Permutation Games for the Weakly Aconjunctive μ\mu-Calculus

We introduce a natural notion of limit-deterministic parity automata and present a method that uses such automata to construct satisfiability games for the weakly aconjunctive fragment of the $\mu$-calculus. To this end we devise a method that determinizes limit-deterministic parity automata of size $n$ with $k$ priorities through limit-deterministic B\"uchi automata to deterministic parity automata of size $\mathcal{O}((nk)!)$ and with $\mathcal{O}(nk)$ priorities. The construction relies on limit-determinism to avoid the full complexity of the Safra/Piterman-construction by using partial permutations of states in place of Safra-Trees. By showing that limit-deterministic parity automata can be used to recognize unsuccessful branches in pre-tableaux for the weakly aconjunctive $\mu$-calculus, we obtain satisfiability games of size $\mathcal{O}((nk)!)$ with $\mathcal{O}(nk)$ priorities for weakly aconjunctive input formulas of size $n$ and alternation-depth $k$. A prototypical implementation that employs a tableau-based global caching algorithm to solve these games on-the-fly shows promising initial results

Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-calculus based on intuitionistic logic is trivial, every $\mu$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal

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

Local Model-Checking of Modal Mu-Calculus on Acyclic Labeled Transition Systems

Model-checking is a popular technique for verifying finite-state concurrent systems, the behaviour of which can be modeled using Labeled Transition Systems (Ltss). In this report, we study the model-checking problem for the modal mu-calculus on acyclic Ltss. This has various applications of practical interest such as trace analysis, log information auditing, run-time monitoring, etc. We show that on acyclic Ltss, the full mu-calculus has the same expressive power as its alternation-free fragment. We also present two new algorithms for local model-checking of mu-calculus formulas on acyclic Ltss. Our algorithms are based upon a translation to boolean equation systems and exhibit a better performance than existing model-checking algorithms applied to acyclic Ltss. The first algorithm handles mu-calculus formulas phi with alternation depth ad (phi) greater or equal than 2 and has time complexity O (|phi|^2 * (|S|+|T|)) and space complexity O (|phi|^2 * |S|), where |S| and |T| are the number of states and transitions of the acyclic Lts and |phi| is the number of operators in phi. The second algorithm handles formulas with alternation depth ad (phi) = 1 and has time complexity O (|phi| * (|S|+|T|)) and space complexity O (|phi| * |S|)