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Ondrej Majer

Publications and source records attributed to Ondrej Majer.

9 recordsLinked to original sources
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↗
arXiv2025

Two-layered logics for probabilities and belief functions over Belnap--Dunn logic

This paper is an extended version of an earlier submission to WoLLIC 2023. We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic. We consider two probabilistic logics that present two perspectives on the probabilities in the Belnap--Dunn logic: $\pm$-probabilities and $\mathbf{4}$-probabilities. In the first case, every event $ϕ$ has independent positive and negative measures that denote the likelihoods of $ϕ$ and $\negϕ$, respectively. In the second case, the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $ϕ$. In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions. They equip events with two measures (positive and negative) with their main difference being whether the negative measure of $ϕ$ is defined as the belief in $\negϕ$ or treated independently as the plausibility of $\negϕ$. We provide a sound and complete Hilbert-style axiomatisation of the logic of $\mathbf{4}$-probabilities and establish faithful translations between it and the logic of $\pm$-probabilities. We also show that the satisfiability problem in all logics is $\mathsf{NP}$-complete.

math.LO↗
arXiv2024

Qualitative reasoning in a two-layered framework

The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality $\Be$ that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.

math.LO↗
arXiv2023

Reasoning with belief functions over Belnap--Dunn logic

We design an expansion of Belnap--Dunn logic with belief and plausibility functions that allow non-trivial reasoning with inconsistent and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in~\cite{FaginHalpernMegiddo1990}. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of Łukasiewicz logic. The second approach is inspired by~\cite{BaldiCintulaNoguera2020}. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.

math.LO↗
arXiv2022

Updating belief functions over Belnap--Dunn logic

Belief and plausibility are weaker measures of uncertainty than that of probability. They are motivated by the situations when full probabilistic information is not available. However, information can also be contradictory. Therefore, the framework of classical logic is not necessarily the most adequate. Belnap-Dunn logic was introduced to reason about incomplete and contradictory information. Klein et al and Bilkova et al generalize the notion of probability measures and belief functions to Belnap-Dunn logic, respectively. In this article, we study how to update belief functions with new pieces of information. We present a first approach via a frame semantics of Belnap-Dunn logic.

cs.AI↗
arXiv2021

The (Relevant) Logic of Scientific Discovery

This paper presents a thoroughgoing interpretation of a weak relevant logic built over the Dunn-Belnap four-valued semantics in terms of the communication of information in a network of sites of knowledge production (laboratories). The knowledge communicated concerns experimental data and the regularities tested using it. There have been many nods to interpretations similar to ours - for example, in Dunn (1976), Belnap (1977). The laboratory interpretation was outlined in Bilkova et al. (2010). Our system is built on the Routley--Meyer semantics for relevant logic equipped with a four-valued valuation of formulas, where labs stand in for situations, and the four values reflect the complexity of assessing results of experiments. This semantics avoids using the Routley star, on the cost of introducing a further relation, required in evaluating falsity assignments of implication. We can however provide a natural interpretation of two accessibility relations - confirmation and refutation of hypotheses are two independent processes in our laboratory setup. This setup motivates various basic properties of the accessibility relations, as well as a number of other possible restrictions. This gives us a flexible modular system which can be adjusted to specific epistemic contexts. As perfect regularities are rarely, or perhaps never, actually observed, we add probabilities to the logical framework. As our logical framework is non-classical, the probability is non-classical as well, satisfying a weaker version of Kolmogorov axioms (cf. Priest 2006). We show that these probabilities allow for a relative frequency as well as for a subjective interpretation (we provide a Dutch book argument). We further show how to update the probabilities and to distinguish conditional probabilities from the probability of conditionals.

math.LO↗
arXiv2020

How to reason with inconsistent probabilistic information?

A recent line of research has developed around logics of belief based on evidence. The approach of Bílková et al understands belief as based on information confirmed by a reliable source. We propose a finer analysis of how belief can be based on information, where the confirmation comes from multiple possibly conflicting sources and is of a probabilistic nature. We use Belnap-Dunn logic and its probabilistic extensions to account for potentially contradictory information on which belief is grounded. We combine it with an extension of Lukasiewicz logic, or a bilattice logic, within a two-layer modal logical framework to account for belief.

cs.LO↗
arXiv2020

Probabilities with Gaps and Gluts

Belnap-Dunn logic (BD), sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes' and Jeffrey updating and also suggest mechanisms for aggregating information from different sources.

math.LO↗