Complexity of validity for propositional dependence logics

We study the validity problem for propositional dependence logic, modal dependence logic and extended modal dependence logic. We show that the validity problem for propositional dependence logic is NEXPTIME-complete. In addition, we establish that the corresponding problem for modal dependence logic and extended modal dependence logic is NEXPTIME-hard and in NEXPTIME^NP.Comment: In Proceedings GandALF 2014, arXiv:1408.556

On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

An Introduction to Mechanized Reasoning

Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics journals. We introduce mechanized reasoning to economists in three ways. First, we introduce mechanized reasoning in general, describing both the techniques and their successful applications. Second, we explain how mechanized reasoning has been applied to economic problems, concentrating on the two domains that have attracted the most attention: social choice theory and auction theory. Finally, we present a detailed example of mechanized reasoning in practice by means of a proof of Vickrey's familiar theorem on second-price auctions