Proofsysteme

Proofsysteme, or proof systems, are formal frameworks used to certify the truth of statements in logic and computer science. A proof system describes how a verifier checks a candidate proof for a given statement and, ideally, how short proofs can be for true statements.

Formally, a proof system for a language L ⊆ {0,1}* is a polynomial-time predicate V(x,y) that takes a statement x and a candidate proof y and outputs true or false. The system is complete if for every x in L there exists a y such that V(x,y) = true, and sound if for every x not in L, no y satisfies V(x,y) = true. A pair (x,y) with V(x,y) = true is a proof or certificate of x.

This framework underpins the class NP: a language L is in NP if there exists a polynomial-time predicate V such that x ∈ L iff there exists a y with V(x,y) true. The string y is a certificate of x. Many proof systems aim to minimize proof length, giving rise to notions of succinct or efficient proofs.

Variants include interactive proof systems, where a prover and verifier exchange messages to convince the verifier of x, and probabilistically checkable proofs (PCPs), where a verifier reads only a few bits of a long proof with high confidence. In logic, concrete proof systems include Frege systems and natural deduction; in cryptography, proof systems underlie zero-knowledge proofs and succinct proofs (SNARKs).

Proofsysteme provide a unifying language for understanding verification, efficiency, and trust in deductive reasoning across mathematics, computer science, and cryptography.