nach oben

Journal of Logic, Language and Information

Erschienen in:

01.07.2013

The Dynamification of Modal Dependence Logic

verfasst von: Pietro Galliani

Erschienen in: Journal of Logic, Language and Information | Ausgabe 3/2013

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We examine the transitions between sets of possible worlds described by the compositional semantics of Modal Dependence Logic, and we use them as the basis for a dynamic version of this logic. We give a game theoretic semantics, a (compositional) transition semantics and a power game semantics for this new variant of modal Dependence Logic, and we prove their equivalence; and furthermore, we examine a few of the properties of this formalism and show that Modal Dependence Logic can be recovered from it by reasoning in terms of reachability. Then we show how we can generalize this approach to a very general formalism for reasoning about transformations between pointed Kripke models.

Vorheriger Artikel On the Origin of Ambiguity in Efficient Communication

Nächster Artikel An Analytic Tableaux Model for Deductive Mastermind Empirically Tested with a Massively Used Online Learning System

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

In Sect. 2.2, however, we will describe a simple variant of this semantics for a dynamic version of this logic.

For simplicity and analogy with the other logics which we will examine in this work, we will only consider formulas in negation normal form and we will not admit negated dependence atoms. In any case, in Modal Dependence Logic a negated dependence atom holds in a set of worlds only if its set is empty, so for all purposes $\lnot =\!\!(p_1 \ldots p_n, q)$ can be seen as equivalent to $q \wedge \lnot q$.

A standard example of a “Donkey Sentence” is: “Every farmer who owns a donkey beats it.”

As an aside, the fact that in Dynamic Predicate Logic existential quantifiers can have an effect even beyond their syntactic scope was one of the main reasons why this semantics can be used to interpret natural language statements in which pronouns refer to nouns which lie beyond their apparent scopes, as in the famous example

$$\begin{aligned} \text{(A } \text{ man) }_1 \text{ walks } \text{ in } \text{ the } \text{ park. }\,\, \text{(He) }_1 \text{ whistles. } \end{aligned}$$

We refer to (Groenendijk and Stokhof 1991) for further details.

Or, equivalently, $(((\Box _1;\Diamond _2);\Box _1);(p \vee q))$. Indeed, as we will see, our concatenation operator is associative.

Intuitively speaking, we may assume that the initial position is selected randomly among this set, or that all of these positions are played in parallel.

The usual choice in the study of the Game Theoretic Semantics of Dependence Logic and Modal Dependence Logic is to add the semantic conditions for literals directly to the definition of the game, and to define uniformity conditions over strategies in order to take care of dependence atoms. However, we prefer here to deal with literals and dependence atoms together.

We include $\psi = p$ in this condition, for $k = 0$. The same holds for the next cases.

The choice of $Z$ is irrelevant here, as it merely affects which plays are winning.

We thank a referee for pointing this out.

In fact, it is more than that—it is nonconfluent, in the sense that for all of its subformulas of the form $\psi _1;\psi _2$, $\psi _1$ does not contain disjunctions.

Note that if $\phi $ is confluent only on flats, applying any of the distribution rules described of the lemma preserves this property.

These games are the modal analogues of the power games for Dependence Logic of Väänänen (2007).

Of course, nothing in principle prevents one from “flattening” the two levels and representing a Kripke team as the disjoint union of its elements. But, at least in the opinion of the author, this seems not to be advantageous.

Abramsky, S. (2007). A compositional game semantics for multi-agent logics of partial information. In J. van Bentham, D. Gabbay, & B. Lowe (Eds.), Interactive logic, volume 1 of texts in logic and games (pp. 11–48). Amsterdam: Amsterdam University Press.

Abramsky, S., & Väänänen, J. (2009). From IF to BI. Synthese, 167, 207–230. doi:10.1007/s11229-008-9415-6.CrossRef

Baltag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. Proceedings of the 7th conference on theoretical aspects of rationality and knowledge, TARK ’98 (pp. 43–56). San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.

Bradfield, J. (2000). Independence: Logics and concurrency. In P. Clote & H. Schwichtenberg (Eds.), Computer science logic volume 1862 of Lecture Notes in Computer Science (pp. 247–261). Berlin/Heidelberg: Springer.

Dekker, P. (2008). A guide to dynamic semantics. ILLC Prepublication Series, (PP-2008-42).

Durand, A., & Kontinen, J. (2011). Hierarchies in dependence logic. CoRR, abs/1105.3324.

Ebbing, J., & Lohmann, P. (2012). Complexity of model checking for modal dependence logic. In M. Bielikov, G. Friedrich, G. Gottlob, S. Katzenbeisser, & G. Turn (Eds.), SOFSEM 2012: theory and practice of Computer Science volume 7147 of Lecture Notes in Computer Science (pp. 226–237). Berlin/Heidelberg: Springer.

Engström, F. (2012). Generalized quantifiers in dependence logic. Journal of Logic, Language and Information, 21, 299–324. doi:10.1007/s10849-012-9162-4.CrossRef

Galliani, P. (2012a). Inclusion and exclusion dependencies in team semantics: On some logics of imperfect information. Annals of Pure and Applied Logic, 163(1), 68–84.CrossRef

Galliani, P. (2012b, September). The dynamics of imperfect information. PhD thesis, University of Amsterdam.

Grädel, E., & Väänänen, J. (2013). Dependence and Independence. Studia Logica, 101(2), 399–410. doi:10.1007/s11225-013-9479-2.

Groenendijk, J., & Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14(1), 39–100.CrossRef

Henkin, L. (1961). Some remarks on infinitely long formulas. In Infinitistic Methods. Proceedings of symposium on foundations of mathematics (pp. 167–183). Pergamon Press.

Hintikka, J., & Sandu, G. (1989). Informational independence as a semantic phenomenon. In J. E. Fenstad, I. T. Frolov, & R. Hilpinen, (Eds.), Logic, methodology and philosophy of science (pp. 571–589). Amsterdam: Elsevier.

Hodges, W. (1997). Compositional semantics for a language of imperfect information. Journal of the Interest Group in Pure and Applied Logics, 5(4), 539–563.

Kontinen, J., & Väänänen, J. (2009). On definability in dependence logic. Journal of Logic, Language and Information, 3(18), 317–332.CrossRef

Kontinen, J., & Väänänen, J. (2011). A remark on negation of dependence logic. Notre Dame Journal of Formal Logic, 52(1), 55–65.CrossRef

Lohmann, P., & Vollmer, H. (2010). Complexity results for modal dependence logic. In A. Dawar, & H. Veith (Eds.), Computer science logic, volume 6247 of Lecture Notes in Computer Science (pp. 411–425). Berlin/Heidelberg: Springer.

Plaza, J. (2007). Logics of public communications. Synthese, 158, 165–179. doi:10.1007/s11229-007-9168-7.

Sevenster, M. (2009). Model-theoretic and computational properties of modal dependence logic. Journal of Logic and Computation, 19(6), 1157–1173.CrossRef

Tulenheimo, T. (2004). Independence-friendly Modal Logic. PhD thesis, University of Helsinki.

Väänänen, J. (2007). Dependence Logic. Cambridge: Cambridge University Press.CrossRef

Väänänen, J. (2008). Modal Dependence Logic. In K. R. Apt & R. van Rooij (Eds.), New perspectives on games and interaction. Amsterdam: Amsterdam University Press.

van Benthem, J. (1997). Exploring logical dynamics. Stanford, CA, USA: Center for the Study of Language and Information.

van Benthem, J. (2006). The epistemic logic of IF games. In R. E. Auxier, & L. E. Hahn (Eds.), The Philosophy of Jaakko Hintikka, volume 30 of Library of living philosophers, chapter 13 (pp. 481–513). Open Court Publishers.

Van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied NonClassical Logics, 17(2), 129–155.CrossRef

van Eijck, J., & Visser, A. (2012). Dynamic semantics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. (Winter ed.). http://plato.stanford.edu/archives/win2012/entries/dynamic.

Titel: The Dynamification of Modal Dependence Logic
verfasst von: Pietro Galliani
Publikationsdatum: 01.07.2013
Verlag: Springer Netherlands
Erschienen in: Journal of Logic, Language and Information / Ausgabe 3/2013
Print ISSN: 0925-8531
Elektronische ISSN: 1572-9583
DOI: https://doi.org/10.1007/s10849-013-9175-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 3/2013

On the Origin of Ambiguity in Efficient Communication

An Analytic Tableaux Model for Deductive Mastermind Empirically Tested with a Massively Used Online Learning System

Inferential Conditionals and Evidentiality

Linguistic and Visual Cognition: Verifying Proportional and Superlative Most in Bulgarian and Polish

Premium Partner