We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as corollary. 1

This paper presents a new model construction for a natural cut-free infinitary version K + ω (µ) of the propositional modal µ-calculus. Based on that the completeness of K + ω (µ) and the related system Kω(µ) can be established directly – no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cut-free sound and complete system for the propositional modal µ-calculus. 1

We provide a purely syntactical treatment of simultaneous fixpoints in the modal µ-calculus by proving directly in Kozen’s axiomatisation their properties as greatest and least fixpoints, that is, the fixpoint axiom and the induction rule. Further, we apply our result in order to get a completeness result for characteristic formulae of finite pointed transition systems. Keywords: Modal µ-calculus, proof theory, Kozen’s axiomatisation, simultaneous fixpoints

In this paper we propose a formal language for writing electronic contracts, based on the deontic notions of obligation, permission, and prohibition. We take an ought-to-do approach, where deontic operators are applied to actions instead of state-of-a airs. We propose an extension of the µ-calculus in order to capture the intuitive meaning of the deontic notions and to express concurrent actions. We provide a translation of the contract language into the logic, the semantics of which faithfully captures the meaning of obligation, permission and prohibition. We also show how our language captures most of the intuitive desirable properties of electronic contracts, as well as how it avoids most of the classical paradoxes of deontic logic. We nally show its applicability on a contract example.

This thesis provides an overview of my work on the proof theory for modal fixed point logics. In particular, it summarizes the main results of the following papers. 1. G. Jäger, M. Kretz, and T. Studer. Cut-free common knowledge.

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