Unranked Probabilistic Theory
Project Presentation
Anriette Bishara1, Lia Kurtanidze2, Mikheil Rukhaia1 and Lali Tibua1
1I.Vekua Institute of Applied Mathematics, I.Javakhishvili Tbilisi State University
2Georgian National University SEU

Introduction

Objectives

I Since the early days of Artificial Intelligence logical and probabilistic
methods have been independently used to solve tasks requiring some
sort of “intelligence”.

Develop unranked probabilistic logic (UPL) formalism
and study its soundness and completeness properties.

I Probability theory deals with the challenges posed by uncertainty,
while logic is more often used for reasoning with perfect knowledge.
I Modern AI is area where computer science, mathematics (probability
theory, numerical analysis, etc.), and logic come together.
I Researchers started combining logical and probabilistic methods in asingle framework and developed several formalisms and programming
tools.

Develop reasoning
methods for UPL
and study wellbehaved fragments.

I All probabilistic logic formalisms studied so far are either propositional,or permit only individual variables that can be instantiated by a singleterm.
I On the other hand, there are very useful theories of symbolic logic,
which are using sequence variables and unranked function/predicatesymbols.

Proof-theoretic study of
UPL and corresponding
reasoning methods.

Prototype implementation and experiments.

Methods

Aims

I Following Bruno Buchberger, we identify computing, solving, andproving as three major activities in mathematical reasoning.

I Develop a novel theory, where sequence variables, unranked terms,
and probabilistic primitives will be available together.

I Computing: provide an expression E (term, formula, program) and“evaluate” the expression for a value v , E[x 7→ v] (rewriting,simplification).

I Such formalism is interesting from a theoretical point of view as well asfrom a practical one since it provides a very flexible and expressive
platform to model various problems coming from real-world
applications.

I Solving: provide an expression E with free variable x and find allvalues v such that E[x 7→ v] holds (unification and constraint solving).
I Proving: provide an expression E with free variable x and generate a
proof (or disprove) that for all values v , E[x 7→ v] holds (reasoningmethods).

Unranked Terms
I The unranked term is a first-order term, where the same function
symbol can occur in different places with different numbers ofarguments.

I The expected results are both of theoretical and practical character.

I Sequence variables can be instantiated by finite (possibly empty)
sequence of (unranked) terms.

I Theoretical results: development of a new, unranked probabilistic logic
and the corresponding reasoning methods.

I Example: f (g , f (S), g(a , x)) and σ = [S 7→ (g(a), x), x 7→ f (a)]I Practical applications: expected results can be used for knowledge
modelling in the areas of semantic web and web mining, medicine,
activity recognition, transportation systems, e-commerce and the like.

Sfg7→

gfaSgxx7→aExpected Results

fax

Referencesf (g , f (S), g(a , x))σ = f (g , f (g(a), x), g(a , f (a)))
fggfg
ax

afa[1] Temur Kutsia and Bruno Buchberger.
Predicate logic with sequence variables and sequence function symbols.
In International Conference on Mathematical Knowledge Management, pages
205–219. Springer, 2004.
[2] Zoran Ognjanović et al.
Probabilistic Extensions of Various Logical Systems.
Springer, 2020.
[3] Bruno Buchberger.
Symbolic computation in software science: My personal view.
arXiv preprint arXiv:2109.02806, 2021.

Probabilistic Logics
Acknowledgements
I There are two kinds of formal probabilistic logic languages having:. Probabilistic operator P≥s α : the probability of holding α is at least s.
. Probabilistic quantifier (Px > r)α (x): the probability of the set{x | α (x)} is greater than r.
I Semantics: there is a probability that a sentence is true, but thesentence itself can be either true or false and no other value ispossible (unlike many-valued logics).
I Properties: compactness usually fails and even more, the strong
completeness is not available in some of probabilistic logics.

http://viam.science.tsu.ge/new/

This project was supported by Shota Rustaveli National Science Foundation of Georgia under the grant №FR-22-4254.

Contact Information
I Email: mrukhaia@logic.at
I Web: https://www.viam.science.tsu.ge/new/index.
php?lang=eng&page=projects&subpage=125

mrukhaia@logic.at