Notices

2015 Bulletin of Symbolic Logic  
Mints' research spanned several areas, with many interconnections. The grand unifying theme in his oeuvre was proof theory and constructivism, but entangled with this came a lifelong interest in computational logic as well as special attention for two specific areas: intuitionistic logic and modal logic. Mints was trained in the Russian school of constructive mathematics of Markov and of his teacher Nikolai Shanin. In this paradigm, all mathematical objects should be defined in a finite
more » ... , and every proof of existence should provide an algorithm, where properties of the algorithm are to be proved by constructive principles including Markov's schema. In tandem with this foundational view concerning mathematics, Shanin's group at Steklov also worked on practical automated reasoning, with an emphasis on generating natural proofs, where 'natural' included making sense from the perspective of a human agent. This mixture of proof theory and computation led to inventions such as the 'inverse method', a style of analyzing provability similar to resolution, developed in 1964 by Sergei Maslov, a close friend of Mints. Combining proof theory and computation naturally leads to a study of intuitionistic logic, and the most significant results of Mints' first period concern this interface. They include an extension of Herbrand's theorem to intuitionistic predicate logic (1962), and a proof of the undecidability of intuitionistic predicate logic with a single unary predicate (1965, a classical paper joint with Maslov and Orevkov). Some of these results are still under active investigation. Mints' paper of 1968 on decidability of a certain fragment of the Gentzen system LJ introduced what is now called the "Mints hierarchy" of intuitionistic first-order formulas. Complexity properties of Mints classes are still under investigation (e.g., in recent works by Schubert, Urzyczyn, et al.), since they are important to understanding constructive type-theoretic proof assistants such as Coq. Also noteworthy is Mints' proof in 1974 that familiar procedures for extracting programs from existence proofs in intuitionistic arithmetic produce equivalent algorithms. In subsequent years, Mints turned to the study of relationships between proof theory and category theory. He used proof-theoretic methods to simplify proofs in category theory and to prove new theorems. a clear exposition of Lambek's results connecting Cartesian closed categories and intuitionistic logic using a novel technique of encoding proofs as explicit proof-terms, an idea that has entered categorical logic itself, and that reemerged in modern manifestations such as Artemov's justification logic. Important results obtained by Mints with this style of analysis include normalization and coherence theorems. c 2015, Association for Symbolic Logic 1079-8986/15/2101-0005
doi:10.1017/bsl.2015.4 fatcat:hsefhckb5bajbpnbriwmg6dgcq