On p-optimal proof systems and logics for PTIME

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More on Descriptive Complexity of Second-Order HORN Logics

This paper concerns Gradel's question asked in 1992: whether all problems which are in PTIME and closed under substructures are definable in second-order HORN logic SO-HORN. We introduce revisions of SO-HORN and DATALOG by adding first-order universal quantifiers over the second-order atoms in the bodies of HORN clauses and DATALOG rules. We show that both logics are as expressive as FO(LFP), the least fixed point logic. We also prove that FO(LFP) can not define all of the problems that are in PTIME and closed under substructures. As a corollary, we answer Gradel's question negatively

Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields

Worst-case Optimal Query Answering for Greedy Sets of Existential Rules and Their Subclasses

The need for an ontological layer on top of data, associated with advanced reasoning mechanisms able to exploit the semantics encoded in ontologies, has been acknowledged both in the database and knowledge representation communities. We focus in this paper on the ontological query answering problem, which consists of querying data while taking ontological knowledge into account. More specifically, we establish complexities of the conjunctive query entailment problem for classes of existential rules (also called tuple-generating dependencies, Datalog+/- rules, or forall-exists-rules. Our contribution is twofold. First, we introduce the class of greedy bounded-treewidth sets (gbts) of rules, which covers guarded rules, and their most well-known generalizations. We provide a generic algorithm for query entailment under gbts, which is worst-case optimal for combined complexity with or without bounded predicate arity, as well as for data complexity and query complexity. Secondly, we classify several gbts classes, whose complexity was unknown, with respect to combined complexity (with both unbounded and bounded predicate arity) and data complexity to obtain a comprehensive picture of the complexity of existential rule fragments that are based on diverse guardedness notions. Upper bounds are provided by showing that the proposed algorithm is optimal for all of them

Preliminary results on Ontology-based Open Data Publishing

Despite the current interest in Open Data publishing, a formal and comprehensive methodology supporting an organization in deciding which data to publish and carrying out precise procedures for publishing high-quality data, is still missing. In this paper we argue that the Ontology-based Data Management paradigm can provide a formal basis for a principled approach to publish high quality, semantically annotated Open Data. We describe two main approaches to using an ontology for this endeavor, and then we present some technical results on one of the approaches, called bottom-up, where the specification of the data to be published is given in terms of the sources, and specific techniques allow deriving suitable annotations for interpreting the published data under the light of the ontology

On the Complexity of ATL and ATL* Module Checking

Module checking has been introduced in late 1990s to verify open systems, i.e., systems whose behavior depends on the continuous interaction with the environment. Classically, module checking has been investigated with respect to specifications given as CTL and CTL* formulas. Recently, it has been shown that CTL (resp., CTL*) module checking offers a distinctly different perspective from the better-known problem of ATL (resp., ATL*) model checking. In particular, ATL (resp., ATL*) module checking strictly enhances the expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*) model checking. In this paper, we provide asymptotically optimal bounds on the computational cost of module checking against ATL and ATL*, whose upper bounds are based on an automata-theoretic approach. We show that module-checking for ATL is EXPTIME-complete, which is the same complexity of module checking against CTL. On the other hand, ATL* module checking turns out to be 3EXPTIME-complete, hence exponentially harder than CTL* module checking.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Horn rewritability vs PTime query evaluation for description logic TBoxes

We study the following question: if τ is a TBox that is formulated in an expressive DL L and all CQs can be evaluated in PTime w.r.t. τ, can τ be replaced by a TBox τ' that is formulated in the Horn-fragment of L and such that for all CQs and ABoxes, the answers w.r.t. τ and τ' coincide? Our main results are that this is indeed the case when L is the set of ALCHI or ALCIF TBoxes of quantifier depth 1 (which covers the majority of such TBoxes), but not for ALCHIF and ALCQ TBoxes of depth 1