22 research outputs found
On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components
For every positive integer k we consider the class SCCk of all finite graphs
whose strongly connected components have size at most k. We show that for every
k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level
Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1
and Pi1). This contrasts with the class of all graphs, where
Delta2=Comp(Sigma1,Pi1)
On Modal μ -Calculus and Gödel-Löb Logic
We show that the modal μ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modalμ ~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formul
The Modal mu-Calculus and The Gödel-Löb Logic
We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [vBe06]. Further, we introduce the modal µ∼-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula
Ruitenburg's Theorem mechanized and contextualized
In 1984, Wim Ruitenburg published a surprising result about periodic
sequences in intuitionistic propositional calculus (IPC). The property
established by Ruitenburg naturally generalizes local finiteness
(intuitionistic logic is not locally finite, even in a single variable).
However, one of the two main goals of this note is to illustrate that most
"natural" non-classical logics failing local finiteness also do not enjoy the
periodic sequence property; IPC is quite unique in separating these properties.
The other goal of this note is to present a Coq formalization of Ruitenburg's
heavily syntactic proof. Apart from ensuring its correctness, the formalization
allows extraction of a program providing a certified implementation of
Ruitenburg's algorithm.Comment: This note has been prepared for the informal (pre-)proceedings of
FICS 2024. The version to be submitted to the post-proceedings volume is
going to be significantly different, focusing on the Coq formalization, as
requested by referees and the P
The Modal μ-Calculus Hierarchy on Restricted Classes of Transition Systems
We discuss the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment
A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic
We define a infinitary labelled sequent calculus for PDL, G3PDL^{\infty}. A
finitarily representable cyclic system, G3PDL^{\omega}, is then given. We show
that both are sound and complete with respect to standard models of PDL and,
further, that G3PDL^{\infty} is cut-free complete. We additionally investigate
proof-search strategies in the cyclic system for the fragment of PDL without
tests