PM5M: A Formal Axiomatic Language for Probability Theory with Memory-Dependent Limit Theorems
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2025
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| author | Nikitin Leonid Vladimirovich, Nikitin Leonid Vladimirovich |
| author_facet | Nikitin Leonid Vladimirovich, Nikitin Leonid Vladimirovich |
| contents | <p><strong>Description</strong><br>This work presents the completed formal axiomatic language <strong>PM5M</strong> (“Probability with Memory, 5 Modules”) and its application to solving memory–dependent limit theorems for sequences of dependent events. PM5M extends classical probability theory by introducing explicit <em>memory weights</em> αn\alpha_nαn into the probabilistic framework, enabling rigorous handling of long–range dependencies and decaying influence of past events.</p> <p>The paper includes:</p> <ul> <li> <p><strong>Full syntax and semantics</strong> of PM5M in BNF form, with precise derivation rules and model–theoretic interpretation of memory.</p> </li> <li> <p><strong>Weighted Borel–Cantelli–type theorems</strong> under Kochen–Stone–style covariance decay, yielding necessary and sufficient criteria for almost–sure infinitely–often occurrence in processes with memory.</p> </li> <li> <p><strong>Reduction proofs</strong> showing that PM5M collapses to classical probability when αn\alpha_nαn is constant or equal to 1.</p> </li> <li> <p><strong>Internal consistency analysis</strong> ensuring that the axioms are non–contradictory and compatible with standard probability theory.</p> </li> <li> <p><strong>Empirical validation</strong> via Monte Carlo simulations for multiple dependency profiles (ρ\rhoρ) and weight/probability decay laws, confirming the theoretical predictions in both divergent and convergent regimes.</p> </li> </ul> <p>This framework closes a long–standing gap in handling the interaction between event dependency and memory in probability theory. It is both <strong>formally checkable</strong> and <strong>empirically verifiable</strong>, making it applicable to theoretical research, stochastic process modeling, and AI systems that require probabilistic reasoning with memory effects.<br><br>This release presents the original <strong>PM5M probabilistic–modal framework with memory</strong>, created and authored by <strong>Leonid V. Nikitin (Independent Researcher, Kazan, Russia)</strong>.<br>It implements a complete, runnable model of memory-dependent probability accumulation, integrating formal logical structures from Nikitin’s Theorems I–II.<br>The system combines classical and memory-augmented probability modes, parameter presets, interactive tuning, automated report generation, and secure signature/timestamp embedding.<br>It is intended for research in probability theory, logic, and AI systems, as well as for practical modeling of repeated-event inevitability in domains such as forecasting, risk assessment, and intelligent decision-making.<br>All mathematical concepts and the PM5M formalism are original to the author and protected under the chosen license.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16785445 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | PM5M: A Formal Axiomatic Language for Probability Theory with Memory-Dependent Limit Theorems Nikitin Leonid Vladimirovich, Nikitin Leonid Vladimirovich Probability theory Modal logic Dependent events Covariance decay Memory effects Kochen–Stone theorem Asymptotic probability <p><strong>Description</strong><br>This work presents the completed formal axiomatic language <strong>PM5M</strong> (“Probability with Memory, 5 Modules”) and its application to solving memory–dependent limit theorems for sequences of dependent events. PM5M extends classical probability theory by introducing explicit <em>memory weights</em> αn\alpha_nαn into the probabilistic framework, enabling rigorous handling of long–range dependencies and decaying influence of past events.</p> <p>The paper includes:</p> <ul> <li> <p><strong>Full syntax and semantics</strong> of PM5M in BNF form, with precise derivation rules and model–theoretic interpretation of memory.</p> </li> <li> <p><strong>Weighted Borel–Cantelli–type theorems</strong> under Kochen–Stone–style covariance decay, yielding necessary and sufficient criteria for almost–sure infinitely–often occurrence in processes with memory.</p> </li> <li> <p><strong>Reduction proofs</strong> showing that PM5M collapses to classical probability when αn\alpha_nαn is constant or equal to 1.</p> </li> <li> <p><strong>Internal consistency analysis</strong> ensuring that the axioms are non–contradictory and compatible with standard probability theory.</p> </li> <li> <p><strong>Empirical validation</strong> via Monte Carlo simulations for multiple dependency profiles (ρ\rhoρ) and weight/probability decay laws, confirming the theoretical predictions in both divergent and convergent regimes.</p> </li> </ul> <p>This framework closes a long–standing gap in handling the interaction between event dependency and memory in probability theory. It is both <strong>formally checkable</strong> and <strong>empirically verifiable</strong>, making it applicable to theoretical research, stochastic process modeling, and AI systems that require probabilistic reasoning with memory effects.<br><br>This release presents the original <strong>PM5M probabilistic–modal framework with memory</strong>, created and authored by <strong>Leonid V. Nikitin (Independent Researcher, Kazan, Russia)</strong>.<br>It implements a complete, runnable model of memory-dependent probability accumulation, integrating formal logical structures from Nikitin’s Theorems I–II.<br>The system combines classical and memory-augmented probability modes, parameter presets, interactive tuning, automated report generation, and secure signature/timestamp embedding.<br>It is intended for research in probability theory, logic, and AI systems, as well as for practical modeling of repeated-event inevitability in domains such as forecasting, risk assessment, and intelligent decision-making.<br>All mathematical concepts and the PM5M formalism are original to the author and protected under the chosen license.</p> |
| title | PM5M: A Formal Axiomatic Language for Probability Theory with Memory-Dependent Limit Theorems |
| topic | Probability theory Modal logic Dependent events Covariance decay Memory effects Kochen–Stone theorem Asymptotic probability |
| url | https://doi.org/10.5281/zenodo.16785445 |