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Probabilistic Disjunctive Normal Forms in Temporal Logic and Automata Theory

Alexander Kuznetsov

Abstract

This article introduces probabilistic disjunctive normal forms (PDNFs) as a framework for representing and reasoning about uncertainty in logical systems. Unlike classical DNFs, PDNFs assign real-valued weights to variables, encoding probabilistic information about their presence, absence, or negation. Then we construct a vector space of PDNFs that allows algebraic evidence combination. PDNFs are interpreted as probability distributions over venjunctions (temporal logic constructs) and as integrable functions over partitioned intervals, where the integrals determine variable probabilities. This dual perspective allows for a Banach space structure and the application of functional analysis. We demonstrate that, under exponential parametrisation, PDNF addition aligns with Bayesian evidence fusion and derive bounds for outcome identification from random samples. The formalism thus bridges logic, numerical methods, and continuous probability.

Probabilistic Disjunctive Normal Forms in Temporal Logic and Automata Theory

Abstract

This article introduces probabilistic disjunctive normal forms (PDNFs) as a framework for representing and reasoning about uncertainty in logical systems. Unlike classical DNFs, PDNFs assign real-valued weights to variables, encoding probabilistic information about their presence, absence, or negation. Then we construct a vector space of PDNFs that allows algebraic evidence combination. PDNFs are interpreted as probability distributions over venjunctions (temporal logic constructs) and as integrable functions over partitioned intervals, where the integrals determine variable probabilities. This dual perspective allows for a Banach space structure and the application of functional analysis. We demonstrate that, under exponential parametrisation, PDNF addition aligns with Bayesian evidence fusion and derive bounds for outcome identification from random samples. The formalism thus bridges logic, numerical methods, and continuous probability.
Paper Structure (17 sections, 6 theorems, 102 equations, 5 figures, 1 table)

This paper contains 17 sections, 6 theorems, 102 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Probabilistic Disjunctive Normal Forms and their Properties
  3. Transition from discrete structure to continuos functions
  4. Structural similarity between PDNFs, classical DNFs, venjunction, and asynchronous sequential logic
  5. Probability Measure on the Space of Encoders
  6. PDNFs as Generators of Finite Languages
  7. Probabilistic Disjunctive Normal Forms as Automata Arrays
  8. Connection of PDNFs and Markov Processes
  9. Strong measurability.
  10. Integrability of the norm.
  11. Conclusion
  12. Discussion on the PDNFs Algebra
  13. Discussion on the PDNF Structure
  14. Discussion on the PDNFs Measure
  15. PDNFs and Bayesian Fusions
  16. ...and 2 more sections

Key Result

Theorem 5.2

Let $A$ be a deterministic finite automaton with $m$ output variables $x_1,\dots,x_m$, observed by two independent agents $ag_1$ and $ag_2$ over $n$ synchronised time steps. Suppose each agent records its observations as a PDNF $Z_k \in \mathcal{B}^m_{\mathbf{F}}(n)$ ($k=1,2$), using the same family Then the PDNF $Z = Z_1 + Z_2$ (with addition defined by (eqsum)) represents the combined evidence f

Figures (5)

  • Figure 1: Piecewise linear encoders $Z_S\in \mathcal{E}^8_{\mathbf{F}}(3)$ (A) and $U_S\in \mathcal{E}_{\mathbf{F}}(3\times 8)$ (B) for the "Soviet Story" PDNF
  • Figure 2: Comparison of the classical DNF $x\wedge \overline{y}\vee x\wedge y$ (a) and the PDNF $x^{\xi_1}\wedge y^{\eta_1}\vee x^{\xi_2}\wedge y^{\eta_2}$ (b)
  • Figure 3: The "temperature" components $F_{-1}^x$ (a), $F_0^x$ (b), $F_1^x$ (c) of the mapping $\mathbf{F}$
  • Figure 4: Comparison of averaged values of the mapping components $F_{-1}^x$ (A), $F_0^x$ (B), $F_1^x$ (C) (circles) and the corresponding values $\chi^x_{-1}(t)$, $\chi^x_{0}(t)$, $\chi^x_{1}(t)$ (triangles)
  • Figure 5: Comparison of averaged values of the mapping component $(F_{-1}^y, F_0^y, F_1^y)$ (circles) and the tuple $(\chi^y_{-1}(t), \chi^y_{0}(t),\chi^y_{1}(t))$ (triangles)

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 3.1
  • Example 4.1
  • Definition 5.1: Bayesian fusion for ternary outcomes
  • ...and 15 more