In memoriam Lex Hendriks

(02-03-1952–13-05-2023)

Dick de Jongh

September 2023

Lex was born as Alex Hendriks in Deventer on March 2, 1952 in a blue collar neighborhood. Already at an early age his acute intelligence was obvious. He was the first child from his neighborhood and his grade school to enter the HBS. This type of high school without Latin and Greek had been successfully introduced at the end of the 19th century with the intention to draw a wider social segment of children to a higher education. It was very influential in raising the level of science in the Netherlands. In his final year at the HBS the lasting relationship with his classmate and future wife Hannie Bijkerk was initiated during his extra mathematics lessons to her.

His family as well as his surrounding neighborhood raised a political interest at a young age. Lex joined the ANJV, the youth organisation of the Communist Party. In the Second World War that party had established an impressive record in the resistance. He became an active member. A previous well-known communist with an active interest in intuitionistic logic had been Gerrit Mannoury, professor at the University of Amsterdam and a well-known national figure in the interbellum.

It was never a question of going to study anything but mathematics. In 1969 he started his study at the UvA, and finished his bachelor's degree in 1974. Lex had married Hannie the year before. He continued his mathematics studies towards a master majoring in logic (taking some philosophy courses as well). He distinguished himself as a student and for a number of years functioned as a student assistant, in different capacities like teaching logic to philosophy students, first with Johan van Benthem, later with Dick de Jongh. Already during these studies he joined De Waarheid, the communist newspaper, in 1977, as an editor. His outside activities, not only political, meant that his studies didn't go very fast but he always made steady progress.

At this time Dick de Jongh had just rekindled the interest from his own studies in programming intuitionistic logic with Hans Kamp. One of Lex's co-students, Henk van Riemsdijk, had already finished a master thesis on this topic. This subject triggered Lex's imagination and it quickly became the subject that he has been identified with at the ILLC. It always had a focus on using the programs to study fragments containing only a subset of the connectives. This first led to his master thesis in 1980, and later to his doctoral dissertation as well, and to a lifelong wider research interest in the subject of programming propositional logics.

The fact that the fragment of {\sf IPC} containing only $\to$ is locally finite (Diego's Theorem), meaning that finitely many variables can only lead tofinitely many non-equivalent formulas, had already fascinated Anne Troelstra and Dick de Jongh before as students, and many others. Studying this locally finite fragment became a model for studying other fragments, and for researching related subjects.

Since his undergraduate studies Lex had always in principle worked on logic one day a week next to another job. This became more difficult for a while after he finished his master thesis. This changed when another researcher became fascinated by Diego's Theorem, Gerard Renardel de Lavalette, at the time first a master and then a PhD student of Anne Troelstra. He finished his dissertation in 1985. After Lex's master thesis Gerard took an active interest in Lex's research and inspired Lex anew. Both he and Lex appreciated the very large numbers that arise in the size of the fragments and calculated a number of them exactly. This was aided by the fact that Gerard and Lex had discovered the work of de Bruyn on Diego's Theorem and the exact models corresponding to the n-variable implicational fragments. De Bruyn was a famous Dutch mathematician, an analyst, and later the originator of Automath, a language for proving mathematical theorems which pioneered what is now athriving branch of computational logic. De Bruyn investigated this subject for a while apparently as a hobby, and made a start with the construction of so-called exact models. Lex was able to pick his one day a week schedule again consistently. During 1991 he spent his one day in Utrecht where Gerard had a position. He was able to improve on de Bruyn's models theoretically, and also practically, using them to speed up his programs. When Gerard became a professor in Groningen, Dick and he were able to obtain from the NWO a one day a week PhD position for Lex to work at the ILLC. From 1992 until his death Lex was officially a guest researcher at the ILLC. The thesis was finished properly in 1996.

Throughout his life Lex always worked outside the university as well, even in the period when he wrote his PhD, first at De Waarheid, later, when doubts arose in him about his effectiveness in politics, in a number of different positions in the IT sector, a natural extension of his scientific interests. His final occupation in that area from 1998 to 2016 was with the company EXIN. This company prepares IT exams for official use. In fact, Lex had already prepared exams for EXIN from 1992 onward. During his period in the company he concentrated on higher level questions such as what kind of exams are needed and where they should be used.

During the period when he wrote his PhD in his one day a week at the ILLC he shared an office with Dick de Jongh at the Plantage Muidergracht. Dick remembers well his immense concentration, as well as his characteristic hat and pipe. The latter had to be banished from the office. Very impressive was his wide reading in logic and surrounding areas, far surpassing the one day a week officially allocated by him to logic. The Plantage Muidergracht had as a bonus that the typical Amsterdam Hartog bakery was close by; Lex always took bread from there home with him.

Lex's 1996 dissertation and the subsequent article about that work, finally published in 2012 with Gerard and Dick, can be seen as the hallmark of the area of research into fragments of intuitionistic logic. In the introduction Lex sketched the history of the subject starting with Rieger, Diego and others.

During the period following his PhD his now unpaid logic work led to a number of interesting articles, mostly in the general area of intuitionistic logic. In his paper, `Intuitionistic propositional logic with only equivalence has no interpolation', he used his programs successfully to obtain his counterexample. A widely read article, `Characterization of strongly equivalent logic programs in intermediate logics', with Dick de Jongh owed its existence to a large extent to the fact that the wide reading by Lex included a considerable amount of literature on Logic Programming (a more theoretical field than its name suggests).

After his retirement from EXIN in 2016 he was able to profit from his own adagium ``Choose for your future something you really like doing and you will never be bored". He became much more active at the ILLC after dropping the restriction on his work in logic. In many ways this became the most fruitful period of his research. He worked much more independently than before. For example, in that period he produced eight articles on logic and design, a subject otherwise untouched by members of the ILLC. Also, in that period he successfully participated, using his programs, in projects with master students. His final two articles, as yet unfinished, are in collaboration with Joop Niekus and with Dick de Jongh. Those articles are among the best work he produced. For Joop Niekus he created a logic that exactly fits the Kripke models Joop devised in 1982 to support his new ideas about Brouwer's creating subject, and comes exactly at the right time to give logical support to Joop's definitive work on the subject. With Dick de Jongh he cooperated on a study of subminimal logic, for which the ideas reach back to, among others, Haskell Curry, incidentally a former director of the Instituut voor Grondslagenonderzoek. Only a few days before his death he was in contact with both of his co-authors about the two papers. The papers will be finished; his part of the work is essentially done.

Lex's death was sudden, due to an unsuccessful operation, but not unanticipated, he had been ill for a number of years. Still it came as quite a shock to his close collaborators and friends at the ILLC. He was not known personally to many members of the institute because both his subject and the time he spent on it at the ILLC were non-mainstream. Those who did know him, such as the members of the Monday morning coffee meetings that flourished for many years, respected him highly for his wide interests and his pleasant unassuming character, not pressing his opinions on others. In his educational and research collaborations he was very active and very much part of the team. He will be remembered and missed.

**Personal statement by Johan van Benthem **

Lex Hendriks was one of the ‘hidden gems’ of the ILLC: an unusual outside talent attracted by the work going on in our circle, always setting his own agendas, but finding results of broader interest. His theoretical investigations into small fragments of logics made him a pioneer in two directions of still growing importance: the use of computational techniques as a general tool for researchers in logic, and the rise of automated deduction for mathematical reasoning.

For me, as an insecure beginning teacher in the 1970s, Lex was crucially important as a teaching assistant. The place of logic in the philosophy curriculum was under intense pressure from radical students: as being outdated, and not geared toward improving the lot of the working class. Lex was a beacon of calm and support for his young professor, and given his credentials in the Dutch communist party, nobody could accuse him of being on the wrong side of history.

I also have a later recollection, perhaps apocryphal, that Lex used the early computers of the Social Security instance where he worked in their downtime at night to solve logic problems programmed by him. Even if this were just a legend, it does sound just like one of the original things he was capable of.

Lex’s qualities persisted throughout his life. I always found encounters with this quiet thoughtful person both informative and conducive to my peace of mind.