Noam Zeilberger

Assistant Professor in Computer Science

Laboratoire d'informatique de l'École Polytechnique

email: noam.zeilberger@polytechnique.edu (or @gmail.com)

Background

I am an assistant professor at LIX and a member of the PARTOUT team within the Proofs and Algorithms pole. Previously, I was a Birmingham Fellow at the University of Birmingham for three years, and a member of the vibrant Theory Group in the School of Computer Science. I moved back to France in 2020 to rejoin my wife, after a few years of tricky navigation between Birmingham and Paris.

Before that, I was an itinerant postdoc for around seven years, first in Paris at Laboratoire PPS (now IRIF) with a fellowship of the Fondation Sciences Mathématiques de Paris, then at IMDEA Software in Madrid, then at the Institute for Advanced Study in Princeton during the special year on Univalent Foundations, and then back to Paris at the MSR-Inria Joint Centre and as a member of Team Parsifal.

And before all that I was a PhD student at CMU SCS, spending six great years in Pittsburgh, Pennsylvania.

Research Interests

I am interested broadly in connections between logic, language, mathematics, and computation, and my specific focuses of attention have shifted over the years as I've gone down different rabbit-holes. During my PhD studies I fell in love with ideas from programming languages theory, proof theory, and type theory, which later evolved into a love for category theory and an appreciation for combinatorics (especially map theory, using "map" in the sense of the "four color map theorem") and automata theory. I do like theory! But also concrete examples, diagrams, and computations.

Recent and Upcoming News and Events (circa mid-2023)

Our ANR project LambdaComb is ongoing! You can visit the project page to find out more about this "cartographic quest between lambda-calculus, logic, and combinatorics". In particular, you can check out the page of our January 2024 meeting to see some recent work by project members.

I am very proud of my now former PhD students, Nicolas Blanco and Alexandros Singh, who both recently defended their dissertations!

Recent and not-so-recent papers

The categorical contours of the Chomsky-Schützenberger representation theorem.: With Paul-André Melliès. 29 December 2023. [hal]
Convolution Products on Double Categories and Categorification of Rule Algebras.: With Nicolas Behr and Paul-André Melliès. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). [doi]
Parsing as a lifting problem and the Chomsky-Schützenberger representation theorem.: With Paul-André Melliès. In Proceedings of the 38th Conference on Mathematical Foundations of Programming Semantics (MFPS 2022). [hal] [arXiv:2212.09060]
Asymptotic Distribution of Parameters in Trivalent Maps and Linear Lambda Terms.: With Olivier Bodini and Alexandros Singh. [arXiv:2106.08291]
Deductive systems and coherence for skew prounital closed categories.: With Tarmo Uustalu and Niccolò Veltri. Proceedings of the 15th International Workshop on Logical Frameworks and Metalanguages: Theory and Practice (LFMTP 2020). [arXiv:2102.03809]
Proof theory of partially normal skew monoidal categories.: With Tarmo Uustalu and Niccolò Veltri. In Proceedings of the 3rd Applied Category Theory Conference (ACT 2020). [arXiv:2101.10487]
Bifibrations of polycategories and classical linear logic.: With Nicolas Blanco. In Proceedings of the 36th Conference on Mathematical Foundations of Programming Semantics (MFPS 2020). [doi]
Eilenberg-Kelly reloaded.: With Tarmo Uustalu and Niccolò Veltri. In Proceedings of the 36th Conference on Mathematical Foundations of Programming Semantics (MFPS 2020). [doi]
The sequent calculus of skew monoidal categories (extended version).: With Tarmo Uustalu and Niccolò Veltri. To appear in Lambek Memorial Volume, C. Casadio and P. J. Scott (editors), Outstanding Contributions to Logic, Springer, 2020. [arXiv:2003.05213] [Agda code]
A sequent calculus for a semi-associative law (extended version).: Logical Methods in Computer Science, volume 15, issue 1, 2019, pp. 9:1-9:23. Appeared in a special issue of selected papers from FSCD 2017. [journal link] [arXiv:1803.10080]. [some related code]
Two Questions about the Fractional Counting of Partitions.: With Doron Zeilberger. October 30, 2018. [paper url] [arXiv:1810.12701].
The sequent calculus of skew monoidal categories.: With Tarmo Uustalu and Niccolò Veltri. In Proceedings of the 34th Conference on Mathematical Foundations of Programming Semantics (MFPS 2018). [paper url with Agda code]
A theory of linear typings as flows on 3-valent graphs.: In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2018). [doi] [arXiv:1804.10540]
A sequent calculus for a semi-associative law.: In Proceedings of the Second International Conference on Formal Structures for Computation and Deduction (FSCD 2017). [Note this article mostly supersedes arXiv:1701.02917, and is superseded by arXiv:1803.10080.]
Connected chord diagrams and bridgeless maps.: With Julien Courtiel and Karen Yeats. Electronic Journal of Combinatorics, volume 26, issue 4, 2019. [doi]
A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine.: With Paul-André Melliès. In Proceedings of the Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016). [doi] [arXiv:1601.06098]
Towards a mathematical science of programming.: Research statement, circa 2016.
Linear lambda terms as invariants of rooted trivalent maps.: Journal of Functional Programming, volume 26, 2016. [doi] [arXiv:1512.06751]
Counting isomorphism classes of β-normal linear lambda terms.: September 25, 2015. [arXiv:1509.07596]
An Isbell Duality Theorem for Type Refinement Systems.: With Paul-André Melliès. Mathematical Structures in Computer Science 28:6, 736-774, 2018. [doi] [arXiv:1501.05115]
Balanced polymorphism and linear lambda calculus.: Presented at TYPES 2015.
Functors are Type Refinement Systems.: With Paul-André Melliès. In Proceedings of the 42nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL 2015). [doi]
A correspondence between rooted planar maps and normal planar lambda terms.: With Alain Giorgetti. Logical Methods in Computer Science, Vol. 11(3:22)2015, pp. 1-39. [doi] [arXiv:1408.5028]
Type refinement and monoidal closed bifibrations.: With Paul-André Melliès. October 1, 2013. [arXiv:1310.0263]
Polarity and the logic of delimited continuations.: In Proceedings of the Twenty-Fifth Annual IEEE Symposium on Logic in Computer Science (LICS 2010). [doi] [twelf code] [slides]
Defunctionalizing focusing proofs.: Presented at the 2009 International Workshop on Proof-Search in Type Theories. [twelf code] [more twelf]
Refinement types and computational duality.: In Proceedings of the 2009 Workshop on Programming Languages meets Program Verification (PLPV 09). [agda code]
Focusing on binding and computation.: With Dan Licata and Bob Harper. In Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science (LICS 08). [doi] [tech report]
Focusing and higher-order abstract syntax.: In Proceedings of the 35th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL 08). [doi] [coq code] [notes]
On the unity of duality.: Annals of Pure and Applied Logic 153:1 (2008), special issue on "Classical logic and computation". [doi]

Recent and not-so-recent talks

The combinatorics of free bifibrations: December 15, 2023, at the 17th CLA workshop in Kraków. Based on joint work with Bryce Clarke and Gabriel Scherer. See also our extended abstract.
Context-free grammars and finite-state automata over categories: December 1, 2023, at the Birmingham Theoretical Computer Science Seminar.
On the complexity of normalization for the planar lambda calculus: July 2, 2023, at TLLA 2023. Based on joint work with Anupam Das, Damiano Mazza, and Tito Nguyen.
Generalizing and abstracting the notion of context-free language: June 28, 2023, at the Lambda Pros day.
Type refinement systems: combining intrinsic and extrinsic views of typing, categorically: December 7, 2022, at the 3rd online workshop on Proofs, Computation and Meaning.
Type refinement as a unifying principle.: August 1, 2022, for the Special Session for Frank Pfenning jointly organized by LFMTP and FSCD at FLoC 2022.
Parsing as a lifting problem and the Chomsky-Schützenberger representation theorem.: June 30, 2022, for the Topos Institute Colloquium. [video] [2022.07.21 version at the CMU POP seminar] [2023.04.21 version, invited talk at Journée GT DAAL 2023]
A quick introduction to species, operads, and closed multicategories.: Apr 11, 2022, at the LambdaComb kickoff meeting.
A cartographic quest between lambda calculus, logic, and combinatorics.: Mar 17, 2022, at the LIX Seminar.
Fibrational views on typing, proving, and parsing.: Sep 1, 2021, at the special session on Categorical Type Theory at MFPS 2021.
Complexity of normalization for subsystems of untyped linear lambda calculus.: June 28, 2021, at the Structure meets Power Workshop. Warning: We realized (~Nov 2022) that our proof of PTIME-hardness contains a gap! The complexity of normalizing planar lambda terms remains open... Update 2023: now we again believe it is PTIME-complete. See TLLA 2023 talk above, which we will hopefully eventually turn into a peer-reviewed paper.
Skew monoidal categories and the proof-theoretic anatomy of associativity (and unitality).: February 9, 2021, at the Bath Mathematical Foundations Seminars.
Untyped Linear Lambda Calculus and the Combinatorics of 3-valent graphs.: December 2, 2020, contributed talk at Combinatorics and Arithmetic for Physics: special days (CAP7). [video]
Higher connectivity in linear lambda-terms as 3-valent graphs.: September 4, 2019, invited talk at the Fifth Symposium on Compositional Structures (SYCO 5).
A proof-theoretic analysis of the rotation lattice of binary trees.: January 9, 2019, joint Combi-Parsifal seminar at École Polytechnique.
A proof-theoretic analysis of the rotation lattice of binary trees.: September 13, 2018, at the Rutgers Experimental Mathematics Seminar.
A theory of linear typings as flows on 3-valent graphs.: July 9, 2018, at LICS 2018.
Lambda calculus and the Four Colour Theorem.: June 22, 2018, at the OASIS seminar in Oxford.
Type refinement systems and the categorical perspective on type theory.: June 19, 2018, at the workshop on Practical and Foundational Aspects of Type Theory at the University of Kent.
Lambda calculus and the Four Colour Theorem.: November 21, 2017, at the Edinburgh LFCS Seminar.
Skew-closed objects, typings of linear lambda terms, and flows on trivalent graphs.: September 9, 2017, at STRING 2017.
A sequent calculus for a semi-associative law.: September 4, 2017, at FSCD 2017.
Some bridges between lambda calculus and graphs on surfaces.: July 6, 2017, at the 26th British Combinatorial Conference.
Some enumerative, topological, and algebraic aspects of linear lambda calculus.: May 18, 2017, at CLA 2017.
A Categorical Perspective on Type Refinement Systems.: December 9, 2016, at the Logic and Semantics Seminar of the Cambridge Computer Laboratory.
A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine.: July 7, 2016, at LICS 2016.
Principles of Type Refinement.: Lectures given at OPLSS 2016, June 29-July 2, 2016.
Linear lambda calculus and the combinatorics of embedded graphs.: October 14, 2015, at Journées Nationales GEOCAL-LAC-LTP 2015. [pdf]
Balanced polymorphism and linear lambda calculus.: May 18, 2015, at TYPES 2015.
A connection between lambda calculus and maps.: January 18, 2015, at OBT 2015.
Functors are Type Refinement Systems.: January 15, 2015, at POPL 2015
Polarity in Proof Theory and Programming.: August 30, 2013, at the Summer School on Linear Logic and Geometry of Interaction in Torino, Italy.
HOPE for a type-theoretic understanding of zero-knowledge.: September 9, 2012, at the 1st ACM SIGPLAN workshop on Higher-Order Programming with Effects. (Note: the slides seem to render funny with Firefox -- best viewed in Chrome or Safari.)
BONUS: slides for the "15 minute" version I gave October 4th at the IAS postdoc seminar series.
Towards a non-commutative logic of effects.: December 2010, presented at the Oxford OASIS seminar and at MSR Cambridge. As it says on one of the first slides, the ideas here were "X% baked for some X probably significantly below 50". Still, I'm posting the talk here for historical reasons, since this was near the time Paul-André and I first started thinking about substructural types as presheaves over context categories (with helpful exchanges with Jonas!), an idea that eventually developed into our work on type refinement systems. The connection between focusing/polarity and Isbell duality was elaborated upon in our MSCS paper, though I think there is still more to be said.

Dissertation

PhD in computer science, 2009, Carnegie Mellon University
The Logical Basis of Evaluation Order and Pattern-Matching.
Committee: Peter Lee (co-advisor), Frank Pfenning (co-advisor), Robert W. Harper, Paul-André Melliès (external member)

Techniques from linear logic and infinitary proof theory (connected to the old idea of a "proof-theoretic semantics" of logic) yield new insights into seemingly extra-logical features of modern programming languages. By applying the Curry-Howard correspondence to focusing proofs, we develop a polarized type theory in which evaluation order is explicitly reflected at the level of types, and which has built-in support for pattern-matching. This framework provides an elegant, uniform account of both untyped and intrinsically well-typed computation, and moreover can be extended with an extrinsic (Curry-style) type system to express and enforce more refined semantic properties of programs. We apply these ideas to explore the theory of typing and subtyping for intersection and union types in the presence of effects, giving a simplified explanation of some of the unusual artifacts of existing systems.

Software

LinLam: a Haskell library for experimental linear lambda calculus.

Students

Former PhD students

Nicolas Blanco: Bifibrations of polycategories and classical multiplicative linear logic; School of Computer Science, University of Birmingham, February 2023.; Co-advised with Paul Levy.
Alexandros Singh: A unified approach to the combinatorics of the λ-calculus and maps: bijections and limit properties; Laboratoire d’Informatique de Paris Nord, Université Paris XIII, November 2022.; Co-advised with Olivier Bodini.

Former Masters students

George Kaye: a visualiser for lambda-terms as rooted maps (MSci project, 2019)

Tingyu Huang: visualization of combinatorial maps (MSc project, 2019)

Farzad Jafarrahmani: categorical semantics of induction and subtyping (M2 internship, 2019)

Teaching

Current DIX: CSE102 - Computer Programming; CSE301 - Functional Programming; INF412 - Fondements de l’Informatique: logique, modèles, calculs
Former DIX: CSE103 - Introduction to Algorithms; INF371 - Mécanismes de la programmation orientée-objet
Former Birmingham: Functional Programming; Advanced Topics in Functional Programming; Introduction to Mathematics for Computer Science

Et cetera

Rapport sur la Thèse de Dov Tamari: Report on the thesis of Dov Tamari, signed by his advisor Paul Dubreil (scanned by phone at Sophie Germain Library).
nLab and OEIS: Knowledge is a collaborative effort.
And Quiet Flows the Mon: A photography project from the dark days after the 2004 U.S. Presidential Elections. 2017 update: whoops, it seems that my dystopian vision of US politics was a little bit too optimistic!
Archive of papers by John Reynolds: Direct link to the CMU AFS directory, since the ftp server is sometimes down.
In Tune With Fun: A true-life story about learning the accordion.
Nathack: a natural language interface for nethack.: January, 2003. With Cassia Martin, David Molnar, and Dev Purkayastha. As the title suggests, this was a natural language interface for nethack! Done with a mix of prolog, embedded lua, and scary hacking within nethack's internal C source.

We may just be cockroaches at the base of a very large garbage mountain.
Dana Scott (on mathematics)

And tell me Margaret, when I’m gone, what will I want,
To be left at the bottom of a garbage bin, or dusted off and pulled up onto stage?
Jason Webley