Proof-graphs for Minimal Implicational Logic

Proof-graphs for Minimal Implicational Logic

It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific ...

Every super-polynomial proof in purely implicational minimal logic has a polynomially sized proof in classical implicational propositional logic

In this note we show how any formula δ with a proof of super-polynomial size in M→ has a poly-sized proof in K→ . This fact entails that any propositional classical tautology has short proofs, i.e., NP=CoNP. 1 Every classical tautology has short proofs This article reports the results on the investigation on the existence of short proofs for every Classical Propositional Logic tautology. Firstl...

Embedding classical in minimal implicational logic

Consider the problem which set V of propositional variables suffices for StabV `i A whenever `c A, where StabV := {¬¬P → P | P ∈ V }, and `c and `i denote derivability in classical and intuitionistic implicational logic, respectively. We give a direct proof that stability for the final propositional variable of the (implicational) formula A is sufficient; as a corollary one obtains Glivenko’s t...

Unique Normal Proof Property for Implicational Minimal Formulas in the Intuitionistic Logic

A formula is said to have unique-normal proof property and unique-normal proof property ii it has a unique-normal proof and a-normal proof in NJ, respectively. In this report a condition of implicational minimal formulas in the intuitionistic logic for unique-normal proof property is presented making use of unique-normal proof property for negatively non-duplicated formulas. It is shown that th...

Proof Search in Minimal Logic