$\mathbf {LPP_2}$, a Propositional Probability Logic Without Iterations of Probability Operators

Abstract

The probability logic denoted $LPP_2$ is described with the aim to give a clear, step-by-step introduction to the field and the main proof techniques that will be used elsewhere in the book. The logic enriches propositional calculus with probabilistic operators of the form $P_{\ge s}$ with the intended meaning “probability is at least s”. In $LPP_2$ the operators are applied to propositional formulas, while iterations of probability operators are not allowed. Possible world semantics with a finitely additive probability measure on sets of worlds definable by formulas is defined, so that formulas remain true or false. The corresponding axiomatization is provided. The axiom system is infinitary. It contains an infinitary rule with countable many premisses and one conclusion. The rule is related to the Archimedean property of real numbers. The logic $LPP_2$ is not compact: there are unsatisfiable sets of formulas that are finitely satisfiable. Some of the consequences of non-compactness are described. Then, soundness and strong completeness of the logic is proved with respect to several classes of probability models. This is followed by a proof of decidability of PSAT, the satisfiability problem for $LPP_2$, which is NP-complete. Finally, a heuristic approach to PSAT is presented. This Chapter covers some results from Ikodinović et al., Int J Approx Reason, (55):1830–1842, 2014, [3], Jovanović et al., Variable neighborhood search for the probabilistic satisfiability problem, 2007, [4], Kokkinis et al., Logic J. IGPL, (23):662–687, 2015, [5], Ognjanović, J. Logic Comput, (9):181–195, 1999, [6], Ognjanović et al., Theor Comput Sci, (247):191–212, 2000, [7], Ognjanović et al., A genetic algorithm for satisfiability problem in a probabilistic logic, 2001, [8],Ognjanović et al., A Genetic Algorithm for Probabilistic SAT Problem, 2004, [9], Ognjanović et al., A Hybrid Genetic and Variable Neighborhood Descent for Probabilistic SAT Problem, 2005, [10], Ognjanović et al., Zbornik Radova, Subseries Logic in Computer Science, 2009, [11], Rašković and Ognjanović, Some propositional probabilistic logics, 1996, [12], Rašković and Ognjanović, A first order probability logic, $LP_Q$, 1999, [13], Stojanović et al., Appl Soft Comput, (31):339–347, 2015, [14].