115 research outputs found
Synthetic Completeness for a Terminating Seligman-Style Tableau System
Hybrid logic extends modal logic with nominals that name worlds. Seligman-style tableau systems for hybrid logic divide branches into blocks named by nominals to achieve a local proof style. We present a Seligman-style tableau system with a formalization in the proof assistant Isabelle/HOL. Our system refines an existing system to simplify formalization and we claim termination from this relationship. Existing completeness proofs that account for termination are either analytic or based on translation, but synthetic proofs have been shown to generalize to richer logics and languages. Our main result is the first synthetic completeness proof for a terminating hybrid logic tableau system. It is also the first formalized completeness proof for any hybrid logic proof system
Completeness of Tableau Calculi for Two-Dimensional Hybrid Logics
Hybrid logic is one of the extensions of modal logic. The many-dimensional
product of hybrid logic is called hybrid product logic (HPL). We construct a
sound and complete tableau calculus for two-dimensional HPL. Also, we made a
tableau calculus for hybrid dependent product logic (HdPL), where one dimension
depends on the other. In addition, we add a special rule to the tableau
calculus for HdPL and show that it is still sound and complete. All of them
lack termination, however.Comment: Version 2. 27 pages. 5 figures. This is a preprin
Modal Hybrid Logic
This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory
Automated Synthesis of Tableau Calculi
This paper presents a method for synthesising sound and complete tableau
calculi. Given a specification of the formal semantics of a logic, the method
generates a set of tableau inference rules that can then be used to reason
within the logic. The method guarantees that the generated rules form a
calculus which is sound and constructively complete. If the logic can be shown
to admit finite filtration with respect to a well-defined first-order semantics
then adding a general blocking mechanism provides a terminating tableau
calculus. The process of generating tableau rules can be completely automated
and produces, together with the blocking mechanism, an automated procedure for
generating tableau decision procedures. For illustration we show the
workability of the approach for a description logic with transitive roles and
propositional intuitionistic logic.Comment: 32 page
Incorrect Responses in First-Order False-Belief Tests:A Hybrid-Logical Formalization
In the paper (Braüner, 2014) we were concerned with logical formalizations of the reasoning involved in giving correct responses to the psychological tests called the Sally-Anne test and the Smarties test, which test children’s ability to ascribe false beliefs to others. A key feature of the formal proofs given in that paper is that they explicitly formalize the perspective shift to another person that is required for figuring out the correct answers – you have to put yourself in another person’s shoes, so to speak, to give the correct answer. We shall in the present paper be concerned with what happens when answers are given that are not correct. The typical incorrect answers indicate that children failing false-belief tests have problems shifting to a perspective different from their own, to be more precise, they simply reason from their own perspective. Based on this hypothesis, we in the present paper give logical formalizations that in a systematic way model the typical incorrect answers. The remarkable fact that the incorrect answers can be derived using logically correct rules indicates that the origin of the mistakes does not lie in the children’s logical reasoning, but rather in a wrong interpretation of the task
Complexity Results and Practical Algorithms for Logics in Knowledge Representation
Description Logics (DLs) are used in knowledge-based systems to represent and
reason about terminological knowledge of the application domain in a
semantically well-defined manner. In this thesis, we establish a number of
novel complexity results and give practical algorithms for expressive DLs that
provide different forms of counting quantifiers.
We show that, in many cases, adding local counting in the form of qualifying
number restrictions to DLs does not increase the complexity of the inference
problems, even if binary coding of numbers in the input is assumed. On the
other hand, we show that adding different forms of global counting restrictions
to a logic may increase the complexity of the inference problems dramatically.
We provide exact complexity results and a practical, tableau based algorithm
for the DL SHIQ, which forms the basis of the highly optimized DL system iFaCT.
Finally, we describe a tableau algorithm for the clique guarded fragment
(CGF), which we hope will serve as the basis for an efficient implementation of
a CGF reasoner.Comment: Ph.D. Thesi
A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic
We define a infinitary labelled sequent calculus for PDL, G3PDL^{\infty}. A
finitarily representable cyclic system, G3PDL^{\omega}, is then given. We show
that both are sound and complete with respect to standard models of PDL and,
further, that G3PDL^{\infty} is cut-free complete. We additionally investigate
proof-search strategies in the cyclic system for the fragment of PDL without
tests
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Incremental decision procedures for modal logics with nominals and eventualities
This thesis contributes to the study of incremental decision procedures for modal logics with nominals and eventualities. Eventualities are constructs that allow to reason about the reflexive-transitive closure of relations. Eventualities are an essential feature of temporal logics and propositional dynamic logic (PDL). Nominals extend modal logics with the possibility to reason about state equality. Modal logics with nominals are often called hybrid logics. Incremental procedures are procedures that can potentially solve a problem by performing only the reasoning steps needed for the problem in the underlying calculus. We begin by introducing a class of syntactic models called demos and showing how demos can be used for obtaining nonincremental but worst-case optimal decision procedures for extensions of PDL with nominals, converse and difference modalities. We show that in the absence of nominals, such nonincremental procedures can be refined into incremental demo search procedures, obtaining a worst-case optimal decision procedure for modal logic with eventualities. We then develop the first incremental decision procedure for basic hybrid logic with eventualities, which we eventually extend to deal with hybrid PDL. The approach in the thesis suggests a new principled design of modular, incremental decision procedures for expressive modal logics. In particular, it yields the first incremental procedures for modal logics containing both nominals and eventualities.Diese Dissertation untersucht inkrementelle Entscheidungsverfahren für Modallogiken mit Nominalen und Eventualities. Eventualities sind Konstrukte, die erlauben, über den reflexiv-transitiven Abschluss von Relationen zu sprechen. Sie sind ein Schlüsselmerkmal von Temporallogiken und dynamischer Aussagenlogik (PDL). Nominale erweitern Modallogik um die Möglichkeit, über Gleichheit von Zuständen zu sprechen. Modallogik mit Nominalen nennt man Hybridlogik. Inkrementell ist ein Verfahren dann, wenn es ein Problem so lösen kann, dass für die Lösung nur solche Schritte in dem zugrundeliegenden Kalkül gemacht werden, die für das Problem relevant sind. Wir führen zunächst eine Klasse syntaktischer Modelle ein, die wir Demos nennen. Wir nutzen Demos um nichtinkrementelle aber laufzeitoptimale Entscheidungsverfahren für Erweiterungen von PDL zu konstruieren. Wir zeigen, dass im Fall ohne Nominale solche Verfahren durch algorithmische Verfeinerung zu inkrementellen Verfahren ausgebaut werden können. Insbesondere erhalten wir so ein optimales Verfahren für Modallogik mit Eventualities. Anschließend entwickeln wir das erste inkrementelle Verfahren für Hybridlogik mit Eventualities, welches wir schließlich auf hybrides PDL erweitern. Die Dissertation vermittelt einen neuen Ansatz zur Konstruktion modularer, inkrementeller Entscheidungsverfahren für expressive Modallogiken. Insbesondere liefert der Ansatz die ersten inkrementellen Verfahren für Modallogiken mit Nominalen und Eventualities
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