Density of Tautologies in Logics with One Variable

Abstract

In the present paper we estimate the ratio of the number of tautologies and the number of formulae of length n by determining the asymptotic density of tautologies in different kinds of logics with one variable. The logics under consideration are the ones with a single connective (nand or nor); negation with a connective (disjunction or conjunction); and several connectives.

Computing the density of tautologies in propositional logic by solving system of quadratic equations of generating functions

“…So the coefficients of I −;B cannot have a larger growth rate than those of W . Since √ m+1 , then we may apply Szegő lemma as [3] and [1], by computing…”

“…This system of equations satisfies the conditions of Drmota-Lalley-Woods Theorem ( [4], p.489), so we have proper analytic functions h A;B such that I A;B (z) = h A;B ( 1 − z/r) for some r > 0. This shows that our generating functions are ready to apply Szegő lemma as [3] and [1], which states that for two generating functions…”

“…just one variable. For the one variable case, algebraic analysis on generating functions and limits of ratio between their coefficients are already done in [1] with ¬ and →, and in [3] with ¬ and ∨, etc. But since the equation for I ∅ is not written anywhere, we present it without a proof.…”