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Emre Yolcu is a computer scientist whose research includes proof complexity, Boolean satisfiability and automated deduction. These subjects distinguish the existence of a short formal proof from the computational task of finding it.^[ref]

With Xinyu Wu and Marijn Heule, he studied Mycielski graphs through clausal proofs of their colouring requirements. Their work constructed short proofs in a propagation-redundancy system and compared their availability with the behaviour of automated proof search.^[ref]

Homepage: emreyolcu.com/

Proof complexity measures resources needed to express proofs in a specified propositional proof system. Typical quantities include the total number of symbols, the number of inference steps and the space needed to retain intermediate formulas. A short theorem statement can require a much longer proof, and changing the allowed rules can change the required length dramatically.

For example, a resolution refutation of an unsatisfiable clause set derives the empty clause by resolving pairs of clauses on opposite literals. Verifying a supplied derivation is a local computation on its lines; searching for those lines can be much harder. A lower bound must therefore identify the proof rules and the family of clause sets to which it applies.

More generally, a propositional proof system can be represented by a polynomial-time computable map whose range is exactly the propositional tautologies. Every tautology then has a proof, and every produced formula is valid. If one such system had polynomial-length proofs for every tautology, tautologies would have polynomially checkable certificates, implying $\mathrm{NP}=\mathrm{coNP}$. Conversely that equality would provide such a certificate-based system. This connects proof length to computational verification.

A propositional proof system specifies efficiently checkable certificates for logical tautologies. In the functional formulation, it is a polynomial-time computable surjection from finite binary strings onto the encoded tautologies: the input is a proof and the output is the formula proved. Soundness means no false formula is output; completeness means every logical tautology has some proof.

For example, a truth table can certify $p\vee \neg p$ by checking its two assignments. The same method for $n$ variables may require $2^n$ rows, so efficient verification in the proof's length does not imply short proofs in the formula's length. Proof complexity asks which formulas require long certificates in particular systems. Searching for a proof and checking a supplied proof are different computational tasks.^[ref]

Robert A. Reckhow coauthored the 1979 paper establishing a general computational framework for propositional proof systems with Stephen Cook. Their formulation represents a proof system by a polynomial-time computable surjection onto the logical tautologies, separating verification cost from the length of the shortest proof.^[ref]

The paper compares systems using efficient translations of proofs. If proofs in one system can be converted to proofs in another by a polynomial-time procedure, the translated proofs can grow only polynomially in length. Such a comparison is stronger than observing that both systems eventually prove the same formulas: completeness alone places no useful bound on the cost of converting evidence. This gives proof complexity a way to compare proof formats as computational representations.^[ref]

Stephen Cook is a computer scientist whose 1971 paper established a foundational connection between the class NP and Boolean satisfiability. The Cook-Levin theorem is expressed in modern terminology by saying that Boolean satisfiability is NP-complete.^[ref]

The proof encodes the possible steps of a polynomially bounded Nondeterministic Turing machine as a Boolean formula. A satisfying assignment describes an accepting computation, and the formula has polynomial size. This makes polynomial-time reduction a way to compare apparently different computational tasks: an efficient general solver for the encoded problem would solve every problem in NP. Cook's broader work also connects mathematical logic with proof complexity.^[ref]