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Doctorial Thesis
Dissertation advisor: prof. Sebastian Rudolph and prof. Emanuel Kieroński, Date of
defence: June 25, 2024

Formal ontologies are of significant importance in artificial intelligence, playing a central role in the Semantic Web, ontology-based information integration, or peer-to-peer data management. In such scenarios, an especially prominent role is played by description logics (DLs) – a robust family of logical formalisms used to describe ontologies and serving as the logical underpinning of contemporary standardised ontology languages. To put knowledge bases to full use as core part of intelligent information systems, much attention is being devoted to the area of ontology-based data-access, with conjunctive queries and their generalisations such as positive conjunctive two-way regular path queries being employed as a fundamental querying formalism. The most expressive exemplars of description logics feature advanced constructors for roles and path expressions. Among the most powerful knowledge representation formalisms on the verge of decidability, are the DLs from the Z family. For its most expressive proponent, ZOIQ (a.k.a. ALCHb Self reg OIQ), featuring nominals (O), role inverses (I), and number restrictions (Q), querying is undecidable and even decidability of knowledge-base satisfiability is open, owing to the intricate interplay of the three mentioned features. Restricting the interaction of O, I, and Q however (or excluding one of the features altogether) leads to beneficial model-theoretic properties, which give rise to upper bounds of ExpTime for knowledge-base satisfiability and 2ExpTime for querying. Aiming for better understanding of the “expressive power versus computational complexity” trade-off for the Z family of DLs, we provide a more fine-grained complexity analysis for the query entailment problem over ontologies. In the thesis we focus on tame fragments of ZOIQ, namely the fragments in which either one the three features from {I, O, Q} is dropped or the class of models is restricted to the so-called quasi-forests. We employ the query languages ranging from (unions of) conjunctive queries ((U)CQs) to positive two-way regular path queries (P2RPQs). We mostly follow the classical semantics of entailment, but we also provide several results in the “finite-model” scenario. The most important results of the thesis are summarised below. 1. We provide a complete classification of the complexity of the query entailment problem (for various query languages discussed above) for tamed fragments of ZOIQ under the classical semantics. This involves several new ingredients such as: (i) a uniform exponential-time algorithm based on Lutz’s spoiler technique for the entailment of unions of conjunctive queries for ALCHbreg, (ii) new lower bounds for (rooted and unrooted) conjunctive query entailment over ALCSelf ontologies, and (iii) a novel reduction from the entailment of P2RPQs to the satisfiability problem for tamed ZOIQ, yielding a uniform 2ExpTime upper bound for all the considered logics. As a preliminary step towards lifting the above results to the realm of data complexity, we establish that the satisfiability of tamed ZOIQ is NP-complete. 2. Under the finite model semantics, we focus on UCQs only. With the proviso that the finite satisfiability problem for ZIQ is ExpTime-complete, we also provide a complete picture of the complexity of the query entailment problems. The key insight is that ZOI and ZOQ are finitely controllable. 3. We conclude the thesis by investigating the decidability border of further extensions of the Z family of DLs. Our goal is to understand whether the class of regular languages present in path expressions in Z is maximal for guaranteeing decidability of the underlying logic. We provide a series of undecidability results involving simple, non-regular languages (a subclass of visibly pushdown languages). Our proofs rely on well-established model- and graph-theoretic definitions. What is more, most of them generalise (in a uniform way) and solidify multiple results known by the DL community. Our proofs are also easily adjustable to freshly defined logics, without the need to reproduce nearly-identical proofs.

Journal Papers
Proceedings of the 33rd International Joint Conference on Artificial Intelligence (IJCAI 2024) **To appear.**

We investigate the data complexity of the satisfiability problem for the very expressive description logic ZOIQ (a.k.a. ALCHb^self_regOIQ) over quasi-forests and establish its NP-completeness. This completes the data complexity landscape for decidable fragments of ZOIQ, and reproves known results on decidable fragments of OWL2 (SR family). Using the same technique, we establish coNExpTime-completeness (w.r.t. the combined complexity) of the entailment problem of rooted queries in ZIQ.

Journal Papers
Logical Methods in Computer Science (LMCS). **Under revision.**

Order-invariant first-order logic is an extension of first-order logic (FO) where formulae can make use of a linear order on the structures, under the proviso that they are order-invariant, i.e. that their truth value is the same for all linear orders. We continue the study of the two-variable fragment of order-invariant first-order logic initiated by Zeume and Harwath, and study its complexity and expressive power. We first establish coNExpTime-completeness for the problem of deciding if a given two-variable formula is order-invariant, which tightens and significantly simplifies the coN2ExpTime proof by Zeume and Harwath. Second, we address the question of whether every property expressible in order-invariant two-variable logic is also expressible in first-order logic without the use of a linear order. While we were not able to provide a satisfactory answer to the question, we suspect that the answer is ``no''. To justify our claim, we present a class of finite tree-like structures (of unbounded degree) in which a relaxed variant of order-invariant two-variable FO expresses properties that are not definable in plain FO. On the other hand, we show that if one restricts their attention to classes of structures of bounded degree, then the expressive power of order-invariant two-variable FO is contained within FO.

Journal Papers
Logical Methods in Computer Science (LMCS)

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.

Journal Papers
Journal of Artificial Intelligence Research (JAIR)

It is commonly known that the conjunctive query entailment problem for certain ex- tensions of (the well-known ontology language) ALC is computationally harder than their knowledge base satisfiability problem while for others the complexities coincide, both under the standard and the finite-model semantics. We expose a uniform principle behind this divide by identifying a wide class of (finitely) locally-forward description logics, for which we prove that (finite) query entailment problem can be solved by a reduction to exponentially many calls of the (finite) knowledge base satisfiability problem. Consequently, our algorithm yields tight ExpTime upper bounds for locally-forward logics with ExpTime-complete knowledge base satisfiability problem, including logics between ALC and μALCHbregQ (and more), as well as ALCSCC with global cardinality constraints, for which the complexity of querying remained open. Moreover, to make our technique applicable in future research, we provide easy-to-check sufficient conditions for a logic to be locally-forward based on several novel versions of the model-theoretic notion of unravellings. Together with existing results, this provides a nearly complete classification of the “benign” vs. “malign” primitive modelling features extending ALC, missing out only the Self operator. We then show a rather surprising result, namely that the conjunctive entailment problem for ALCSelf is exponentially harder than for ALC. This places the seemingly innocuous Self operator among the “malign” modelling features, like inverses, transitivity or nominals.

Conference Papers
Proceedings of the 18th Edition of the European Conference on Logics in Artificial Intelligence (JELIA 2023)

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics. Our primary object of interest is ALCvpl, an extension of ALC with path expressions using visibly-pushdown languages, which was shown to be decidable by Löding et al. in 2007. The paper present a series of undecidability results. We prove undecidability of ALCvpl with the seemingly innocent Self operator. Then, we consider the simplest non-regular (visibly-pushdown) language r#s# := {rnsn | n ∈ N}. We establish un- decidability of the concept satisfiability problem for ALCreg extended with nominals and r#s#, as well as of the query entailment problem for ALC-TBoxes, where such non-regular atoms are present in queries.

Conference Papers
Proceedings of the 50th EATCS International Colloquium on Automata, Languages and Programming (ICALP 2023)

We define the adjacent fragment AF of first-order logic, obtained by restricting the sequences of variables occurring as arguments in atomic formulas. The adjacent fragment generalizes (after a routine renaming) two-variable logic as well as the fluted fragment. We show that the adjacent fragment has the finite model property, and that its satisfiability problem is no harder than for the fluted fragment (and hence is Tower-complete). We further show that any relaxation of the adjacency condition on the allowed order of variables in argument sequences yields a logic whose satisfiability and finite satisfiability problems are undecidable. Finally, we study the effect of the adjacency requirement on the well-known guarded fragment (GF) of first-order logic. We show that the satisfiability problem for the guarded adjacent fragment (GA) remains 2ExpTime-hard, thus strengthening the known lower bound for GF.

Journal Papers
ACM Transactions on Computational Logic

We study the expressivity and complexity of two modal logics interpreted on inite forests and equipped with standard modalities to reason on submodels. The logic ML( ) extends the modal logic K with the composition operator from ambient logic, whereas ML(∗) features the separating conjunction ∗ from separation logic. Both operators are second-order in nature. We show that ML( ) is as expressive as the graded modal logic GML (on trees) whereas ML(∗) is strictly less expressive than GML. Moreover, we establish that the satisfiability problem is Tower-complete for ML(∗), whereas it is (only) AExpPolcomplete for ML( ), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.

Journal Papers
Logical Methods in Computer Science

Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.

Conference Papers
Proceedings of the 28th Workshop on Logic, Language, Information and Computation (WOLLIC 2022)

We introduce two versions of Presburger Automata with the Büchi acceptance condition, working over infinite, finite-branching trees. These automata, in addition to the classical ones, allow nodes for checking linear inequalities over labels of their children. We establish tight NP and ExpTime bounds on the complexity of the non-emptiness problem for the presented machines. We demonstrate the usefulness of our automata models by polynomially encoding the two-variable guarded fragment extended with Presburger constraints, improving the existing triply-exponential upper bound to a single exponential.

Conference Papers
Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it. We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the relative expressive power of ordered logics, and to characterise our logics as bisimulation-invariant fragments of FO à la van Benthem. Afterwards, we study the Craig Interpolation Property (CIP). We refute yet another claim from the infamous work by Purdy, by showing that the fluted and forward fragments do not enjoy CIP. We complement this result by showing that the ordered fragment and the guarded ordered logics enjoy CIP. These positive results rely on novel and quite intricate model constructions, which take full advantage of the ``forwardness'' of our logics.

Workshop Papers
Proceedings of the Description Logics Workshop 2022

A categorical approach to study model comparison games in terms of comonads was recently initiated by Abramsky et al. In this work, we analyse games that appear naturally in the context of description logics and supplement them with suitable game comonads. More precisely, we consider expressive sublogics of ALCSelf IbO, namely, the logics that extend ALC with any combination of inverses, nominals, safe boolean roles combinations and the Self operator. Our construction augments and modifies the so-called modal comonad by Abramsky and Shah. The approach that we took heavily relies on the use of relative monads, which we leverage to encapsulate additional capabilities within the bisimulation games in a compositional manner.

Conference Papers
Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI 2022)

In the last few years the field of logic-based knowledge representation took a lot of inspiration from database theory. A vital example is that the finite model semantics in description logics (DLs) is reconsidered as a desirable alternative to the classical one and that query entailment has replaced knowledge-base satisfiability (KBSat) checking as the key inference problem. However, despite the considerable effort, the overall picture concerning finite query answering in DLs is still incomplete. In this work we study the complexity of finite entailment of local queries (conjunctive queries and positive boolean combinations thereof) in the Z family of DLs, one of the most powerful KR formalisms, lying on the verge of decidability. Our main result is that the DLs ZOQ and ZOI are finitely controllable, i.e. that their finite and unrestricted entailment problems for local queries coincide. This allows us to reuse recently established upper bounds on querying these logics under the classical semantics. While we will not solve finite query entailment for the third main logic in the Z family, ZIQ, we provide a generic reduction from the finite entailment problem to the finite KBSat problem, working for ZIQ and some of its sublogics. Our proofs unify and solidify previously established results on finite satisfiability and finite query entailment for many known DLs.

Conference Papers
Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI 2022)

Abstract: In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be EXPTIME-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We contribute to this line of results by showing the surprising fact that even extending ALC by just the Self operator – which proved innocuous in many other contexts – increases the complexity of CQ entailment to 2EXPTIME. As common for this type of problem, our proof establishes a reduction from alternating Turing machines running in exponential space, but several novel ideas and encoding tricks are required to make the approach work in that specific, restricted setting.

Conference Papers
Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning. In this work we investigate the extensions of these classical decidable logics with percentage quantifiers, specifying how frequently a formula is satisfied in the indented model. We show, surprisingly, that all the mentioned decidable fragments become undecidable under such extension, sharpening the existing results in the literature. Our negative results are supplemented by decidability of the two-variable guarded fragment with even more expressive counting, namely Presburger constraints. Our results can be applied to infer decidability of various modal and description logics, e.g. Presburger Modal Logics with Converse or ALCI, with expressive cardinality constraints.

Journal Papers
Theory and Practice of Logic Programming

A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames have already been established. By ”traditional” classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

We show that the consistency problem for Statistical EL ontologies, defined by Penaloza and Potyka in SUM 2017, is ExpTime-hard. Together with existing ExpTime upper bounds, we conclude ExpTime-completeness of the logic. Our proof goes via a reduction from the consistency problem for EL extended with negation of atomic concepts.

Conference Papers
**[Best student paper award]** Proceedings of the 17th Edition of the European Conference on Logics in Artificial Intelligence (JELIA 2021)

We study the complexity of two standard reasoning problems for Forward Guarded Logic (FGF), obtained as a restriction of the Guarded Fragment in which variables appear in atoms only in the order of their quantification. We show that FGF enjoys the higher-arity-forest- model property, which results in ExpTime-completeness of its (finite and unrestricted) knowledge-base satisfiability problem. Moreover, we show that FGF is well-suited for knowledge representation. By employing a generalisation of Lutz’s spoiler technique, we prove that the conjunctive query entailment problem for FGF remains in ExpTime. We find that our results are quite unusual as FGF is, up to our knowledge, the first decidable fragment of First-Order Logic, extending standard description logics like ALC, that offers unboundedly many variables and higher-arity relations while keeping its complexity surprisingly low.

Conference Papers
Proceedings of the 24th International Conference on Foundations of Software Science and Computation Structures (FOSSACS 2021).

Linear Temporal Logic (LTL) interpreted on finite traces is a robust specification framework popular in formal verification. However, despite the high interest in the logic in recent years, the topic of their quantitative extensions is not yet fully explored. The main goal of this work is to study the effect of adding weak forms of percentage constraints (e.g. that most of the positions in the past satisfy a given condition, or that sigma is the most-frequent letter occurring in the past) to fragments of LTL. Such extensions could potentially be used for the verification of influence networks or statistical reasoning. Unfortunately, as we prove in the paper, it turns out that percentage extensions of even tiny fragments of LTL have undecidable satisfiability and model-checking problems. Our undecidability proofs not only sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple. We also show that the undecidability can be avoided by restricting the allowed usage of the negation, and briefly discuss how the undecidability results transfer to first-order logic on words.

Conference Papers
Proceedings of the 27th International Symposium on Temporal Representation and Reasoning (TIME 2020).

We consider the satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers, interpreted over finite words with data, denoted here with C2[≤,succ,∼,πbin]. In our scenario, we allow for using arbitrary many uninterpreted binary predicates from ,πbin, two navigational predicates ≤ and succ over word positions as well as a data-equality predicate ~. We prove that the obtained logic is undecidable, which contrasts with the decidability of the logic without counting by Montanari, Pazzaglia and Sala [MontanariPS16]. We supplement our results with decidability for several sub-fragments of C2[≤,succ,∼,πbin], e.g., without binary predicates, without successor succ, or under the assumption that the total number of positions carrying the same data value in a data-word is bounded by an a priori given constant.

Conference Papers
Proceedings of the 29th International Joint Conference on Artificial Intelligence and the 17th Pacific Rim International Conference on Artificial Intelligence (IJCAI 2020).

The chase is a famous algorithmic procedure in database theory with numerous applications in ontology-mediated query answering. We consider static analysis of the chase termination problem, which asks, given set of TGDs, whether the chase terminates on all input databases. The problem was recently shown to be undecidable by Gogacz et al. for sets of rules containing only ternary predicates. In this work, we show that undecidability occurs already for sets of single-head TGD over binary vocabularies. This question is relevant since many real-world ontologies, e.g., those from the Horn fragment of the popular OWL, are of this shape.

Conference Papers
Proceedings of the 29th International Joint Conference on Artificial Intelligence and the 17th Pacific Rim International Conference on Artificial Intelligence (IJCAI 2020).

Description logics are well-known logical formalisms for knowledge representation. We propose to enrich knowledge bases (KBs) with dynamic axioms that specify how the satisfaction of statements from the KBs evolves when the interpretation is decomposed or recomposed, providing a natural means to predict the evolution of interpretations. Our dynamic axioms borrow logical connectives from separation logics, well-known specification languages to verify programs with dynamic data structures. In the paper, we focus on ALC and EL augmented with dynamic axioms, or to their subclass of positive dynamic axioms. The knowledge base consistency problem in the presence of dynamic axioms is investigated, leading to interesting complexity results, among which the problem for EL with positive dynamic axioms is tractable, whereas EL with dynamic axioms is undecidable.

Conference Papers
Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2020

We investigate the expressivity and computational complexity of two modal logics on finite forests equipped with operators to reason on submodels. The logic ML(|) extends the basic modal logic ML with the composition operator | from static ambient logic, whereas ML(∗) contains the separating conjunction ∗ from separation logic. Though both operators are second-order in nature, we show that ML(|) is as expressive as the graded modal logic GML (on finite trees) whereas ML(∗) lies strictly between ML and GML. Moreover, we establish that the satisfiability problem for ML(∗) is Tower-complete, whereas for ML(|) is (only) AExpPol-complete.As a by-product, we solve several open problems related to sister logics, such as static ambient logic, modal separation logic, and second-order modal logic on finite trees.s

Conference Papers
Proceedings of the 24th European Conference on Artificial Intelligence (ECAI 2020).

Abstract. 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. We prove that the added expressivity does not increase the complexity of satisfiability checking and other standard inference problems. However, reasoning in ALCSCC++ becomes undecidable if inverse roles are added or conjunctive query entailment is considered. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. In this setting, query entailment can be shown to be EXPTIME-complete and hence not harder than reasoning in ALC.

We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove NP-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.

Conference Papers
Proceedings of the 28th International Joint Conference on Artificial Intelligence, IJCAI 2019

Among the most expressive knowledge representation formalisms are the description logics of the Z family. For well-behaved fragments of ZOIQ, entailment of positive two-way regular path queries is well known to be 2EXPTIMEcomplete under the proviso of unary encoding of numbers in cardinality constraints. We show that this assumption can be dropped without an increase in complexity and EXPTIME-completeness can be achieved when bounding the number of query atoms, using a novel reduction from query entailment to knowledge base satisfiability. These findings allow to strengthen other results regarding query entailment and query containment problems in very expressive description logics. Our results also carry over to GC2 , the two-variable guarded fragment of firstorder logic with counting quantifiers, for which hitherto only conjunctive query entailment has been investigated.

Workshop Papers
Proceedings of the Description Logics Workshop 2019

Among the most expressive knowledge representation formalisms are the description logics of the Z family. For well-behaved fragments of ZOIQ, entailment of positive two-way regular path queries is well known to be 2EXPTIMEcomplete under the proviso of unary encoding of numbers in cardinality constraints. We show that this assumption can be dropped without an increase in complexity and EXPTIME-completeness can be achieved when bounding the number of query atoms, using a novel reduction from query entailment to knowledge base satisfiability. These findings allow to strengthen other results regarding query entailment and query containment problems in very expressive description logics. Our results also carry over to GC2 , the two-variable guarded fragment of firstorder logic with counting quantifiers, for which hitherto only conjunctive query entailment has been investigated.

Workshop Papers
Proceedings of the Description Logics Workshop 2019

We consider an expressive description logic (DL) in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we show that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, 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.

Conference Papers
Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019

Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability whereas CTL with propositional quantification under the tree semantics (written QCTLt) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of QCTLt as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that QCTLt restricted to the temporal operator EX is already Tower-hard, which is unexpected as, for instance, EX can only enforce local properties. When QCTLt restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExppol-complete; AExppol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As significant consequences of our proof method, we prove Tower-hardness of QCTLt restricted to EF or to EXEF and of the well-known modal logics K, KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees.

Conference Papers
Proceedings of the 16th Edition of the European Conference on Logics in Artificial Intelligence (JELIA 2019)

A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By 'traditional' classes of frames we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

Conference Papers
Proceedings of the 37th IARCS Annual Conference on Foundations of
Software Technology and Theoretical Computer Science, FSTTCS 2017,
December 11–15, 2017, Kanpur, India. LIPIcs 93, Schloss Dagstuhl -
Leibniz-Zentrum fuer Informatik 2017, pages 12:1–12:16.

We consider the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees. We prove a small-model property of this logic, which gives a technique for deciding the satisfiability problem. In the case of words this gives a new proof of EXPSPACE upper bound, and in the case of trees it gives a 2EXPTIME algorithm. This algorithm is optimal: we prove a matching lower bound by a generic reduction from alternating Turing machines working in exponential space; the reduction involves a development of a new version of tiling games.

Conference Papers
Proceedings of the 26th EACSL Annual Conference on Computer Science
Logic, CSL 2017, August 20-24, 2017, Stockholm, Sweden. LIPIcs 82,
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017, pages
11:1-11:20

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting quantifiers. It turns out that the two logics are equally expressive, although counting quantifiers do add expressive power in the restricted case of unordered trees.

Conference Papers
Proceedings of ESSLLI 2017 Student Session, 29th European Summer
School in Logic, Language & Information, July 17-28, 2017,
Toulouse, France, pages 7-13

We consider the one-variable fragment of first-order logic extended with modulo counting quantifiers. We prove NPTime-completeness of such fragment by presenting an optimal algorithm for solving its finite satisfiability problem.

Bachelor Thesis
Dissertation advisor: prof. dr hab. Witold Charatonik, Date of
defence: Feb. 15, 2017

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We show that adding counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem.