arXiv subjects

Wolfgang Poiger

Publications and source records attributed to Wolfgang Poiger.

7 recordsLinked to original sources

arXiv2026

Many-valued coalgebraic dynamic logics: Safety and strong completeness via reducibility

We present a coalgebraic framework for studying generalisations of dynamic modal logics such as PDL and game logic in which both the propositions and the semantic structures can take values in an algebra $\mathbf{A}$ of truth-degrees. More precisely, we work with coalgebraic modal logic via $\mathbf{A}$-valued predicate liftings and interpret actions (abstracting programs and games) as $\mathsf{F}$-coalgebras where the functor $\mathsf{F}$ represents some type of $\mathbf{A}$-weighted system. We also allow combinations of crisp propositions with $\mathbf{A}$-weighted systems and vice versa. We introduce coalgebra operations and tests, with a focus on operations that are reducible in the sense that modalities for composed actions can be reduced to compositions of modalities for the constituent actions. We prove that reducible operations are safe for bisimulation and behavioural equivalence, and prove a general strong completeness result, from which we obtain new strong completeness results for $\mathbf{2}$-valued iteration-free PDL with $\mathbf{A}$-weighted accessibility relations when $\mathbf{A}$ is a finite chain, and for many-valued iteration-free game logic with many-valued strategies based on finite Lukasiewicz logic.

cs.LO↗

arXiv2026

Pointed Modal Abelian Logic, Algebraically

In this article, we investigate the pointed modal logic of reals. We first establish its relational (Kripke) semantics with bounded valuations in the Abelian l-group of real numbers with the distinguished negative constant -1. To study this logic algebraically, we introduce the variety of negatively pointed modal Abelian l-groups, in particular we focus on the strongly pointed members thereof. Constructing complex algebras and canonical frames, we establish a Truth Lemma connecting the relational and algebraic frameworks. Since finitary axiomatizations cannot fully capture the Kripke validities of the reals we introduce further algebraic constraints, in particular including an infinitary Archimedean-style rule. Finally, we prove a corresponding `infinitary algebraic completeness' result for pointed modal Abelian logic with respect to the variety of pointed modal Abelian l-groups.

cs.LO↗

arXiv2026

Knowledge on a Budget

In various computational systems, accessing information incurs time, memory or energy costs. However, standard epistemic logics usually model the acquisition of evidence as a cost-free process, which restricts their applicability in environments with limited resources. In this paper, we bridge the gap between qualitative epistemic reasoning and quantitative resource constraints by introducing semiring-annotated topological spaces (seats). Building on Topological Evidence Logic (TEL), we extend the representation of evidence as open sets, adding an annotation function that maps evidence to semiring ideals, representing the resource budgets sufficient for observation. This framework allows us to reason not only about what is observable in principle, but also about what is affordable given a specific budget. We develop a family of seat-based epistemic logics with resource-indexed modalities and provide sound, strongly complete axiomatisations for these logics. Furthermore, we introduce suitable notions of bisimulation and disjoint union to delineate the expressive power of our framework.

cs.LO↗

arXiv2024

Many-valued coalgebraic logic over semi-primal varieties

We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.

cs.LO↗

arXiv2023

Natural dualities for varieties generated by finite positive MV-chains

We provide a simple natural duality for the varieties generated by the negation- and implication- free reduct of a finite MV-chain. We study these varieties through the dual equivalence thus obtained. For example, we fully characterize their algebraically closed, existentially closed and injective members. We also explore the relationship between this natural duality and Priestley duality in terms of distributive skeletons and Priestley powers.

math.RA↗

arXiv2023

New perspectives on semi-primal varieties

We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.

math.LO↗

arXiv2022

The Minor Order of Homomorphisms via Natural Dualities

We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.

math.CO↗