Extending Two-Variable Logic on Trees

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

A NExpTime-Complete Description Logic Strictly Contained in C²

We examine the complexity and expressivity of the combination of the Description Logic ALCQI with a terminological formalism based on cardinality restrictions on concepts. This combination can naturally be embedded into C², the two variable fragment of predicate logic with counting quantifiers. We prove that ALCQI has the same complexity as C² but does not reach its expressive power.An abriged version of this paper has been submitted to CSL'9

Two-variable Logic with Counting and a Linear Order

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata

One-Dimensional Fragment Over Words and Trees

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one variable free. We investigate this fragment over words and trees, presenting a complete classification of the complexity of its satisfiability problem for various navigational signatures and comparing its expressive power with other important formalisms. These include the two-variable fragment with counting and the unary negation fragment.Peer reviewe

One-Dimensional Logic over Trees

A one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. This fragment contains two-variable logic, and it is known that over words both formalisms have the same complexity and expressive power. Here we investigate the one-dimensional fragment over trees. We consider unranked unordered trees accessible by one or both of the descendant and child relations, as well as ordered trees equipped additionally with sibling relations. We show that over unordered trees the satisfiability problem is ExpSpace-complete when only the descendant relation is available and 2ExpTime-complete with both the descendant and child or with only the child relation. Over ordered trees the problem remains 2ExpTime-complete. Regarding expressivity, we show that over ordered trees and over unordered trees accessible by both the descendant and child the one-dimensional fragment is equivalent to the two-variable fragment with counting quantifiers

Lifted Algorithms for Symmetric Weighted First-Order Model Sampling

Weighted model counting (WMC) is the task of computing the weighted sum of all satisfying assignments (i.e., models) of a propositional formula. Similarly, weighted model sampling (WMS) aims to randomly generate models with probability proportional to their respective weights. Both WMC and WMS are hard to solve exactly, falling under the $\#\mathsf{P}$-hard complexity class. However, it is known that the counting problem may sometimes be tractable, if the propositional formula can be compactly represented and expressed in first-order logic. In such cases, model counting problems can be solved in time polynomial in the domain size, and are known as domain-liftable. The following question then arises: Is it also the case for weighted model sampling? This paper addresses this question and answers it affirmatively. Specifically, we prove the domain-liftability under sampling for the two-variables fragment of first-order logic with counting quantifiers in this paper, by devising an efficient sampling algorithm for this fragment that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of cardinality constraints. To empirically verify our approach, we conduct experiments over various first-order formulas designed for the uniform generation of combinatorial structures and sampling in statistical-relational models. The results demonstrate that our algorithm outperforms a start-of-the-art WMS sampler by a substantial margin, confirming the theoretical results.Comment: 47 pages, 6 figures. An expanded version of "On exact sampling in the two-variable fragment of first-order logic" in LICS23, submitted to AIJ. arXiv admin note: substantial text overlap with arXiv:2302.0273

The Complexity of Reasoning with Cardinality Restrictions and Nominals in Expressive Description Logics

We study the complexity of the combination of the Description Logics ALCQ and ALCQI with a terminological formalism based on cardinality restrictions on concepts. These combinations can naturally be embedded into C^2, the two variable fragment of predicate logic with counting quantifiers, which yields decidability in NExpTime. We show that this approach leads to an optimal solution for ALCQI, as ALCQI with cardinality restrictions has the same complexity as C^2 (NExpTime-complete). In contrast, we show that for ALCQ, the problem can be solved in ExpTime. This result is obtained by a reduction of reasoning with cardinality restrictions to reasoning with the (in general weaker) terminological formalism of general axioms for ALCQ extended with nominals. Using the same reduction, we show that, for the extension of ALCQI with nominals, reasoning with general axioms is a NExpTime-complete problem. Finally, we sharpen this result and show that pure concept satisfiability for ALCQI with nominals is NExpTime-complete. Without nominals, this problem is known to be PSpace-complete

Efficient reasoning about data trees via integer linear programming

Data trees provide a standard abstraction of XML documents with data values: they are trees whose nodes, in addition to the usual labels, can carry labels from an infinite alphabet (data). Therefore, one is interested in decidable formalisms for reasoning about data trees. While some are known – such as the two-variable logic – they tend to be of very high complexity, and most decidability proofs are highly nontrivial. We are therefore interested in reasonable complexity formalisms as well as better techniques for proving decidability. Here we show that many decidable formalisms for data trees are subsumed – fully or partially – by the power of tree automata together with set constraints and linear constraints on cardinalities of various sets of data values. All these constraints can be translated into instances of integer linear programming, giving us an NP bound on the complexity of the reasoning tasks. We prove that this bound, as well as the key encoding technique, remain very robust, and allow the addition of features such as counting of paths and patterns, and even a concise encoding of constraints, without increasing the complexity. We also relate our results to several reasoning tasks over XML documents, such as satisfiability of schemas and data dependencies and satisfiability of the two-variable logic