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Tracking Redexes in the Lambda Calculus

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The French School of Programming

Abstract

Residuals of redexes keep track of redexes along reductions in the lambda calculus. Families of redexes keep track of redexes created along these reductions. In this chapter, we review these notions and their relation to a labeled \(\lambda \)-calculus introduced here in a systematic way. These properties may be extended to combinatory logic, term rewriting systems, process calculi and proofnets of linear logic.

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Acknowledgements

As an intern, I started at Inria on October 1968 in the Operating Systems group. Inria (then named IRIA) had no more than 20 researchers. On summers 1969–1970, I attented two fantastic CEA-EDF schools about Programming and about Interactive Graphics at the Bréau-ss-Nappe with notably Dijkstra and Newman.

In 1971, I was introduced to theoretical computer science by Jean Vuillemin (still at Stanford University) and Jean-Marie Cadiou (just back from Stanford). Gilles Kahn came (from Stanford) to Inria a few months later. With Gilles, we started by making small formal proofs with Scott computational induction in the spirit of Manna-Ness-Vuillemin. Gilles also worked on coroutines and on the non-deterministic multiplexer. On summer 1972, Inria hosted the first ICALP conference with worldwide researchers in this new area of theoretical computer science. Huge event! quite challenging !!

In parallel, a new connection started with Edinburgh. Robin Milner (also back from Stanford) and Chris Wadsworth visited Inria in 1972–1973. Robin destroyed our attempt with Jean-Marie to formalize parallel programs with oracles. Anyhow our paper was accepted at SWAT’73! Gilles gave me Church’s monography about the Calculi of Lambda Conversion. I read it in a café at the bottom of avenue de Wagram. Crystal clear ! Then I tried to read Barendregt’s thesis and Nederpelt’s Ph.D. Not so easy ! I went to Gilles and said there were too many theorems. Gilles looked at it and said “oh! he is dutch!”…Maurice Nivat (who joined Inria in 1971) was excellent at inviting current stars of theoretical computer science. Many foreigners visited our building. At Pentecôte 1973, I extended Vuillemin’s results about safe computations in recursive program schemes to the \({\lambda }\)-calculus. Immediately, Maurice asked me to register for a Thèse de 3ème cycle and to speak at the seminar on next Tuesday!!

In 1973, Gilles and I started a small implementation of Milner’s LCF logic, in APL on the IBM 360/91 in CEA-Saclay (a big failure), and later in AlgolW on another IBM 360/67 machine in Grenoble (more successfully). At fall 1973, Gilles took me to visit David Park in Warwick. I left him to travel to Edinburgh on November. David said “speak with Gordon Plotkin, and at Edinburgh, you’ll be freezing like a lolly”. I indeed met Robin and Gordon, and many others in the CS department at the King’s buildings and the AI group at Hope Park Square. Discussions were great. I remember Robin jumping to the whiteboard and making my proofs before I could present them. Gordon showed me on a ridiculously small blackboard how Scott’s continuous functions were unable to capture the sequentiality induced by leftmost-outermost evaluations. Thus Scott’s denotational semantics was not fully abstract. Gordon also showed his \(\omega \)-incompleteness counterexample to a conjecture of Barendregt. I also saw the 3-bar heater in Gordon’s office where only one bar was on.

At winter 1973, Maurice told me that Peter Welch, a student of David Park, was trying to show the completeness of inside-out reductions in the \({\lambda }\)-calculus. He had a 60-page long proof! I tried to read it, and some while later, I strangely felt that Maurice would not sign my dissertation if I was not able to prove it by my own means. Fortunately, I could make a simple proof, 3 months later! I defended my Thèse de 3ème cycle on summer 1974, with J.-Y. Girard in the jury. Maurice did advice me to visit Jean-Yves at University of Paris 7. Another great moment! Jean-Yves was lecturing in front of no more than 5 students. But when finished, we quickly went to the café in front of Jussieu and he showed me on a paper napkin the reductibility method for proving strong normalisation in typed calculi. On September 1974, I was invited to the Swansea mini-conference on \({\lambda }\)-calculus. I met there Roger Hindley, Henk Barendregt, Corrado Böhm, Gordon Plotkin, Peppe Longo, Anne Troelstra, \(\ldots \) and Haskell Curry. I was a newbie in the world of logicians. Later, I did my first talk in public at the Rome 1975 Symposium about \({\lambda }\)-calculus and Computer Science Theory where I presented my result about inside-out completeness. Dana Scott asked me loudly a question about the completeness of transfinite reductions! Goodness! I could not imagine reductions going further than infinity. I was stuck. (I now know that Dana is the kindest person.)

I went on with the \({\lambda }\)-calculus until my Thèse d’État on January 1978. Some of the results were in common with Gérard Berry (who joined Inria in 1973). In the present chapter, part of them is presented. For me, that was the end of the glorious times in the theory of programming at building 8 in Inria-Rocquencourt. Of course, I forgot to quote many other interesting events, schools and curious meetings with exceptional people. We indeed had lot of fun and I was extremely lucky to participate to these times.

Afterward, the building grew up. I zigzaged in term rewriting systems with Gérard Huet (who joined Inria in 1972), CAD for VLSI, system programming, parallel programming, mobility, and formal proofs. I also enjoyed teaching at Ecole polytechnique in Palaiseau and had the fantastic opportunity to meet excellent (PhD) students.

Finally, I cannot thank all the people who influenced my work or helped me. Among them, my mentors Gilles Kahn, Jean-Marie Cadiou, Jean Vuillemin, Maurice Nivat and my friends Gérard Huet, Gérard Berry. Also for sure Gordon Plotkin, Robin Milner, Henk Barendregt and Jan Willem Klop. I am deeply sorry for the very many, French or aliens, that I love, but did not mention.

I also thank the two brave reviewers of this (theoretical) chapter, who greatly contributed to clarify many obscure points.

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  1. Irif & Inria, Paris, France

    Jean-Jacques Lévy

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Correspondence to Jean-Jacques Lévy .

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  1. Constructor Institute, Schaffhausen, Switzerland

    Bertrand Meyer

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Lévy, JJ. (2024). Tracking Redexes in the Lambda Calculus. In: Meyer, B. (eds) The French School of Programming. Springer, Cham. https://doi.org/10.1007/978-3-031-34518-0_10

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