Tautologies over implication with negative literals

@article{Fournier2010TautologiesOI,
  title={Tautologies over implication with negative literals},
  author={Herv{\'e} Fournier and Dani{\`e}le Gardy and Antoine Genitrini and Marek Zaionc},
  journal={Mathematical Logic Quarterly},
  year={2010},
  volume={56}
}
We consider logical expressions built on the single binary connector of implication and a finite number of literals (Boolean variables and their negations). We prove that asymptotically, when the number of variables becomes large, all tautologies have the following simple structure: either a premise equal to the goal, or two premises which are opposite literals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 

The fraction of large random trees representing a given Boolean function in implicational logic

The relation between the probability of a function and its complexity is obtained when random expressions are drawn according to a critical branching process and it is proved that most expressions computing any given function in this system are “simple”.

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

A fundamental relation between generating functions' values on the singularity point and ratios of coefficients is discovered, which can be understood as another intepretation of the Szegő lemma for certain quadratic systems.

On counting untyped lambda terms

A sprouting tree model for random boolean functions

A new probability distribution for Boolean functions of k variables of size n is defined, which is compared with two previously‐known distributions induced by two other random trees: the Catalan tree and the Galton‐Watson tree.

The Growing Trees Distribution on Boolean Functions

A probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions is defined, inspired by the growth model of Binary Search Trees, and is called the growing tree law.

Enumerating lambda terms by weighted length of their De Bruijn representation

References

SHOWING 1-10 OF 22 REFERENCES

Classical and Intuitionistic Logic Are Asymptotically Identical

This paper considers logical formulas built on the single binary connector of implication and a finite number of variables and proves that asymptotically, all classical tautologies are intuitionistic.

Intuitionistic vs. Classical Tautologies, Quantitative Comparison

It is proved that the limit of that fraction of formulas which differ only in the naming of variables is 1 when n tends to infinity.

Quantitative Comparison of Intuitionistic and Classical Logics - Full Propositional System

Two different approaches are applied, to estimate the asymptotic fraction of intuitionistic tautologies among classical tautology, obtaining the same results for both.

On the Asymptotic Density of Tautologies in Logic of Implication and Negation

The paper solves the problem of finding the asymptotic probability of the set of tautologies of classical logic with one propositional variable, implication and negation and proves the existence of this limit for classical (and at the same time intuitionistic) logic of implication built with exactly one variable.

Probability distribution for simple tautologies

On the density and the structure of the Peirce-like formulae †

Within the language of propositional formulae built on implication and a finite number of variables k, we analyze the set of formulae which are classical tautologies but not i ntuitionistic (we call

And/or tree probabilities of Boolean functions

Two probability distributions on Boolean functions defined in terms of their representations by $\texttt{and/or}$ trees (or formulas) are considered, with special attention being paid to the constant function $\textit{True}$ and read-once functions in a fixed number of variables.

Statistical properties of simple types

From the lower and upper bounds presented, it is deduced that at least 1/3 of classical tautologies are intuitionistic, or the density or asymptotic probability of provable intuitionistic propositional formulas in the set of all formulas.

Asymptotic Density for Equivalence

And/Or Trees Revisited

A detailed analysis of the functions enumerating some sub-families of trees, and of their radius of convergence, allows us to improve on the upper bound of $P(f)$, established by Lefmann and Savický.