# Transition semantics: the dynamics of dependence logic

## Abstract

We examine the relationship between dependence logic and game logics. A variant of dynamic game logic, called Transition Logic, is developed, and we show that its relationship with dependence logic is comparable to the one between first-order logic and dynamic game logic discussed by van Benthem. This suggests a new perspective on the interpretation of dependence logic formulas, in terms of assertions about reachability in games of imperfect information against Nature. We then capitalize on this intuition by developing expressively equivalent variants of dependence logic in which this interpretation is taken to the foreground.

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1. 1.

In general, we will assume through this whole work that all first-order models which we are considering have at least two elements. As one-element models are trivial, this is not a very onerous restriction.

2. 2.

The main difference between this version and the one of Parikh’s original paper lies in the absence of the iteration operator $$\gamma ^*$$ from our formalism. In this, we follow van Benthem (2003) and van Benthem et al. (2008).

3. 3.

This requirement is nothing but a formal version of Zermelo’s Theorem: if one of the players cannot force the outcome of the game to belong to a set of “winning outcomes” $$X$$, this implies that the other player can force it to belong to the complement of $$X$$.

4. 4.

van Benthem et al. (2008) gives the following alternative condition for the powers of the universal player:

• $$s \rho ^A X$$ if and only if $$X = Z_1 \cup Z_2$$ for two $$Z_1$$ and $$Z_2$$ such that $$s \rho ^A_1 Z_1$$ and $$s \rho ^A_2 Z_2$$.

It is trivial to see that, if our games satisfy the monotonicity condition, this rules is equivalent to the one we presented.

5. 5.

The term “trump” is taken from Hodges (1997), where it is used to describe the set of all teams which satisfy a given formula.

6. 6.

Here we write $$A \uplus B$$ for the disjoint union of the sets $$A$$ and $$B$$.

7. 7.

That is, such that $$X(x)$$ is a set of states of the transition model.

8. 8.

A moment’s thought shows that, by downwards closure, the condition $$z \not = y$$ in the second disjunct can be removed, but for simplicity we will keep it.

9. 9.

For example, consider the formula $$(\exists x Px) \wedge Qx$$: by the rules given, it is easy to see that $$M \models _s (\exists x Px) \wedge Qx$$ if and only if $$P^M \cap Q^M \not = \emptyset$$, that is, if and only if $$M \models _s \exists x (Px \wedge Qx)$$, differently from the case of Tarski’s semantics.

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## Acknowledgments

The author wishes to thank Johan van Benthem and Jouko Väänänen for a number of useful suggestions and insights. Furthermore, he wishes to thank the reviewers for a number of highly useful suggestions and comments.

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Galliani, P. Transition semantics: the dynamics of dependence logic. Synthese 191, 1249–1276 (2014). https://doi.org/10.1007/s11229-013-0327-8

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### Keywords

• Dependence logic
• Game logic
• Imperfect information