Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

Zero-one laws for provability logic:Axiomatizing validity in almost all models and almost all frames

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5. In this paper, we prove zero-one laws for provability logic with respect to both model and frame validity. Moreover, we axiomatize validity in almost all relevant finite models and in almost all relevant finite frames, leading to two different axiom systems. In the proofs, we use a combinatorial result by Kleitman and Rothschild about the structure of almost all finite partial orders. On the way, we also show that a previous result by Halpern and Kapron about the axiomatization of almost sure frame validity for S4 is not correct. Finally, we consider the complexity of deciding whether a given formula is almost surely valid in the relevant finite models and frames

What Drives People's Choices in Turn-Taking Games, if not Game-Theoretic Rationality?

In an earlier experiment, participants played a perfect information game against a computer, which was programmed to deviate often from its backward induction strategy right at the beginning of the game. Participants knew that in each game, the computer was nevertheless optimizing against some belief about the participant's future strategy. In the aggregate, it appeared that participants applied forward induction. However, cardinal effects seemed to play a role as well: a number of participants might have been trying to maximize expected utility. In order to find out how people really reason in such a game, we designed centipede-like turn-taking games with new payoff structures in order to make such cardinal effects less likely. We ran a new experiment with 50 participants, based on marble drop visualizations of these revised payoff structures. After participants played 48 test games, we asked a number of questions to gauge the participants' reasoning about their own and the opponent's strategy at all decision nodes of a sample game. We also checked how the verbalized strategies fit to the actual choices they made at all their decision points in the 48 test games. Even though in the aggregate, participants in the new experiment still tend to slightly favor the forward induction choice at their first decision node, their verbalized strategies most often depend on their own attitudes towards risk and those they assign to the computer opponent, sometimes in addition to considerations about cooperativeness and competitiveness.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Balancing Selfishness and Efficiency in Mobile Ad-hoc Networks:An Agent-based Simulation

We study wireless ad-hoc networks from an agent-based perspective. In our model agents with different strategies such as being selfish, tit-for-tat or battery-based compete and cooperate. If only different levels of selfishness are allowed then being selfish is clearly the dominant strategy. However, introduction of more advanced strategies allows to some extent to combat selfishness. In particular we present a battery-based approach and a hybrid of battery-based and tit-for-tat approaches. The findings give hope that the introduction of widely available ad-hoc networks might at some point be possible. Even when users are given full control of their devices, effective strategies allow for the networks overall to be effective and feasible