PARTOUT

PARTOUT - 2023

2023Activity reportProject-TeamPARTOUT

RNSR: 201923495K

Research center Inria Saclay Centre at Institut Polytechnique de Paris
In partnership with:CNRS, Institut Polytechnique de Paris
Team name: Proof Automation and RepresenTation: a fOundation of compUtation and deducTion
In collaboration with:Laboratoire d'informatique de l'école polytechnique (LIX)
Domain:Algorithmics, Programming, Software and Architecture
Theme:Proofs and Verification

Keywords

Computer Science and Digital Science

A2.1. Programming Languages
A2.2. Compilation
A2.4. Formal method for verification, reliability, certification
A4.5. Formal methods for security
A7.2. Logic in Computer Science
A7.2.1. Decision procedures
A7.2.2. Automated Theorem Proving
A7.2.3. Interactive Theorem Proving
A7.2.4. Mechanized Formalization of Mathematics
A7.3.1. Computational models and calculability
A8.1. Discrete mathematics, combinatorics
A8.11. Game Theory

1 Team members, visitors, external collaborators

Research Scientists

Lutz Strassburger [Team leader, INRIA, Senior Researcher, HDR]
Beniamino Accattoli [INRIA, Researcher]
Kaustuv Chaudhuri [INRIA, Researcher]
Ian Mackie [CNRS, Researcher]
Dale Miller [INRIA, Senior Researcher]
Gabriel Scherer [INRIA, Researcher]

Faculty Members

Ambroise Lafont [ECOLE POLY PALAISEAU, from Oct 2023]
Benjamin Werner [ECOLE POLY PALAISEAU]
Noam Zeilberger [ECOLE POLY PALAISEAU]

Post-Doctoral Fellow

Bryce Clarke [INRIA, Post-Doctoral Fellow]

PhD Students

Farah Al Wardani [INRIA]
Pablo Donato [INRIA, from Nov 2023]
Pablo Donato [ECOLE POLY PALAISEAU, until Sep 2023]
Adrienne Lancelot [INRIA]
Olivier Martinot [INRIA]
Marianela Evelyn Morales Elena [Inria, until Sep 2023]
Giti Omidvar [INRIA, until Sep 2023]
Adonis Rima [INRIA, from Dec 2023]
Antoine Sere [ECOLE POLY PALAISEAU, until Sep 2023]
Jui-Hsuan Wu [IP PARIS]

Interns and Apprentices

Ruben Bueno [ECOLE POLY PALAISEAU, Intern, from Jul 2023 until Sep 2023]
Thea Li [ECOLE POLY PALAISEAU, Intern, from Jun 2023 until Jul 2023]

Administrative Assistant

Michael Barbosa [INRIA]

External Collaborator

Wendlasida Ouedraogo [SIEMENS MOBILITY, until Sep 2023]

2 Overall objectives

There is an emerging consensus that formal methods must be used as a matter of course in software development. Most software is too complex to be fully understood by one programmer or even a team of programmers, and requires the help of computerized techniques such as testing and model checking to analyze and eliminate entire classes of bugs. Moreover, in order for the software to be maintainable and reusable, it not only needs to be bug-free but also needs to have fully specified behavior, ideally accompanied with formal and machine-checkable proofs of correctness with respect to the specification. Indeed, formal specification and machine verification is the only way to achieve the highest level of assurance (EAL7) according to the ISO/IEC Common Criteria.1

Historically, achieving such a high degree of certainty in the operation of software has required significant investment of manpower, and hence of money. As a consequence, only software that is of critical importance (and relatively unchanging), such as monitoring software for nuclear reactors or fly-by-wire controllers in airplanes, has been subjected to such intense scrutiny. However, we are entering an age where we need trustworthy software in more mundane situations, with rapid development cycles, and without huge costs. For example: modern cars are essentially mobile computing platforms, smart-devices manage our intensely personal details, elections (and election campaigns) are increasingly fully computerized, and networks of drones monitor air pollution, traffic, military arenas, etc. Bugs in such systems can certainly lead to unpleasant, dangerous, or even life-threatening incidents.

The field of formal methods has stepped up to meet this growing need for trustworthy general purpose software in recent decades. Techniques such as computational type systems and explicit program annotations/contracts, and tools such as model checkers and interactive theorem provers, are starting to become standard in the computing industry. Indeed, many of these tools and techniques are now a part of undergraduate computer science curricula. In order to be usable by ordinary programmers (without PhDs in logic), such tools and techniques have to be high level and rely heavily on automation. Furthermore, multiple tools and techniques often need to marshaled to achieve a verification task, so theorem provers, solvers, model checkers, property testers, etc. need to be able to communicate with—and, ideally, trust—each other.

With all this sophistication in formal tools, there is an obvious question: what should we trust? Sophisticated formal reasoning tools are, generally speaking, complex software artifacts themselves; if we want complex software to undergo rigorous formal analysis we must be prepared to formally analyze the tools and techniques used in formal reasoning itself. Historically, the issue of trust has been addressed by means of relativizing it to small and simple cores. This is the basis of industrially successful formal reasoning systems such as Coq, Isabelle, HOL4, and ACL2. However, the relativization of trust has led to a balkanization of the formal reasoning community, since the Coq kernel, for example, is incompatible with the Isabelle kernel, and neither can directly cross-validate formal developments built with the other. Thus, there is now a burgeoning cottage industry of translations and adaptations of different formal proof languages for bridging the gap. A number of proposals have also been made for universal or retargetable proof languages (e.g., Dedukti, ProofCert) so that the cross-platform trust issues can be factorized into single trusted checkers.

Beyond mutual incompatibility caused by relativized trust, there is a bigger problem that the proof evidence that is accepted by small kernels is generally far too detailed to be useful. Formal developments usually occurs at a much higher level, relying on algorithmic techniques such as unification, simplification, rewriting, and controlled proof search to fill in details. Indeed, the most reusable products of formal developments tend to be these algorithmic techniques and associated collections of hand-crafted rules. Unfortunately, these techniques are even less portable than the fully detailed proofs themselves, since the techniques are often implemented in terms of the behaviors of the trusted kernels. We can broadly say that the problem with relativized trust is that it is based on the operational interpretation of implementations of trusted kernels. There still remains the question of meta-theoretic correctness. Most formal reasoning systems implement a variant of a well known mathematical formalism (e.g., Martin-Löf type theory, set theory, higher-order logic), but it is surprising that hardly any mainstream system has a formalized meta-theory.2 Furthermore, formal reasoning systems are usually associated with complicated checkers for side-conditions that often have unclear mathematical status. For example, the Coq kernel has a built-in syntactic termination checker for recursive fixed-point expressions that is required to work correctly for the kernel to be sound. This termination checker evolves and improves with each version of Coq, and therefore the most accurate documentation of its behavior is its own source code. Coq is not special in this regard: similar trusted features exist in nearly every mainstream formal reasoning system.

The Partout project is interested in the principles of deductive and computational formalisms. In the broadest sense, we are interested in the question of trustworthy and verifiable meta-theory. At one end, this includes the well studied foundational questions of the meta-theory of logical systems and type systems: cut-elimination and focusing in proof theory, type soundness and normalization theorems in type theory, etc. The focus of our research here is on the fundamental relationships behind the the notions of computation and deduction. We are particularly interested in relationships that go beyond the well known correspondences between proofs and programs.3 Indeed, interpreting computation in terms of deduction (as in logic programming) or deduction in terms of computation (as in rewrite systems or in model checking) can often lead to fruitful and enlightening research questions, both theoretical and practical.

From another end, Partout works on the question of the essential nature of deductive or computational formalisms. For instance, we are interested in the question of proof identity that attempts to answer the following question: when are two proofs of the same theorem the same? Surprisingly, this very basic question is left unanswered in proof theory, the branch of mathematics that supposedly treats proofs as algebraic objects of interest. We also pay particular attention to the combinatorial and complexity-theoretic properties of the formalisms. Indeed, it is surprising that until very recently the $λ$ -calculus, which is the de facto basis of every functional programming language, lacked a good complexity-theoretic foundation, i.e., a cost model that would allow us to use the $λ$ -calculus directly to define complexity classes.

To put trustworthy meta-theory to use, the Partout project also works on the design and implementations of formal reasoning tools and techniques. We study the mathematical principles behind the representations of formal concepts ( $λ$ -terms, proofs, abstract machines, etc.), with the goal of identifying the relationships and trade-offs. We also study computational formalisms such as higher-order relational programming that is well suited to the specification and analysis of systems defined in the structural operational semantics (SOS) style. We also work on foundational questions about induction and co-induction, which are used in intricate combinations in metamathematics.

3 Research program

Software and hardware systems perform computation (systems that process, compute and perform) and deduction (systems that search, check or prove). The makers of those systems express their intent using various frameworks such as programming languages, specification languages, and logics. The Partout project aims at developing and using mathematical principles to design better frameworks for computation and reasoning. Principles of expression are researched from two directions, in tandem:

Foundational approaches, from theories to applications: studying fundamental problems of programming and proof theory.

Examples include studying the complexity of reduction strategies in lambda-calculi with sharing, or studying proof representations that quotient over rule permutations and can be adapted to many different logics.
Empirical approaches, from applications to theories: studying systems currently in use to build a theoretical understanding of the practical choices made by their designers.

Examples include studying realistic implementations of programming languages and proof assistants, which differ in interesting ways from their usual high-level formal description (regarding of sharing of code and data, for example), or studying new approaches to efficient automated proof search, relating them to existing approaches of proof theory, for example to design proof certificates or to generalize them to non-classical logics.

One of the strengths of Partout is the co-existence of a number of different expertise and points of view. Many dichotomies exist in the study of computation and deduction: functional programming vs logic programming, operational semantics vs denotational semantics, constructive logic vs classical logic, proof terms vs proof nets, etc. We do not identify with any one of them in particular, rather with them as a whole, believing in the value of interaction and cross-fertilization between different approaches. Partout defines its scope through the following core tenets:

An interest in both computation and logic.
The use of mathematical formalism as our core scientific method, paired with practical implementations of the systems we study.
A shared belief in the importance of good design when creating new means of expression, iterating towards simplicity and elegance.

More concretely, the research in Partout will be centered around the following four themes:

Foundations of proof theory as a theory of proofs. Current proof theory is not a theory of proofs but a theory of proof systems. This has many practical consequences, as a proof produced by modern theorem provers cannot be considered independent from the tool that produced it. A central research topic here is the quest for proof representations that are independent from the proof system, so that proof theory becomes a proper theory of proofs.
Program Equivalence We intend to use our proof theoretical insights to deepen our understanding of the structure of computer programs by discovering canonical representations for functional programming languages, and to apply these to the problems of program equivalence checking and program synthesis.
Reasoning with relational specifications of formal systems. Formal systems play a central role for proof checkers and proof assistants that are used for software verification. But there is usually a large gap between the specification of those formal systems in concise informal mathematical language and their implementation in ML or C code. Our research goal is to close that gap.
Foundations of complexity analysis for functional programs. One of the great merits of the functional programming paradigm is the natural availability of high-level abstractions. However, these abstractions jeopardize the programmer's predictive control on the performance of the code, since many low-level steps are abstracted away by higher-order functions. Our research goal is to regain that control by developing models of space and time costs for functional programs.

4 Application domains

4.1 Automated Theorem Proving

The Partout team studies the structure of mathematical proofs, in ways that often makes them more amenable to automated theorem proving – automatically searching the space of proof candidates for a statement to find an actual proof – or a counter-example.

(Due to fundamental computability limits, fully-automatic proving is only possible for simple statements, but this field has been making a lot of progress in recent years, and is in particular interested with the idea of generating verifiable evidence for the proofs that are found, which fits squarely within the expertise of Partout.)

4.2 Proof-assistants

Our work on the structure of proofs also suggests ways how they could be presented to a user, edited and maintained, in particular in “proof assistants”, automated tool to assist the writing of mathematical proofs with automatic checking of their correctness.

4.3 Programming language design

Our work also gives insight on the structure and properties of programming languages. We can improve the design or implementation of programming languages, help programmers or language implementors reason about the correctness of the programs in a given language, or reason about the cost of execution of a program.

5 Highlights of the year

5.1 Results

We proved that intuitionistic modal logic S4 is decidable, which was an open problem for 3 decades

5.2 Awards

Miller received the Dov Gabbay Prize for Logic and Foundations in 2023, for “pioneering and agenda-setting research bringing together and advancing logical proof theory and computational logic in the areas of higher-order logic programming and higher-order theorem proving.”
The OCaml programming language received the ACM SIGPLAN Programming Languages Software Award, with Gabriel Scherer listed among the 14 co-recipients.

6 New software, platforms, open data

6.1 New software

6.1.1 Abella

Keyword:
Proof assistant
Functional Description:
Abella is an interactive theorem prover for reasoning about computations given as relational specifications. Abella is particuarly well suited for reasoning about binding constructs.
Release Contributions:

This release includes a major refactoring of the Abella documentation generator. Abella developments can now be easily converted into interactive web-based presentations that can be used without having to run Abella by the reader.

This release also fixes a number of outstanding issues with the 2.0.7 and earlier releases. At least two of these fixes involve soundness issues with regard to higher-order arguments.

Abella is now also independently packaged for MacOS (homebrew), FreeBSD, and OpenBSD.
URL:
https://abella-prover.org/
Contact:
Kaustuv Chaudhuri
Participants:
Dale Miller, Gopalan Nadathur, Kaustuv Chaudhuri, Mary Southern, Matteo Cimini, Olivier Savary-Bélanger, Yuting Wang
Partner:
Department of Computer Science and Engineering, University of Minnesota

6.1.2 Actema

Name:
Actema
Keywords:
Higher-order logic, First-order logic, Proof assistant, GUI (Graphical User Interface), Man-machine interfaces, User Interfaces
Functional Description:
This is a new approach, aiming at making the building of formal proofs more intuitive and convenient. The system is currently at a prototype stage. An interfacing with the Coq proof-system is under study. The system runs through an html/JS serve.
Release Contributions:
This version can be used online at actema.xyz and comes with explanation videos.
URL:
http://actema.xyz
Publication:
03823357
Contact:
Benjamin Werner
Participants:
Benjamin Werner, Pablo Donato, Pierre-Yves Strub
Partner:
Ecole Polytechnique

6.1.3 DAMF Dispatch

Keywords:
Interactive Theorem Proving, Distributed systems, Verification
Scientific Description:

The Distributed Assertion Management Framework (DAMF) is a proposed collection of formats and techniques to enable heterogeneous formal reasoning systems and users to communicate assertions in a decentralized, reliable, and egalitarian manner. An assertion is a unit of mathematical knowledge—think lemmas, theorems, corollaries, etc.—that is cryptographically signed by its originator.

DAMF is based on content-addressable storage using the InterPlanetary File System (IPFS) network, and uses the InterPlanetary Linked Data (IPLD) data model to represent assertions and all their components.
Functional Description:

Dispatch is an intermediary tool for publishing, retrieval, and trust analysis in the Distributed Assertion Management Framework (DAMF). Dispatch specifies a family of JSON-based formats for DAMF objects and implements the main DAMF processes. It is intended to be usable by both human users and tools.

Dispatch is being developed as part of the exploratory action W3Proof.
Release Contributions:
This initial version has a demonstration proof of a theorem using a combination of Coq, LambdaProlog, and Abella.
URL:
https://distributed-assertions.github.io
Publication:
hal-04167922
Contact:
Kaustuv Chaudhuri
Participants:
Farah Al Wardani, Kaustuv Chaudhuri, Dale Miller

6.1.4 MOIN

Name:
MOdal Intuitionistic Nested sequents
Keywords:
Logic programming, Modal logic
Functional Description:

MOIN is a SWI Prolog theorem prover for classical and intuitionstic modal logics. The modal and intuitionistic modal logics considered are all the 15 systems occurring in the modal S5-cube, and all the decidable intuitionistic modal logics in the IS5-cube. MOIN also provides a protptype implementation for the intuitionistic logics for which decidability is not known (IK4,ID5 and IS4). MOIN is consists of a set of Prolog clauses, each clause representing a rule in one of the three proof systems. The clauses are recursively applied to a given formula, constructing a proof-search tree. The user selects the nested proof system, the logic, and the formula to be tested. In the case of classic nested sequent and Maehara-style nested sequents, MOIN yields a derivation, in case of success of the proof search, or a countermodel, in case of proof search failure. The countermodel for classical modal logics is a Kripke model, while for intuitionistic modal logic is a bi-relational model. In case of Gentzen-style nested sequents, the prover does not perform a countermodel extraction.

A system description of MOIN is available at https://hal.inria.fr/hal-02457240
URL:
http://www.lix.polytechnique.fr/Labo/Lutz.Strassburger/Software/Moin/MoinProver.html
Publication:
hal-02457240
Contact:
Lutz Strassburger

6.1.5 OCaml

Keywords:
Functional programming, Static typing, Compilation
Functional Description:
The OCaml language is a functional programming language that combines safety with expressiveness through the use of a precise and flexible type system with automatic type inference. The OCaml system is a comprehensive implementation of this language, featuring two compilers (a bytecode compiler, for fast prototyping and interactive use, and a native-code compiler producing efficient machine code for x86, ARM, PowerPC, RISC-V and System Z), a debugger, and a documentation generator. Many other tools and libraries are contributed by the user community and organized around the OPAM package manager.
URL:
https://ocaml.org/
Publications:
hal-03146495, hal-03510931, hal-03145030, hal-01929508, hal-03125031, hal-00772993, hal-00914493, hal-00914560, inria-00074804, hal-01499973, hal-01499946
Contact:
Damien Doligez
Participants:
Florian Angeletti, Damien Doligez, Xavier Leroy, Luc Maranget, Gabriel Scherer, David Allsopp, Stephen Dolan, Alain Frisch, Jacques Garrigue, Anil Madhavapeddy, Kc Sivaramakrishnan, Nicolas Ojeda Bar, Leo White

6.1.6 ocaml-boxroot

Keywords:
Interoperability, Library, Ocaml, Rust
Scientific Description:
Boxroot is an implementation of roots for the OCaml GC based on concurrent allocation techniques. These roots are designed to support a calling convention to interface between Rust and OCaml code that reconciles the latter's foreign function interface with the idioms from the former.
Functional Description:
Boxroot implements fast movable roots for OCaml in C. A root is a data type which contains an OCaml value, and interfaces with the OCaml GC to ensure that this value and its transitive children are kept alive while the root exists. This can be used to write programs in other languages that interface with programs written in OCaml.
URL:
https://gitlab.com/ocaml-rust/ocaml-boxroot
Publication:
hal-03910313
Contact:
Guillaume Munch
Participants:
Guillaume Munch, Gabriel Scherer

6.1.7 Profound-Intuitionistic

Name:
Interactive theorem proving by direct manipulation for Intuitionistic Logic
Keywords:
Interactive Theorem Proving, First-order logic
Functional Description:
Profound-Intuitionistic (Profint) is a tool for building formal proofs in intuitionistic logic using an interactive direct manipulation based web-interface. The tool can transform the interactive proof into formal proof objects in a variety of backend provers including: Coq, Lean 3, Lean 4, Isabelle/HOL, HOL4, and Abella.
Release Contributions:

This release adds support for inductive theorem proving using sized relations in the style of Abella.

This release also adds preliminary support for three-dimensional representations of the proof state (using WebGL and the Three.js library).
URL:
https://github.com/direct-manipulation/profint
Contact:
Kaustuv Chaudhuri

6.1.8 YADE

Name:
Yet Another Diagram Editor
Keyword:
Diagram
Functional Description:
This diagram editor can help mechanising diagrammatic categorical proofs by generating Coq proof scripts from a drawn diagram. This is part of the Coreact ANR Project (started in March 2023), which aims at developing a methodology for diagrammatic reasoning in Coq.
URL:
https://amblafont.github.io/graph-editor/index.html
Contact:
Ambroise Lafont
Participant:
Ambroise Lafont

7 New results

7.1 The Complexity of BV and Pomset Logic

Participants: Lutz Straßburger.

External Collaborators: Nguyễn, Lê Thành Dũng (ENS Lyon)

Following our result from last year, where we have shown that BV and pomset logic are different 27, we have continued our efforts and studied the complexity of the two logics. Our results are published in 10.

7.2 Coqlex, an approach to generate verified lexers

Participants: Lutz Straßburger, Wendlasida Ouedraogo, Gabriel Scherer.

External Collaborators: Danko Ilik (Siemens)

A compiler consists of a sequence of phases going from lexical analysis to code generation. Ideally, the formal verification of a compiler should include the formal verification of every component of the tool-chain. In order to contribute to the end-to-end verification of compilers, we implemented a verified lexer generator with usage similar to OCamllex. This software-Coqlex-reads a lexer specification and generates a lexer equipped with Coq proofs of its correctness. Although the performance of the generated lexers does not measure up to the performance of a standard lexer generator such as OCamllex, the safety guarantees it comes with make it an interesting alternative to use when implementing totally verified compilers or other language processing tools.

More details on this work can be found here 11

7.3 Intuitionistic S4 is decidable

Participants: Lutz Straßburger, Marianela Evelyn Morales Elena.

External Collaborators: Marianna Girlando (Amsterdam), Sonia Marin (Birmingham), Roman Kuznets (Vienna)

In this work, published in 20, we demonstrate decidability for the intuitionistic modal logic S4 first formulated by Fischer Servi. This solves a problem that has been open for almost thirty years since it had been posed in Simpson's PhD thesis in 1994. We obtain this result by performing proof search in a labelled deductive system that, instead of using only one binary relation on the labels, employs two: one corresponding to the accessibility relation of modal logic and the other corresponding to the order relation of intuitionistic Kripke frames. Our search algorithm outputs either a proof or a finite counter-model, thus, additionally establishing the finite model property for intuitionistic S4, which has been another long-standing open problem in the area.

7.4 Term representation and proof theory

Participants: Beniamino Accattoli, Dale Miller, Jui-Hsuan Wu.

Structural proof theory has been used to provide a principled approach to designing term representations and operations (such as substitutions) on them. In the past, the negative polarity approach has been the only one used. In our recent work, we have considered, instead, using the positive polarity approach. This has lead us to a principled description of sharing within terms. While sharing has been described previously in various setting, the one based on proof theory has certain advantages since it is less ad hoc and comes with its own notion of substitution (derived immediately from using cut elimination). This approach also provides a natural setting for treating binding structures within programs. Details of this approach are available from our CSL 2023 paper 13. Wu has developed this positive perspective further in his APLAS 2023 paper 21.

7.5 Formal Reasoning using Distributed Assertions

Participants: Farah Al Wardani, Kaustuv Chaudhuri, Dale Miller.

When a proof system checks a formal proof, we can say that its kernel asserts that the formula is a theorem in a particular logic. We describe a general framework in which such assertions can be made globally available so that any proof assistant willing to trust the assertion's creator can use that assertion without rechecking any associated formal proof. This framework, called DAMF, is heterogeneous and allows each participant to decide which tools and operators they are willing to trust in order to accept external assertions. This framework can also be integrated into existing proof systems by making minor changes to the input and output subsystems of the prover. DAMF achieves a high level of distributivity using off-the-shelf technologies such as IPFS, IPLD, and public key cryptography. We illustrate the framework by describing an implemented tool, called DISPATCH, for validating and publishing assertion objects and a modified version of the Abella theorem prover that can use and publish such assertions. The details of the DAMF system can be found in the FroCos 2023 paper 17 and the earlier report 24.

7.6 A foundation for proof theory based on searching for proofs

Participants: Dale Miller.

In 1935, Gentzen captured the way human think about formal proofs using the natural deduction proof system. This system was based on describing complete proofs and is based on a kind of annotated tree structure. In practice, however, it is also natural to consider the process of extending partial proofs until they become complete. In his LICS 2023 paper 12, Miller describes a simply motivated way to describe proofs based on proof search based on multiset rewriting of various annotated formula objects. This proof framework, called PSF, incorporates many structural features of proof (many, comming from Linear Logic) that were not part of Gentzen's original natural deduction proof systems.

7.7 Convolution Products on Double Categories and Categorification of Rule Algebras

Participants: Noam Zeilberger.

External Collaborators: Nicolas Behr (IRIF), Paul-André Melliès (IRIF)

Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product. These results were published at FSCD 2023 18.

7.8 The Algebraic Weak Factorisation System for Delta Lenses

Participants: Bryce Clarke.

Delta lenses are functors equipped with a suitable choice of lifts, and are used to model bidirectional transformations between systems. In this paper, we construct an algebraic weak factorisation system whose R-algebras are delta lenses. Our approach extends a semi-monad for delta lenses previously introduced by Johnson and Rosebrugh, and generalises to any suitable category equipped with an orthogonal factorisation system and an idempotent comonad. We demonstrate how the framework of an algebraic weak factorisation system provides a natural setting for understanding the lifting operation of a delta lens, and also present an explicit description of the free delta lens on a functor. Published in 19