIME-completeness result for concept satisfiability for the fundamental epistemic multi-agent logic $\mathbf{S5}^{n}$ and its neighbours, and show that some expressive logics that are undecidable with constant domain become decidable (but Ackermann-hard) with expanding domains. Finally, we conduct a fine-grained analysis of decidability of temporal logics.
Algorithms for learning database queries from examples and unique characterisations of queries by examples are prominent starting points for developing automated support for query construction and explanation. 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.
As well known, weak K4 and the difference logic DL do not enjoy the Craig interpolation property. Our concern here is the problem of deciding whether any given implication does have an interpolant in these logics. We show that the nonexistence of an interpolant can always be witnessed by a pair of bisimilar models of polynomial size for DL and of triple-exponential size for weak K4, and so the interpolant existence problems for these logics are decidable in coNP and coN3ExpTime, respectively. We also establish coNExpTime-hardness of this problem for weak K4, which is higher than the PSpace-completeness of its decision problem.
The separation problem for a class Q of database queries is to find a query in Q that distinguishes between a given set of `positive' and `negative' data examples. Separation provides explanations of examples and underpins the query-by-example paradigm to support database users in constructing and refining queries. As the space of all separating queries can be large, it is helpful to succinctly represent this space by means of its most specific (logically strongest) and general (weakest) members. We investigate this extremal separation problem for classes of instance queries formulated in linear temporal logic LTL with the operators conjunction, next, and eventually. Our results range from tight complexity bounds for verifying and counting extremal separators to algorithms computing them.
The interpolant existence problem (IEP) for a logic L is to decide, given formulas P and Q, whether there exists a formula I, built from the shared symbols of P and Q, such that P entails I and I entails Q in L. If L enjoys the Craig interpolation property (CIP), then the IEP reduces to validity in L. Recently, the IEP has been studied for logics without the CIP. The results obtained so far indicate that even though the IEP can be computationally harder than validity, it is decidable when L is decidable. Here, we give the first examples of decidable fragments of first-order logic for which the IEP is undecidable. Namely, we show that the IEP is undecidable for the two-variable fragment with two equivalence relations and for the two-variable guarded fragment with individual constants and two equivalence relations. We also determine the corresponding decidable Boolean description logics for which the IEP is undecidable.
Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022 a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade.
In reverse engineering of database queries, we aim to construct a query from a given set of answers and non-answers; it can then be used to explore the data further or as an explanation of the answers and non-answers. We investigate this query-by-example problem for queries formulated in positive fragments of linear temporal logic LTL over timestamped data, focusing on the design of suitable query languages and the combined and data complexity of deciding whether there exists a query in the given language that separates the given answers from non-answers. We consider both plain LTL queries and those mediated by LTL-ontologies.
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 this problem 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.