A note on the incompleteness of Afshari & Leigh's system Clo

The system $\mathsf{Clo}$ is a cyclic, cut-free proof system for the modal $\mu$-calculus. It was introduced by Afshari & Leigh as an intermediate system in their intent to show the completeness of Kozen's axiomatisation for the modal $\mu$-calculus. We prove that $\mathsf{Clo}$ is incomplete by giving a valid sequent that is not provable in $\mathsf{Clo}$

Proof Systems for the Modal μ\mu-Calculus Obtained by Determinizing Automata

Automata operating on infinite objects feature prominently in the theory of the modal $\mu$-calculus. One such application concerns the tableau games introduced by Niwi\'{n}ski & Walukiewicz, of which the winning condition for infinite plays can be naturally checked by a nondeterministic parity stream automaton. Inspired by work of Jungteerapanich and Stirling we show how determinization constructions of this automaton may be used to directly obtain proof systems for the $\mu$-calculus. More concretely, we introduce a binary tree construction for determinizing nondeterministic parity stream automata. Using this construction we define the annotated cyclic proof system $\mathsf{BT}$, where formulas are annotated by tuples of binary strings. Soundness and Completeness of this system follow almost immediately from the correctness of the determinization method

Ein Fixpunktsatz für Horn-Formelgleichungen

Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersFormula equations are logical equations in which the unknowns are formulas. They naturally occur in many areas like program verification or automated theorem proving. In these communities similar problems are treated independently. Our motivation is to state general theorems about formula equations which then will be used to generalize and simplify results in various different applications. In this thesis we treat the special case of Horn formula equations, this restriction is justified as it covers many applications. For Horn formula equations we are able to compute canonical solutions if a solution exists. This will be achieved by defining an operator such that every solution of the Horn formula equation is a fixed point of it. Now the least fixed point, which exists due to the Knaster-Tarski theorem, turns out to always be a solution if there exists one. This canonical solution will be described by least fixed-point formulas. Furthermore the canonical solution implies every solution of the Horn formula equation. We then use the Horn fixed-point theorem to obtain a first-order approximation of second-order formulas as well as to get a different perspective on Hoare triples in program verification and to get more insights in inductive theorem proving.4