Computing the density of tautologies in propositional logic by solving system of quadratic equations of generating functions
Abstract
In this paper, we will provide a method to compute the density of tautologies among the set of wellformed formulae consisting of $m$ variables, a negation symbol and an implication symbol, which has a possibility to be applied for other logical systems. This paper contains computational numerical values of the density of tautologies for two, three, and four variable cases. Also, for certain quadratic systems, we will introduce the $s$cut concept to make a better approximation when we compute the ratio by bruteforce counting, and discover a fundamental relation between generating functions' values on the singularity point and ratios of coefficients, which can be understood as another intepretation of the Szegő lemma for such quadratic systems. With this relation, we will provide an asymptotic lower bound $m^{1}(7/4)m^{3/2}+O(m^{2})$ of the density of tautologies as $m$ goes to the infinity.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.00511
 Bibcode:
 2020arXiv200300511E
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics
 EPrint:
 34 pages