13 research outputs found
Proceedings of the Workshop on the lambda-Prolog Programming Language
The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases
Higher-order Linear Logic Programming of Categorial Deduction
We show how categorial deduction can be implemented in higher-order (linear)
logic programming, thereby realising parsing as deduction for the associative
and non-associative Lambek calculi. This provides a method of solution to the
parsing problem of Lambek categorial grammar applicable to a variety of its
extensions.Comment: 8 pages LaTeX, uses eaclap.sty, to appear EACL9
Type-driven natural language analysis
The purpose of this thesis is in showing how recent developments in logic programming can be exploited to encode in a computational environment the features of certain linguistic theories. We are in this way able to make available for the purpose of natural language processing sophisticated capabilities of linguistic analysis directly justified by well developed grammatical frameworks.
More specifically, we exploit hypothetical reasoning, recently proposed as one of the possible directions to widen logic programming, to account for the syntax of filler-gap dependencies along the lines of linguistic theories such as Generalized Phrase Structure Grammar and Categorial Grammar. Moreover, we make use, for the purpose of semantic analysis of the same kind of phenomena, of another recently proposed extension, interestingly related to the previous one, namely the idea of replacing first-order terms with the more expressive λ-terms of λ-Calculus
Defining Logical Systems via Algebraic Constraints on Proofs
We comprehensively present a program of decomposition of proof systems for
non-classical logics into proof systems for other logics, especially classical
logic, using an algebra of constraints. That is, one recovers a proof system
for a target logic by enriching a proof system for another, typically simpler,
logic with an algebra of constraints that act as correctness conditions on the
latter to capture the former; for example, one may use Boolean algebra to give
constraints in a sequent calculus for classical propositional logic to produce
a sequent calculus for intuitionistic propositional logic. The idea behind such
forms of reduction is to obtain a tool for uniform and modular treatment of
proof theory and provide a bridge between semantics logics and their proof
theory. The article discusses the theoretical background of the project and
provides several illustrations of its work in the field of intuitionistic and
modal logics. The results include the following: a uniform treatment of modular
and cut-free proof systems for a large class of propositional logics; a general
criterion for a novel approach to soundness and completeness of a logic with
respect to a model-theoretic semantics; and a case study deriving a
model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte
Nominal Logic Programming
Nominal logic is an extension of first-order logic which provides a simple
foundation for formalizing and reasoning about abstract syntax modulo
consistent renaming of bound names (that is, alpha-equivalence). This article
investigates logic programming based on nominal logic. We describe some typical
nominal logic programs, and develop the model-theoretic, proof-theoretic, and
operational semantics of such programs. Besides being of interest for ensuring
the correct behavior of implementations, these results provide a rigorous
foundation for techniques for analysis and reasoning about nominal logic
programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as
of July 23, 200
Goal-directed proof theory
This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape
Hybrid - a definitional two-level approach to reasoning with higher-order abstract syntax
Combining higher-order abstract syntax and (co)-induction in a logical
framework is well known to be problematic.We describe the theory and the practice
of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many
of these difficulties. It allows object logics to be represented using higher-order
abstract syntax, and reasoned about using tactical theorem proving and principles
of (co)induction. Moreover, it is definitional, which guarantees consistency within
a classical type theory. The idea is to have a de Bruijn representation of \u3bb-terms
providing a definitional layer that allows the user to represent object languages using
higher-order abstract syntax, while offering tools for reasoning about them at the
higher level. In this paper we describe how to use Hybrid in a multi-level reasoning
fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly
referencing provability in a middle layer called a specification logic, we solve the
problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical
judgments, which allow very elegant and succinct specifications of object logic
inference rules. We first demonstrate the method on a simple example, formally
proving type soundness (subject reduction) for a fragment of a pure functional
language, using a minimal intuitionistic logic as the specification logic. We then
prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that
serves as the specification layer. This example demonstrates the ease with which
we can incorporate new specification logics, and also illustrates a significantly more
complex object logic whose encoding is elegantly expressed using features of the new
specification logic
Model and Proof Theory of Constructive ALC, Constructive Description Logics
Description logics (DLs) represent a widely studied logical formalism with a significant impact in the field of knowledge representation and the Semantic Web. However, they are equipped with a classical descriptive semantics that is characterised by a platonic notion of truth, being insufficiently expressive to deal with evolving and incomplete information, as from data streams or ongoing processes. Such partially determined and incomplete knowledge can be expressed by relying on a constructive semantics. This thesis investigates the model and proof theory of a constructive variant of the basic description logic ALC, called cALC. The semantic dimension of constructive DLs is investigated by replacing the classical binary truth interpretation of ALC with a constructive notion of truth. This semantic characterisation is crucial to represent applications with partial information adequately, and to achieve both consistency under abstraction as well as robustness under refinement, and on the other hand is compatible with the Curry-Howard isomorphism in order to form the cornerstone for a DL-based type theory. The proof theory of cALC is investigated by giving a sound and complete Hilbert-style axiomatisation, a Gentzen-style sequent calculus and a labelled tableau calculus showing finite model property and decidability. Moreover, cALC can be strengthened towards normal intuitionistic modal logics and classical ALC in terms of sound and complete extensions and hereby forms a starting point for the systematic investigation of a constructive correspondence theory.Beschreibungslogiken (BLen) stellen einen vieluntersuchten logischen Formalismus dar, der den Bereich der Wissensrepräsentation und das Semantic Web signifikant geprägt hat. Allerdings basieren BLen meist auf einer klassischen deskriptiven Semantik, die gekennzeichnet ist durch einen idealisierten Wahrheitsbegriff nach Platons Ideenlehre, weshalb diese unzureichend ausdrucksstark sind, um in Entwicklung befindliches und unvollständiges Wissen zu repräsentieren, wie es beispielsweise durch Datenströme oder fortlaufende Prozesse generiert wird. Derartiges partiell festgelegtes und unvollständiges Wissen lässt sich auf der Basis einer konstruktiven Semantik ausdrücken. Diese Arbeit untersucht die Model- und Beweistheorie einer konstruktiven Variante der Basis-BL ALC, die im Folgenden als cALC bezeichnet wird. Die Semantik dieser konstruktiven Beschreibungslogik resultiert daraus, die traditionelle zweiwertige Interpretation logischer Aussagen des Systems ALC durch einen konstruktiven Wahrheitsbegriff zu ersetzen. Eine derartige Interpretation ist die Voraussetzung dafür, um einerseits Anwendungen mit partiellem Wissen angemessen zu repräsentieren, und sowohl die Konsistenz logischer Aussagen unter Abstraktion als auch ihre Robustheit unter Verfeinerung zu gewährleisten, und andererseits um den Grundstein für eine Beschreibungslogik-basierte Typentheorie gemäß dem Curry-Howard Isomorphismus zu legen. Die Ergebnisse der Untersuchung der Beweistheorie von cALC umfassen eine vollständige und korrekte Hilbert Axiomatisierung, einen Gentzen Sequenzenkalkül, und ein semantisches Tableaukalkül, sowie Beweise zur endlichen Modelleigenschaft und Entscheidbarkeit. Darüber hinaus kann cALC zu normaler intuitionistischer Modallogik und klassischem ALC durch vollständige und korrekte Erweiterungen ausgebaut werden, und bildet damit einen Startpunkt für die systematische Untersuchung einer konstruktiven Korrespondenztheorie