On the Decision Problem for Two-Variable First-Order Logic

Grädel, Erich; Kolaitis, Phokion G.; Vardi, Moshe Y.

doi:10.2307/421196

Cited by 249 publications

(248 citation statements)

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“…The two-variable fragment of first-order logic (two-variable logic or FO 2 for short) is known to be reasonably expressive and its satisfiability and finite satisfiability problems are decidable [Mor75], in fact they are complete for NEXPTIME [GKV97]. Unfortunately many important properties, as for example transitivity, cannot be expressed in two-variable logic.…”

Section: A Note On Two-variable Logicmentioning

confidence: 99%

Small Dynamic Complexity Classes

Zeume

2017

Lecture Notes in Computer Science

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Everything changes and nothing stands still. -Heraclitus (quoted by Platon in Cratylus) PrefaceOn the first few days of my PhD studies in the summer of 2009, my adviser Thomas Schwentick introduced me to dynamic complexity. Very recently Wouter Gelade, Marcel Marquardt and Thomas had obtained a very nice characterization of regular languages in terms of dynamic complexity, and also a lower bound for the dynamic complexity of the alternating reachability problem. Many interesting problems in this area seemed to wait for a solution; and so I started to try to prove a lower bound for reachability. I was not successful. After two (at the end frustrating) months, I abandoned this project.In the following two and a half years I almost forgot about dynamic complexity. Decidability issues for the two-variable fragment of first-order logic had turned out to be a much more accessible and fruitful field. Thomas and I had obtained several results, and a PhD in this field seemed not to be too far away. This was the moment when Thomas asked whether I would be interested in applying for funds from the DFG. If successful, such funding could relieve me from my teaching obligations.We decided to have a second, deeper look into dynamic complexity, and to apply for funds for doing an extensive study of the power of logics in dynamic settings. At that time, the decision to spend more time on dynamic complexity was not easy for me. I was in the third year of my PhD and already had results and further ideas for two-variable logics; and it was not clear whether an application for funding would be successful. On the other hand, now I had more experience, which might turn out to be helpful to attack the very same problems that I had tried to solve at the beginning of my PhD. I do not regret the decision.With the thesis at hand I want to document the progress in dynamic complexity that we have made in the last two and a half years. The focus of this thesis is on small dynamic descriptive complexity classes, in particular on lower bound methods for them. A short summary of results on decidability issues for two-variable logic is presented at the end of the thesis.

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Section: A Note On Two-variable Logicmentioning

confidence: 99%

Small Dynamic Complexity Classes

Zeume

2017

Lecture Notes in Computer Science

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“…These vectors will form a part of the certificate for ϕ, which we assemble in § 4.6. We remind the reader of the notions of profile and c-spectrum, established in § 4.1, as well as the matrix U relating them, and defined in (6).…”

Section: Clustersmentioning

confidence: 99%

“…The logic L 2 has the finite model property [18], and its satisfiability (= finite satisfiability) problem is NExpTime-complete [6]. The logic C 2 is expressive enough for the finite model property to fail; nevertheless, its satisfiability and finite satisfiability problems remain NExpTime-complete [7,19,22].…”

Section: Introductionmentioning

confidence: 99%

The two‐variable fragment with counting and equivalence

Pratt-Hartmann

2015

Mathematical Logic Qtrly

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We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a single distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NExpTime-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable.

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“…In fact, the translation of a knowledge base uses only variables x and y, and thus yields a formula in the two variable fragment of first-order logic, which is known to be decidable in non-deterministic exponential time [79]. Alternatively, we can use the fact that this translation uses quantification only in a restricted way, and therefore yields a formula in the guarded fragment [2], which is known to be decidable in deterministic exponential time [78].…”

Section: Dls and Predicate Logicmentioning

confidence: 99%

“…Highly optimized systems (FaCT, RACE, and DLP [95,80,133]) showed that tableau-based algorithms for expressive DLs led to a good practical behavior of the system even on (some) large knowledge bases. In this phase, the relationship to modal logics [57,147] and to decidable fragments of first-order logic [33,129,79,77,78] was also studied in more detail, and applications in databases (like schema reasoning, query optimization, and integration of databases) were investigated [45,47,51].…”

Section: Introductionmentioning

confidence: 99%

Chapter 3 Description Logics

Baader¹,

Horrocks²,

Sattler³

2008

Foundations of Artificial Intelligence

152

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In this chapter we will introduce description logics, a family of logic-based knowledge representation languages that can be used to represent the terminological knowledge of an application domain in a structured way. We will first review their provenance and history, and show how the field has developed. We will then introduce the basic description logic ALC in some detail, including definitions of syntax, semantics and basic reasoning services, and describe important extensions such as inverse roles, number restrictions, and concrete domains. Next, we will discuss the relationship between description logics and other formalisms, in particular first order and modal logics; the most commonly used reasoning techniques, in particular tableau, resolution and automata based techniques; and the computational complexity of basic reasoning problems. After reviewing some of the most prominent applications of description logics, in particular ontology language applications, we will conclude with an overview of other aspects of description logic research, and with pointers to the relevant literature. IntroductionDescription logics (DLs) [14,25,50] are a family of knowledge representation languages that can be used to represent the knowledge of an application domain in a structured and formally well-understood way. The name description logics is motivated by the fact that, on the one hand, the important notions of the domain are described by concept descriptions, i.e., expressions that are built from atomic concepts (unary predicates) and atomic roles (binary predicates) using the concept and role constructors provided by the particular DL; on the other hand, DLs differ from their predecessors, such as semantic networks and frames, in that they are equipped with a formal, logic-based semantics.We will first illustrate some typical constructors by an example; formal definitions will be given in Section 3.2. Assume that we want to define the concept of "A man that is married to a doctor, and all of whose children are either doctors or professors." This concept can be described with the following concept description:Human ¬Female (∃married.Doctor) (∀hasChild.(Doctor Professor)).

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On the Decision Problem for Two-Variable First-Order Logic

Cited by 249 publications

References 43 publications

Small Dynamic Complexity Classes

Small Dynamic Complexity Classes

The two‐variable fragment with counting and equivalence

Chapter 3 Description Logics

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