# Interactive Logic: Selected Papers from the 7th Augustus de Morgan Workshop, London

JOHAN VAN BENTHEM
DOV GABBAY
BENEDIKT LÖWE
Pages: 362
https://www.jstor.org/stable/j.ctt45kdbf

1. Front Matter
(pp. 1-4)
(pp. 5-6)
3. Preface
(pp. 7-10)
J.F.A.K.v.B., D.G. and B.L.
4. A Compositional Game Semantics for Multi-Agent Logics of Partial Information
(pp. 11-48)
Samson Abramsky

We begin with the following quote from the manifesto of the 7th Augustus de Morgan workshop:

Traditionally, logic has dealt with the zero-agent notion of truth and the one-agent notion of reasoning. In the last decades, research focus in logic shifted from these topics to the vast field of “interactive logic”, encompassing logics of communication and interaction. The main applications of this move ton-agent notions are logical approaches to games and social software.

However, while there are certainly applications of multi-modal logics to reasoningabout n-person games (see e.g. [Pa7Pa503, Pa702]), the more intimate connections between Games and Logic...

5. Quantificational Modal Operators and Their Semantics
(pp. 49-70)
Stefano Borgo

For about 20 years we have witnessed an increasing interest in the formal study of phenomena comprising several entities which present independent and autonomous behavior like software agents, human beings, biological entities, social organizations, robots and stock markets [We199].

In this research area, the issue of (true) concurrency has special interest since it puts at the center phenomena where several entities act simultaneously perhaps affecting each other. This issue is coupled with the need to formalize group collaboration as well as group dynamics. Another challenge is the formalization of the independence of an agent from the others and from the...

6. A Note on Kuhn’s Theorem
(pp. 71-88)

This note is a homage to Kuhn’s classic theorem on the replacement of mixed by behavior strategies in games [Ku050, Ku053]. It reframes Kuhn’s work as two results in decision theory—i.e., in the context of trees involving a decision maker and Nature. The motivation is to see the meaning of Kuhn’s work at this basic level.

The decision-theoretic framing in this note is in accordance with the so-called epistemic approach to game theory. Under the epistemic approach, a game is a multi-player decision problem—more exactly, a collection of decision problems, one for each player. In line with decision...

7. What Kind of Memory is Needed to Win Infinitary Muller Games?
(pp. 89-116)
Erich Grädel and Łukasz Kaiser

We study two-player games of infinite duration that are played on finite or infinite game graphs. Such a game isdeterminedif, from each position, one of the two players has a winning strategy. On the basis of the axiom of choice it is not difficult to prove that there exist nondetermined games. The classical theory of infinite games in descriptive set theory links determinacy of games with topological properties of the winning conditions. Usually the format of Gale-Stewart games is used where the two players strictly alternate, and in each move a player selects an element of {0, 1};...

8. Logics of Imperfect Information: Why Sets of Assignments?
(pp. 117-134)
Wilfrid Hodges

In 1961 Leon Henkin [He161] extended first-order logic by adding partially ordered arrays of quantifiers. He proposed a semantics for sentences$\varphi$that begin with quantifier arrays of this kind:$\varphi$is true in a structureAif and only if there are a sentence${{\varphi}^{+}}$and a structureA+such that:

${{\varphi}^{+}}$comes from$\varphi$by removing each existential quantifier ∃yin the partially ordered prefix, and replacing each occurrence of the variableyby a term$F(\bar{x})$where$\bar{x}$are the variables universally quantified ‘before’ ∃yin the quantifier prefix (so that the new function symbolsFare...

(pp. 135-158)
Eric Pacuit and Rohit Parikh

The topic “who knew what and when” is not just of interest to epistemic logicians. Often it is the subject of political scandals (both real and imagined). For example, consider the much talked about Valerie Plame affair. A July 2003 column in the Washington Post reported that Plame was an undercover CIA operative. This column generated much controversy due to the fact that such information (the identity of CIA operatives) is restricted to the relevant government officials. Of course, in this situation, we know full well “Who knew what and when”: in July of 2003, Robert Novak (the author of...

10. Epistemic Foundations for Backward Induction: An Overview
(pp. 159-194)
Andrés Perea

Backward induction constitutes one of the oldest concepts in game theory. Its algorithmic definition, which goes back at least to [Ze13], seems so natural at first sight that one might be tempted to argue that every player “should” reason in accordance with backward induction in every game with perfect information. However, on a decision theoretic level the concept is no longer as uncontroversial as it may seem. The problem is that the backward induction algorithm, when applied from a certain decision node on, completely ignores the history that has led to this decision node, as it works from the terminal...

11. Multitape Games
(pp. 195-208)
Brian Semmes

Most of our notation and terminology is standard in descriptive set theory and can be found in [Ke095] or [Mo180]. As usual, for setsAandB,ABdenotes the set of functions fromAtoB. In particular,${}^{\omega}\omega$denotes the set of functions from$\omega$to$\omega$, i.e.${}^{\omega}\omega$is the set of$\omega$-length sequences of natural numbers. The notation${}^{<\omega}$Adenotes the set of finite sequences of elements ofA, so that${}^{<\omega}\omega$denotes the set of finite sequences of natural numbers. We use$^{\le\omega}\omega$to denote$^{<\omega }\omega \,\cup \,{}^{\omega }\omega$. For$s\,\in {{\,}^{<\omega }}\omega$, let$[s]:=\,\{u\ \in \ {}^{\omega }\omega \ :\ s\ \subset \ u\}$...

12. The Complexity of Scotland Yard
(pp. 209-246)
Merlijn Sevenster

The discipline ofcombinatorial game theory(CGT) deals almost exclusively with zero-sum games with perfect information. Although the existence of games with imperfect information is acknowledged in one of CGT’s seminal publications [Be1Co3Gu282, pp. 16–7], only a marginal amount of literature appeared on games with imperfect information. Yet, the number of publications on games with perfect information is abundant and offers a robust picture of the computational behavior of games: One-person games orpuzzlesare usually solvable in NP and many of them turn out to be complete for this class.¹ Famous examples include the games of Minesweeper [Ka500]...

13. Approaches to Independence Friendly Modal Logic
(pp. 247-280)
Tero Tulenheimo and Merlijn Sevenster

Already in the seminal publications onindependence friendly first-order logic(IF logic) [Hi195, Hi196, Hi1Sa489, Sa493], applications were pointed out involving a first-order modal setting. It was argued that the logical form of some natural language sentences is best captured by formulas that allow forslashingrelative to modal operators—marking certain logical operators as independent of modal operators in whose syntactic scope they nevertheless lie. In the first publications that developed an independence friendly modal logic, Bradfield [Br000] together with Fröschle [Br0Fr402a] interpreted the logic’s independence indications using a combination of transition systems withconcurrencyand games ofimperfect...

14. Team Logic
(pp. 281-302)
Jouko Väänänen

Let a vocabulary¹Land anL-structure$\cal M$with universeMbe given. In this paper we studyfunctional dependenciesin$\cal M$. This is in contrast to the traditional approach in logic of studyingrelational dependenciesin$\cal M$. Our atomic dependence relations state the existence of a functional dependence without giving any definition for the function that carries the dependence. This gives the whole topic a second order flavor. It seems to the author that although functional dependence in databases has been studied (starting with [Ar74]), a general theory of more complex types of dependence is new. (For...

15. DEMO A Demo of Epistemic Modelling
(pp. 303-362)
Jan van Eijck

In this introduction we shall demonstrate howDEMO,which is short forDynamic Epistemic MOdelling,¹ can be used to check semantic intuitions about what goes on in epistemic update situations.² For didactic purposes, the initial examples have been kept extremely simple. Although the situation of message passing about just two basic propositions with just three epistemic agents already reveals many subtleties, the reader should bear in mind thatDEMOis capable of modelling much more complex situations.

In a situation where you and I know nothing about a particular aspect of the state of the world (about whetherpand...