Authors: Dautović, Šejla 
Doder, Dragan
Ognjanović, Zoran 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Logics for Reasoning about Knowledge and Conditional Probability
First page: 19
Last page: 21
Related Publication(s): Book of Abstracts of the 10th International Conference Logic and Applications, LAP 2021
Conference: 10th International Conference Logic and Applications, LAP 2021, September 20 - 24, 2020, Dubrovnik, Croatia
Issue Date: Sep-2021
Rank: M34
Epistemic logics are formal models designed in order to reason about the
knowledge of agents and their knowledge of each other’s knowledge. During the
last couple of decades, they have found applications in various fields such as
game theory, the analysis of multi-agent systems in computer science and arti-
ficial intelligence [8, 9, 19]. In parallel, uncertain reasoning has emerged as one
of the main fields in artificial intelligence, with many different tools developed
for representing and reasoning with uncertain knowledge. A particular line of
research concerns the formalization in terms of logic, and the questions of pro-
viding an axiomatization and decision procedure for probabilistic logic attracted
the attention of researchers and triggered investigation about formal systems
for probabilistic reasoning [1, 7, 10, 11, 15, 16]. Fagin and Halpern [6] empha-
sised the need for combining those two fields for many application areas, and
in particular in distributed systems applications, when one wants to analyze
randomized or probabilistic programs. They developed a joint framework for
reasoning about knowledge and probability, proposed a complete axiomatization
and investigated decidability of the framework. Based on the seminal paper by
Fagin, Halpern and Meggido [7], they extended the propositional epistemic lan-
guage with formulas which express linear combinations of probabilities, called
linear weight formulas, i.e., the formulas of the form a1w(α1)+...+akw(αk) ≥r,
where aj ’s and r are rational numbers. They proposed a finitary axiomatization
and proved weak completeness, using a small model theorem.
In this talk, we propose two logics that extend the logic from [6] by also
allowing formulas that can represent conditional probability. First we present
a propositional logic for reasoning about knowledge and conditional probability
from [2]. Then we discuss how to develop its first-order extension. Our languages
contain both knowledge operators Ki (one for each agent i) and conditional
probability formulas of the form
a1wi(α1,β1) + ... + akwi(αk,βk) ≥r.
The expressions of the form wi(α,β) represent conditional probabilities that
agent i places on events according to Kolmogorov definition: P(A|B) = P (A∩B)
P (B) if P(B) > 0, while P(A|B) is undefined when P(B) = 0. The corresponding se-
mantics consists of enriched Kripke models, with a probability measure assigned
to every agent in each world.
Our main results are sound and strongly complete (every consistent set of
formulas is satisfiable) axiomatizations for both logics. We prove strong com-
pleteness using an adaptation of Henkin’s construction, modifying some of our
earlier methods [3, 5, 4, 15, 16]. Our axiom system contains infinitary rules of
inference, whose premises and conclusions are in the form of so called k-nested
implications. This form of infinitary rules is a technical solution already used
in probabilistic, epistemic and temporal logics for obtaining various strong ne-
cessitation results [13, 14, 17, 18]. We also prove that our propositional logic is
decidable, combining the method of filtration [12] and a reduction to a system
of inequalities.
Keywords: Probabilistic logic | Epistemic logic | Completeness
Publisher: University Center Dubrovnik, Croatia
Project: Advanced artificial intelligence techniques for analysis and design of system components based on trustworthy BlockChain technology - AI4TrustBC 

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