On statement-valued random variables

Alexander Kuznetsov

doi:10.1007/s40590-025-00840-7

On statement-valued random variables

2025-12-17

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Alexander Kuznetsov *

DOI：10.1007/s40590-025-00840-7

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The article considers random formulae as random statement-valued variables and introduces the concept of the average formula as a generalisation of mathematical expectation. Two approaches are examined: the first is based on dissimilarity metrics for syntactic graphs (i.e. structures wherein nodes contain operations' symbols and variables connected by edges according to their notation in the formula) and defines the mathematical expectation of a random formula as the central characteristic of the set of graphs of these formulas. This approach is computationally hard. The second approach resembles fuzzy logic and introduces the concept of probabilistic disjunctive normal form, in which variables have weights and occur with probabilities defined by these weights. This approach is computationally efficient, but it significantly degrades the properties of formulae, distancing them from the propositions of classical logic. The problem described by such structures can arise, for example, when identifying a poorly scanned formula in a text, when filtering AI hallucinations, or in decision-making. We give examples of computing the average formula for a set of disjunctive normal forms and the average finite automaton for a set of deterministic finite automata, as well as obtaining realisations of a probabilistic disjunctive form and reconstructing a probabilistic disjunctive form from its realisations in the form of ordinary disjunctive normal forms.

Keywords:

Non-classical logic

Disjunctive normal forms

Dissimilarity measure

Random variables

Graphs

Formulae

Average

Finite automata

On statement-valued random variables

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