Related papers: Flat modal fixpoint logics with the converse modal…
The continuous modal mu-calculus is a fragment of the modal mu-calculus, where the application of fixpoint operators is restricted to formulas whose functional interpretation is Scott-continuous, rather than merely monotone. By…
Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular…
Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
We study the topological $\mu$-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and FMP over general topological spaces, as well as over $T_0$ and $T_D$ spaces. We also investigate…
This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form \gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the language…
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal{M}$, or an axiomatization $S$ thereof, we find a modal…
The modal mu-calculus is obtained by adding least and greatest fixed-point operators to modal logic. Its alternation hierarchy classifies the mu-formulas by their alternation depth: a measure of the codependence of their least and greatest…
Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below…
We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central…
Orthogonality is a notion based on the duality between programs and their environments used to determine when they can be safely combined. For instance, it is a powerful tool to establish termination properties in classical formal systems.…
Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic interpreted over coalgebras for an arbitrary set functor. We then consider invariance under behavioral equivalence of MSO-formulas.…
Fixpoints are an important ingredient in semantics, abstract interpretation and program logics. Their addition to a logic can add considerable expressive power. One general issue is how to define proof systems for such logics. Here we…
The two-way modal mu-calculus is the extension of the (standard) one-way mu-calculus with converse (backward-looking) modalities. For this logic we introduce two new sequent-style proof calculi: a non-wellfounded system admitting infinite…
The coalgebraic approach to modal logic provides a uniform framework that captures the semantics of a large class of structurally different modal logics, including e.g. graded and probabilistic modal logics and coalition logic. In this…
This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in…
We prove a general decomposition theorem for the modal $\mu$-calculus $L_\mu$ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two…
This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed…
We present a fixed point theorem for a class of (potentially) non-monotonic functions over specially structured complete lattices. The theorem has as a special case the Knaster-Tarski fixed point theorem when restricted to the case of…
A $\mu$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,\mu_{x}.f)$ where $\mu_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are…