Method for Interpretation of Functions of Propositional Logic by Specific Binary Markov Processes

Abstract

The current paper proposes a method for interpretation of propositional binary logic functions using multi-binary Markov process. This allows logical concepts ‘true’ and ‘false’ to be treated as stochastic variables, and this in two ways—qualitative and quantitative. In the first case, if the probability of finding a Markov process in a definitely true state of this process is greater than 0.5, it is assumed that the Markov process is in state ‘truth.’ Otherwise the Markov process is in state ‘false.’ In quantitative terms, depending on the chosen appropriate binary matrix of transition probabilities it is possible to calculate the probability of finding the process in one of the states ‘true’ or ‘false’ for each of the steps $n = 0 , { }1 , { }2 , { } \ldots$ of the Markov process. A single-step Markov realization is elaborated for standard logic functions of propositional logic; a series of analytical relations are formulated between the stochastic parameters of the Markov process before and after the implementation of the single-step transition. It has been proven that any logical operation can directly, uniquely, and consistently be described by a corresponding Markov process. Examples are presented and a numerical realization is realized of some functions of propositional logic by binary Markov processes.