Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

Outlier detection in default logics: the tractability/intractability frontier

In default theories, outliers denote sets of literals featuring unexpected properties. In previous papers, we have defined outliers in default logics and investigated their formal properties. Specifically, we have looked into the computational complexity of outlier detection problems and proved that while they are generally intractable, interesting tractable cases can be singled out. Following those results, we study here the tractability frontier in outlier detection problems, by analyzing it with respect to (i) the considered outlier detection problem, (ii) the reference default logic fragme…

Charting the Tractability Frontier of Certain Conjunctive Query Answering

An uncertain database is defined as a relational database in which primary keys need not be satisfied. A repair (or possible world) of such database is obtained by selecting a maximal number of tuples without ever selecting two distinct tuples with the same primary key value. For a Boolean query q, the decision problem CERTAINTY(q) takes as input an uncertain database db and asks whether q is satisfied by every repair of db. Our main focus is on acyclic Boolean conjunctive queries without self-join. Previous work has introduced the notion of (directed) attack graph of such queries, and has pro…

On the Complexity of Existential Positive Queries

We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted to fall into the set; a natural question is then to classify which sentence sets are tractable and which are intractable. With respect to fixed-parameter tractability, we give a general theorem that reduces this classification question to the corresponding question for primitive positive logic, for a variety of representations of structures. This general theo…

Fixed-parameter tractability, definability, and model checking

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tracta…

Complexity and (un)decidability of fragments of $\langle ω^{ω^λ}; \times \rangle$

We specify the frontier of decidability for fragments of the first-order theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of $\langle ω^{ω^λ}; \times, ω, ω+1, ω^2+1 \rangle$ for every ordinal $λ$. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the $\exists^*\forall^{6}$-fragment of $\langle ω^{ω^λ}; \times \rangle$ is undecidable for every ordinal $λ$.

一种新的可计算描述逻辑在范畴语义下

A New Tractable Description Logic under Categorical Semantics

本文提出一种扩展EL的逻辑，通过弱化否定来保留可计算性，同时引入范畴语义以处理负知识。

Complexity Thresholds in Inclusion Logic

Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in these logics are relatively well understood. Inclusion logic is similar to these team-based logical formalisms with the exception that it corresponds to deterministic polynomial time in ordered models. In this article we examine connections between syntactical fragments of inclusion logic and different complexity classes in terms of two computational problems: …

A Parameterized Complexity View on Description Logic Reasoning

Description logics are knowledge representation languages that have been designed to strike a balance between expressivity and computational tractability. Many different description logics have been developed, and numerous computational problems for these logics have been studied for their computational complexity. However, essentially all complexity analyses of reasoning problems for description logics use the one-dimensional framework of classical complexity theory. The multi-dimensional framework of parameterized complexity theory is able to provide a much more detailed image of the complex…

Algorithms and data structures for first-order logic with connectivity under vertex failures

We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer k, we can in fixed-parameter time construct a data structure that can later be used to answer queries of the form: ``are vertices s and t connected via a path that avoids vertices $u_1,..., u_k$?'' in time $2^{2^{O(k)}}$. In the terminology of the literature on data structures, this gives the first deterministic data structure for connectivity under vertex failures where for every fixed number of failures, all operations can be p…