770 research outputs found
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in âA Workshop on Realizability Semantics and Its Applicationsâ, Trento, Italy, June 30âJuly 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
Extended RDF: Computability and Complexity Issues
ERDF stable model semantics is a recently proposed semantics for
ERDF ontologies and a faithful extension of RDFS semantics on RDF graphs.
In this paper, we elaborate on the computability and complexity issues of the
ERDF stable model semantics. Based on the undecidability result of ERDF
stable model semantics, decidability under this semantics cannot be achieved,
unless ERDF ontologies of restricted syntax are considered. Therefore, we
propose a slightly modified semantics for ERDF ontologies, called ERDF #n-
stable model semantics. We show that entailment under this semantics is, in
general, decidable and also extends RDFS entailment. Equivalence statements
between the two semantics are provided. Additionally, we provide algorithms
that compute the ERDF #n-stable models of syntax-restricted and general
ERDF ontologies. Further, we provide complexity results for the ERDF #nstable
model semantics on syntax-restricted and general ERDF ontologies.
Finally, we provide complexity results for the ERDF stable model semantics
on syntax-restricted ERDF ontologies
On Counting Propositional Logic and Wagner's Hierarchy
We introduce an extension of classical propositional logic with counting quantifiers. These forms of quantification make it possible to express that a formula is true in a certain portion of the set of all its interpretations. Beyond providing a sound and complete proof system for this logic, we show that validity problems for counting propositional logic can be used to capture counting complexity classes. More precisely, we show that the complexity of the decision problems for validity of prenex formulas of this logic perfectly match the appropriate levels of Wagner's counting hierarchy
Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory
In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige SchlĂźsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenĂśssischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jĂźngeren Untersuchungen, eine zunehmende Bedeutung als MaĂstab fĂźr Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgefĂźhrt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermĂśglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz fĂźr Unentscheidbarkeit und Unvollständigkeit, der groĂteils den berĂźchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natĂźrliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen fĂźr Nutzung und Beiträge externer Anwender ist
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