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

Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 657)

## 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.

## References

1. 1.
Kleene, S.C.: Mathematical Logic. Wiley, N.Y. (1977)Google Scholar
2. 2.
Zadeh, L.A.: The Concept of a Linguistic Variable and Its Applications to Approximate Reasoning. Elsevier Publ. C, N.Y. (1973)Google Scholar
3. 3.
Stern, A.: Matrix Logic and Mind. North Holland, N.Y. (1992)Google Scholar
4. 4.
Sgurev, V.S.: Network flow approach in logic for problem solving. Int. J. Inf. Theor. Appl. 7 (1995)Google Scholar
5. 5.
Gorodetsky, V.I.: Bayes inference and decision making in artificial intelligence systems. In: Industrial Applications of AI, Elsevier S. Publ. B.V., Amsterdam, pp. 276–281 (1991)Google Scholar
6. 6.
Sgurev, V., Jotsov, V.: Discrete Markov process interpretation of propositional logic. In: Proceedings of IEEE Conference, IS’10, London (2010)Google Scholar
7. 7.
Neapolitan R.E.: Learning Bayesian Networks. Prentice Hall (2004)Google Scholar
8. 8.
Kemeny, J.G., Snell, J.K., Knapp, A.W.: Denumerable Markov Chain. Springer, Heidelberg, Berlin (1976)Google Scholar
9. 9.
Mine, H., Osaki, S.: Markovian Decision Processes. American Elsevier Publ. Co, N.Y. (1970)Google Scholar
10. 10.
Domingos, P., Richardson, M.: Markov Logic: A Unifying Framework for Statistical Relational Learning. In: Getoor, L., Taskar, B. (eds.) Introduction to Statistical Relational Learning, pp. 339–371. MIT Press, Cambridge, MA (2007)Google Scholar
11. 11.
Singla, P., Domingos, P.: Discriminative training of Markov logic networks. In: AAAI-2005, pp. 868–873 (2005)Google Scholar
12. 12.
Huang, S., Zhang, S., Zhou, J. Chen, J.: Coreference Resolution using Markov Logic Network. Advances in Computational Linguistics, Research in Computing Science vol. 41, pp. 157–168 (2009)Google Scholar
13. 13.
Mihalkova, L., Huynh, T., Mooney, R.J.: Mapping and revising Markov logic networks for transfer learning. In: Proceedings of the 22nd Conference on Artificial Intelligence (AAAI-07). pp. 608–614, Vancouver, Canada, July 2007Google Scholar 