S. N. Artemov, L. D. Beklemishev, S. S. Goncharov, Yu. L. Ershov, S. P. Odintsov, V. V. Rybakov, S. O. Speranski, V. B. Shehtman, V. F. Yun, “Larisa L'vovna Maksimova (obituary)”, Russian Math. Surveys, 80:3 (2025), 533

Larisa L’vovna Maksimova (5.Nov.1943–4.Apr.2025) was one of the leading Russian experts in the field of non-classical logics and a bright representative of A. I. Malcev’s school in algebra and logic. She was the first woman in the Russian history who defended a D.Sc. thesis on mathematical logic.

She was born on 5 November 1943 in the small town of Kochenevo, Novosibirsk Oblast. Her father Lev Dmitrievich Maksimov was a geobotanist. He graduated from Leningrad State University in 1932 and defended his Ph.D. thesis in 1937. After that he was the director of the Central Black Earth Nature Reserve in Kursk Oblast, where he met Taisiya Matveevna Maslennikova, who was to become his wife and Larisa’s mother. In 1940 he obtained a position at Tomsk University, so Larisa’s parents moved to Siberia, where their first daughter Nataliya was soon born. When the Soviet Union entered the WW2, the life in Tomsk became more difficult, and in 1941 they went to Novosibirsk Oblast, where Larisa was born in 1943. In the next year the family moved to Novosibirsk, where the father found a job at Novosibirsk Pedagogical Institute. Subsequently, he became the dean of the Faculty of Geography there. Taisiya Matveevna graduated from the same institute in 1948 and became a geography teacher in a school. In 1950 Lev Maksimov died after a serious illness, and Taisiya was left alone with two kids.

In 1960 Larisa graduated from school no. 54 in Novosibirsk with a gold medal and enrolled in the Faculty of Mechanics and Mathematics at Novosibirsk University. The university opened in the previous year, at apparently the same time when the Siberian Branch of the Academy of Sciences of the USSR was organized. Institutes of the academy and the university itself were located in the picturesque Akademgorodok. Most of the staff were researchers from Moscow and Leningrad. Larisa was lucky to listen to lecture courses of prominent researchers, such as S. L. Sobolev, A. I. Malcev, A. M. Budker, A. I. Shirshov, M. I. Kargapolov, G. I. Marchuk, A. A. Lyapunov, A. P. Ershov, A. V. Bitsadze, A. A. Borovkov, L. V. Ovsyannikov, Yu. G. Reshetnyak, and others.

In 1963 she had to decide about specialization and chose the Department of Algebra and Logic, headed by academician Anatolii Ivanovich Malcev. At that time nobody in Novosibirsk worked on non-classical logics. On the other hand, Malcev was interested in the algebraic approach to the investigation of logical calculi, as proposed by A. Tarski and presented in the book The mathematics of metamathematics by H. Rasiowa and R. Sikorski, which was published just in 1963. The first task for Maksimova was to review a paper of V. Donchenko on the calculus of rigorous implication, introduced by W. Ackerman in 1956, at the research seminar “Algebra and logic”. After her talk Malcev stated the first problem for her: examine the independence of the system of axioms of the Ackerman calculus. She solved the problem fully, by identifying the derivable axioms and proving the independence of the others. These results were presented in Maksimova’s first paper, published in the journal Algebra i Logika in 1964. After that, a year before her graduation from the university, Larisa Maksimova was hired to the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR, where she worked all her life since then.

Her Ph.D. thesis, written with Malcev as a scientific advisor, was also devoted to problems in relevant logic. She established analogues of the deduction theorem for the Ackerman calculus and the logic $\mathsf{E}$ of relevant entailment. The logic $\mathsf{E}$, which is a modification of the Ackerman calculus proposed by A. R. Anderson and N. D. Belnap, is a standard system of relevant logic. Maksimova was the first to develop algebraic semantics and relational semantics with ternary accessibility relation for $\mathsf{E}$ and to prove a number of separation theorems for $\mathsf{E}$ and the Ackerman calculus. Finally, she proposed the weaker version $\mathsf{SE}$ of the logic of relevant entailment, which avoids the paradoxes of material implication, coincides with $\mathsf{E}$ on first-degree entailments, but at the same time is decidable. The problem of the development of such a calculus had been stated by A. V. Kuznetsov. In July 1967 Malcev deceased, which was a heavy loss for the Novosibirsk algebraists and logicians. Maksimiova defended her thesis in 1968, without her advisor.

After the defence Maksimova turned to superintuitionistic logics, and in 1971 she showed that there exist precisely three finitely approximable pretabular superintuitionistic logics. In combination with Kuznetsov’s result on the finite approximability of pretabular logics¹, this provided a full description of all pretabular superintuitionistic logics. For this result it was necessary to initiate the development of the representation theory for Heyting algebras by means of partially ordered frames: she established a correspondence between embeddings of Heyting algebras and $p$-morphisms of the representing scales. As a consequence, the property of being tabular is decidable in the class of superintuitionistic logics. In the subsequent studies Maksimova was always concerned with the decidability of properties of logics, that is, with finding algorithms that, given a finite system of axioms, decide whether or not the logic defined by these axioms has the property in question.

There is a close connection between the family of superintuitionistic logics and the family of normal extensions of Lewis’s modal logic $\mathsf{S4}$, based on the Gödel–McKinsey–Tarski translation $T$, which embeds the intuitionistic logic $\mathsf{Int}$ faithfully into $\mathsf{S4}$. The study of this connection was initiated by M. A. Dummett and E. J. Lemmon and then continued by Maksimova with her student V. V. Rybakov in their joint paper of 1974. In particular, Maksimova proved in that paper that each superintuitionistic logic has a maximal modal companion², and the family of all such companions forms an interval in the lattice of extensions of $\mathsf{S4}$. As the first application of these results, all pretabular extensions of the logic S4 (five logics in all) were described by Maksimova in 1975. This result was also independently obtained by L. Esakia and V. Meskhi.

One result that brought international fame to Maksimova (as well as the informal – but honorific – title of the ‘Queen of Interpolation’) was her description of the superintuitionisticic logics having the Craig interpolation property (CIP) obtained in 1977. When this result was announced, many experts did not want to believe that, of the continuum many superintuitionistic logics, only eight logics have CIP. The proof is based on the equivalence, established by Maksimova, between the CIP property of a superintuitionistic logic $L$ and the amalgamation property of the variety of Heyting algebras determining the equivalent algebraic semantics of the logic $L$, and also on the description of all amalgamable varieties of Heyting algebras. Using the same method, in 1979 she proved that only a finite number of extensions of $\mathsf{S4}$ have CIP and, moreover, this property is decidable for finitely axiomatizable extensions of $\mathsf{S4}$. In contrast to $\mathsf{Int}$, the precise number of extensions of $\mathsf{S4}$ with CIP is still not known (it was established in Maksimova’s paper of 1979 that this number does not exceed 38, while in a later paper of 1980 she showed that it is at least 25). Thus, checking the interpolation property in extensions of $\mathsf{S4}$ is a remarkable example of a decidable problem such that the corresponding decision algorithm is not known.

The study of various versions of the interpolation and definability properties in the classes of superintuitionistic, normal modal, temporal, and dynamic logics, finding algebraic analogues of these properties, solving the problems of checking properties of a logic on the basis of its axiomatics were the main lines of Maksimova’s research. The methods that she developed have considerably affected the current state of research on non-classical logics in Russia and overseas.

Most of Maksimova’s results were published in the journals Algebra i Logika³ and Sibirskii Matematicheskii Zhurnal⁴. A number of papers were published in Studia Logica, a journal where Maksimova was a member of the editorial board for many years. In 2005 the book Interpolation and definability written by Maksimova in cooperation with the well-known logician Dov Gabbay was published in the series “Oxford Logic Guides”. It reflects the state of research (as of the beginning of this century) of various versions of the interpolation and definability properties in modal and superintuitionistic logics, propositional and first-order ones alike. To a considerable extent this book is based on results due to Maksimova.

Maksimova was also one of the authors of a unique workbook: I. A. Lavrov and L. L. Maksimova, Problems in set theory, mathematical logic, and theory of algorithms. This book was published in 1975 and reprinted several times since then. It was also translated into Hungarian (1988), English (2002), and Polish (2004). Also these days university students use it in their studies.

Maximova started teaching at Novosibirsk University in 1965. First she taught exercises for the lecture course “Mathematical logic and theory of algorithms”. Then she herself read this lecture course for many years. For a number of years, since 1993 she also read a course on modal logic for undergraduate students of the newly organized Philosophy Faculty at Novosibirsk University. In addition, she created a course on applied logic, which she read originally in 1997. It is still in the curriculum for undergraduate students of the Faculty of Mechanics and Mathematics at Novosibirsk University. In 1970 Maksimova organized the research seminar “Non-standard logics”, which works till now. Since 1972 she read a number of special lecture courses on various areas of non-classical logic. Since 1974 she presided the section on non-classical logics at almost all All-Union conferences on mathematical logic and then at the conference “Malcev Readings”.

In 2009 Larisa Maksimova was awarded the A. I. Malcev Prize of the Russian Academy of Sciences, and in 2010, as a member of a group of Novosibirsk logicians, she was awarded the State Prize of the Russian Federation in the field of education. A book dedicated to her, Larisa Maksimova on implication, interpolation, and definability, was published in 2018 in the Springer series “Outstanding Contributions to Logic”.

Maksimova was always proud to belong to the Siberian school of algebra and logic and had a deep respect for her teacher Anatolii Ivanovich Malcev. Devotion to science, pedagogical talent, and remarkable human qualities were characteristic for her. A whole era ended with Larisa L’vovna Maksimova, and she will be greatly missed.

Citation: S. N. Artemov, L. D. Beklemishev, S. S. Goncharov, Yu. L. Ershov, S. P. Odintsov, V. V. Rybakov, S. O. Speranski, V. B. Shehtman, V. F. Yun, “Larisa L'vovna Maksimova (obituary)”, Russian Math. Surveys, 80:3 (2025), 533–536

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