Abstract We present deterministic model construction algorithms for sets of modal clauses saturated with respect to three refinements of the modal-layered resolution calculus implemented in the prover "Image missing". The model construction algorithms are inspired by the Bachmair-Ganzinger method for constructing a model for a set of ground first-order clauses saturated with respect to ordered resolution with selection. The challenge is that the inference rules of the modal-layered resolution calculus for modal operators are more restrictive than an adaptation of ordered resolution with selection for these would be. While these model construction algorithms provide an alternative means to proving completeness of the calculus, our main interest is the provision of a ‘certificate’ for satisfiable modal formulae that can be independently checked to assure a user that the result of "Image missing" is correct. This complements the existing provision of proofs for unsatisfiable modal formulae.

Abstract We are interested in widening the reasoning support for propositional modal logics in the so-called modal cube. The modal cube consists of extensions of the basic modal logic $$\textsf{K}_{}$$ K with an arbitrary combination of the modal axioms $$\textsf{B}$$ B , $$\textsf{D}$$ D , $$\textsf{T}$$ T , $$\textsf{4}$$ 4 and $$\textsf{5}$$ 5 . We revisit recently developed local reductions from all logics in the modal cube to a normal form comprising sets of clausal formulae with associated modal levels. We extend these reductions further to the basic modal logic $$\textsf{K}_{}$$ K , called definitional reductions. This enables any prover for $$\textsf{K}_{}$$ K to be used to solve the satisfiability problem for all logics in the modal cube. We also present alternative, axiomatic, reductions based on ideas originally proposed by Kracht, providing new theoretical results and improved bounds on the size of the reductions. We compare both sets of reductions combined with state-of-the-art provers for $$\textsf{K}_{}$$ K on a large set of parametric benchmarks for all logics in the modal cube. The results show that the provers perform better with reductions based on the clausal normal form than the axiomatic reductions.

Abstract We present novel reductions of extensions of the basic modal logic $${\textsf {K} }$$ K with axioms $$\textsf {B} $$ B , $$\textsf {D} $$ D , $$\textsf {T} $$ T , $$\textsf {4} $$ 4 and $$\textsf {5} $$ 5 to Separated Normal Form with Sets of Modal Levels $$\textsf {SNF} _{sml}$$ SNF sml . The reductions typically result in smaller formulae than the reductions by Kracht. The reductions to $$\textsf {SNF} _{sml}$$ SNF sml combined with a reduction to $$\textsf {SNF} _{ml}$$ SNF ml allow us to use the local reasoning of the prover $${\text {K}_{\text {S}}}{\text {P}}$$ K S P to determine the satisfiability of modal formulae in the considered logics. We show experimentally that the combination of our reductions with the prover $${\text {K}_{\text {S}}}{\text {P}}$$ K S P performs well when compared with a specialised resolution calculus for these logics, the built-in reductions of the first-order prover SPASS, and the higher-order logic prover LEO-III.

Abstract We present novel reductions of the propositional modal logics "Image missing" , "Image missing" , "Image missing" , "Image missing" and "Image missing" to Separated Normal Form with Sets of Modal Levels. The reductions result in smaller formulae than the well-known reductions by Kracht and allow us to use the local reasoning of the prover "Image missing" to determine the satisfiability of modal formulae in these logics. We show experimentally that the combination of our reductions with the prover "Image missing" performs well when compared with a specialised resolution calculus for these logics and with the b̆uilt-in reductions of the first-order prover SPASS.

This paper analyses models of a spatial logic with path operators based on the class of neighbourhood spaces, also called pretopological or closure spaces, a generalisation of topological spaces. For this purpose, we distinguish two dimensions: the type of spaces on which models are built, and the type of allowed paths. For the spaces, we investigate general neighbourhood spaces and the subclass of quasi-discrete spaces, which closely resemble graphs. For the paths, we analyse the cases of quasi-discrete paths, which consist of an enumeration of points, and topological paths, based on the unit interval. We show that the logic admits finite models over quasi-discrete spaces, both with quasi-discrete and topological paths. Finally, we prove that for general neighbourhood spaces, the logic does not have the finite model property, either for quasi-discrete or topological paths.

Abstract This paper presents an extension of the Predictive Runtime Verification (PRV) paradigm to consider multiple models of the System Under Analysis (SUA). We call this extension Multi-Model PRV. Typically, PRV attempts to predict the satisfaction or violation of a property based on a trace and a (single) formal model of the SUA. However, contemporary node- or component-based systems (e.g. robotic systems) may benefit from monitoring based on a model of each component. We show how a Multi-Model PRV approach can be applied in either a centralised or a compositional way (where the property is compositional), as best suits the SUA. Crucially, our approach is formalism-agnostic. We demonstrate our approach using an illustrative example of a Mars Curiosity rover simulation and evaluate our contribution via a prototype implementation.

We present a bisimulation relation for neighbourhood spaces, a generalisation of topological spaces. We show that this notion, path preserving bisimulation, preserves formulas of the spatial logic SLCS. We then use this preservation result to show that SLCS cannot express standard topological properties such as separation and connectedness. Furthermore, we compare the bisimulation relation with standard modal bisimulation and modal bisimulation with converse on graphs and prove it coincides with the latter.

We study ontology-mediated querying in the case where ontologies are formulated in the guarded fragment of first-order logic (GF) or extensions thereof with counting and where the actual queries are (unions of) conjunctive queries. Our aim is to classify the data complexity and Datalog rewritability of query evaluation depending on the ontology O , where query evaluation w.r.t. O is in PT ime (resp. Datalog rewritable) if all queries can be evaluated in PT ime w.r.t. O (resp. rewritten into Datalog under O ), and co NP-hard if at least one query is co NP-hard w.r.t. O . We identify several fragments of GF that enjoy a dichotomy between Datalog-rewritability (which implies PT ime ) and co NP-hardness as well as several other fragments that enjoy a dichotomy between PT ime and co NP-hardness, but for which PT ime does not imply Datalog-rewritability. For the latter, we establish and exploit a connection to constraint satisfaction problems. We also identify fragments for which there is no dichotomy between PT ime and co NP. To prove this, we establish a non-trivial variation of Ladner’s theorem on the existence of NP-intermediate problems. Finally, we study the decidability of whether a given ontology enjoys PT ime query evaluation, presenting both positive and negative results, depending on the fragment.

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.

In ontology-mediated querying with an expressive description logic L, two desirable properties of a TBox T are (1) being able to replace T with a TBox formulated in the Horn-fragment of L without affecting the answers to conjunctive queries, and (2) that every conjunctive query can be evaluated in PTime w.r.t. T. We investigate in which cases (1) and (2) are equivalent, finding that the answer depends on whether the unique name assumption (UNA) is made, on the description logic under consideration, and on the nesting depth of quantifiers in the TBox. We also clarify the relationship between query evaluation with and without UNA and consider natural variations of property (1).

We study the complexity of ontology-mediated querying when ontologies are formulated in the guarded fragment of first-order logic (GF). Our general aim is to classify the data complexity on the level of ontologies where query evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several large and relevant fragments of GF that enjoy a dichotomy between PTime and coNP, some of them additionally admitting a form of counting. In fact, almost all ontologies in the BioPortal repository fall into these fragments or can easily be rewritten to do so. We then establish a variation of Ladner's Theorem on the existence of NP-intermediate problems and use this result to show that for other fragments, there is provably no such dichotomy. Again for other fragments (such as full GF), establishing a dichotomy implies the Feder-Vardi conjecture on the complexity of constraint satisfaction problems. We also link these results to Datalog-rewritability and study the decidability of whether a given ontology enjoys PTime query evaluation, presenting both positive and negative results.