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Automated deduction uses an algorithm to construct or check a mathematical proof from explicit assumptions and inference rules. The representation might use propositional logic, first-order logic or a specialized theory such as linear arithmetic. Soundness means the rules preserve truth; finding a proof additionally requires a search strategy.

For a small resolution example, the clauses $P\vee Q$, $\neg P$ and $\neg Q$ yield $Q$ and then the empty clause. This is a contradiction, so the original clauses cannot all hold. To prove a conclusion from assumptions, a refutation method adds its negation and searches for such a contradiction. In first-order logic, unification determines which terms can instantiate matching predicates. A procedure that eventually finds every valid proof need not terminate on every invalid claim: this distinction explains why proof search and proof checking have different computational costs.

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/

Jean Gallier is a computer scientist whose research connects mathematical foundations with computer programming and geometric applications. His earlier work includes formal verification, automated deduction and unification; his later interests include computer graphics, computer vision, robotics and the geometry of curves and surfaces.^[ref]

His Curves and Surfaces in Geometric Modeling: Theory and Algorithms by Jean Gallier develops geometric constructions with explicit mathematical foundations. For example, its account of affine maps separates the geometry of points from the linear maps acting on displacement vectors, a distinction useful when describing transformations in computer graphics.^[ref]

Homepage: www.cis.upenn.edu/~jean/

Ugo Montanari coauthored the 1982 unification algorithm with Alberto Martelli. Their paper treats first-order terms as equations and derives an efficient implementation from simpler transformation rules, integrating the acyclicity test needed for finite terms.^[ref]

For example, decomposing $f(x,a)=f(b,y)$ gives $x=b$ and $y=a$, whereas $x=f(x)$ has no finite-term solution and must be rejected. The point is to compute a most general unifier without expanding or repeatedly traversing shared term structure unnecessarily. Such unification supports automated deduction and logic programming.