# MichaelWehar/Propositional-Theorem-Enumerator

A propositional theorem enumerator.
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 Failed to load latest commit information. DynamicInteger.java Formula.java Main.java README.md Scheme.java SchemeMethods.java SchemeNode.java TestIntersection.java

# Background

A statement in propositional calculus might be a tautology, a contradiction, or neither. If it is a tautology, then that means is has a proof. That is, there is a sequence of applications of inference rules and axioms that leads you to the tautology.

But, is there a short proof? How many theorems have short proofs?

There are many different axiom systems for propositional calculus. Whether or not a theorem has a short proof may depend on the axiom system that is used. In addition, the number of theorems with short proofs may vary with the axiom system.

Take a look at some example axiom systems: https://en.wikipedia.org/wiki/List_of_logic_systems

Also, take a look at Gödel's speed-up theorem: https://en.wikipedia.org/wiki/G%C3%B6del's_speed-up_theorem

# What is this project?

We carry out a sort of unification on propositional formulas to create a grammer for generating/enumerating tautologies.

See here for information on the unification algorithm: https://en.wikipedia.org/wiki/Unification_(computer_science)

# Why does it matter?

(1) Determining whether or not a propositional formula is a tautology is a classical coNP-complete problem.

Information on coNP: https://en.wikipedia.org/wiki/Co-NP-complete

The problem of factoring an integer is in coNP. This suggests that the tautology problem might be harder than factoring.

(2) A probabilistic version of this enumerator was used to create hard instance benchmarks for the tautology problem.