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.
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Notes
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.
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Acknowledgments
This work has been supported by the EUROCORES, LogICCC LINT programme, by the Väisälä Foundation and by grant 264917 of the Academy of Finland. The author thanks a referee for a number of very useful suggestions and comments.
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Galliani, P. The Dynamification of Modal Dependence Logic. J of Log Lang and Inf 22, 269–295 (2013). https://doi.org/10.1007/s10849-013-9175-7
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DOI: https://doi.org/10.1007/s10849-013-9175-7