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11th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2022),
January 17-18, 2022,
Philadelphia, PA, USA
11th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2022)
Frontmatter
Welcome from the Chairs
Welcome to the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2022). CPP covers the practical and theoretical topics in all areas that consider formal verification and certification as an essential paradigm for their work. CPP spans areas of computer science, mathematics, logic, and education. CPP 2022 will be held on 17-18 January 2022 in Philadelphia, Pennsylvania, United States. The conference is co-located with POPL 2022, and is sponsored by ACM SIGPLAN in cooperation with ACM SIGLOG.
Invited Talks
The seL4 Verification: The Art and Craft of Proof and the Reality of Commercial Support (Invited Talk)
June Andronick
(Proofcraft, Australia; UNSW, Australia; seL4 Foundation, Australia)
The formal verification of the seL4 microkernel started as a research project in 2004 and has achieved commercial scale now, in the number of properties proven, the supported features and platforms, the adoption and deployment by industry and government organisations. It is supported by an open-source Foundation and a growing ecosystem. In this talk, I will reflect on the seL4 verification journey, past, present and future, and the challenges to combine the art and craft of proof with the reality of meeting industry demand for verified software.
@InProceedings{CPP22p1,
author = {June Andronick},
title = {The seL4 Verification: The Art and Craft of Proof and the Reality of Commercial Support (Invited Talk)},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {1--1},
doi = {10.1145/3497775.3505265},
year = {2022},
}
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Coq’s Vibrant Ecosystem for Verification Engineering (Invited Talk)
Andrew W. Appel
(Princeton University, USA)
Program verification in the large is not only a matter of mechanizing a program logic to handle the semantics of your programming language. You must reason in the mathematics of your application domain--and there are many application domains, each with their own community of domain experts. So you will need to import mechanized proof theories from many domains, and they must all interoperate. Such an ecosystem is not only a matter of mathematics, it is a matter of software process engineering and social engineering. Coq's ecosystem has been maturing nicely in these senses.
@InProceedings{CPP22p2,
author = {Andrew W. Appel},
title = {Coq’s Vibrant Ecosystem for Verification Engineering (Invited Talk)},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {2--11},
doi = {10.1145/3497775.3503951},
year = {2022},
}
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Structural Embeddings Revisited (Invited Talk)
César Muñoz
(AWS, USA)
A semantic embedding is a logical encoding of a formal language, namely the object language, into the specification language of a logical framework. In their seminal paper “Experience with embedding hardware description languages in HOL”, Boulton et al. coined the terms deep and shallow embeddings depending on whether or not the syntax of terms of the target language is represented by a data type in the specification language. Thus, a deep embedding enables reasoning about classes of terms, while a shallow embedding limits reasoning to concrete terms. Embeddings of programming languages are well-known applications of interactive theorem provers, specially of those based on higher-order logic. These embeddings are often intended to support the study of a programming language semantics or to enhance a programming language with the deductive capabilities of the logical framework. A different type of embeddings, here referred to as structural embeddings, are intended to augment specification languages with structural elements of the object language. In a structural embedding, the outermost elements of the object language, i.e., the structural parts, are encoded, either deeply or shallowly, but the internal elements, i.e., the basic expressions, are those of the specification language. Advances in automated reasoning and user interfaces have enabled structural embeddings to enhance usability of interactive theorem provers and to reduce the gap between verification tools and modeling tools used by practitioners. This talk presents an overview of several years of research on theorem proving in safety-critical aerospace systems through the lens of embeddings and, more particularly, structural embeddings. The talk focuses on lessons learned and provides examples of successful applications to automated reasoning, termination analysis, floating-point analysis, and verification of cyber-physical systems. Our main point, which is hardly original, is that interactive theorem provers will serve as intermediate systems that connect a cluster of components. Structural embeddings could then provide the frontend capabilities to access this cluster of components.
@InProceedings{CPP22p12,
author = {César Muñoz},
title = {Structural Embeddings Revisited (Invited Talk)},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {12--12},
doi = {10.1145/3497775.3503949},
year = {2022},
}
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Program Verification
Overcoming Restraint: Composing Verification of Foreign Functions with Cogent
Louis Cheung
,
Liam O'Connor , and Christine Rizkallah
(University of Melbourne, Australia; University of Edinburgh, UK)
Cogent is a restricted functional language designed to reduce the cost of developing verified systems code. Because of its sometimes-onerous restrictions, such as the lack of support for recursion and its strict uniqueness type system, Cogent provides an escape hatch in the form of a foreign function interface (FFI) to C code. This poses a problem when verifying Cogent programs, as imported C components do not enjoy the same level of static guarantees that Cogent does. Previous verification of file systems implemented in Cogent merely assumed that their C components were correct and that they preserved the invariants of Cogent’s type system. In this paper, we instead prove such obligations. We demonstrate how they smoothly compose with existing Cogent theorems, and result in a correctness theorem of the overall Cogent-C system. The Cogent FFI constraints ensure that key invariants of Cogent’s type system are maintained even when calling C code. We verify reusable higher-order and polymorphic functions including a generic loop combinator and array iterators and demonstrate their application to several examples including binary search and the BilbyFs file system. We demonstrate the feasibility of verification of mixed Cogent-C systems, and provide some insight into verification of software comprised of code in multiple languages with differing levels of static guarantees.
@InProceedings{CPP22p13,
author = {Louis Cheung and Liam O'Connor and Christine Rizkallah},
title = {Overcoming Restraint: Composing Verification of Foreign Functions with Cogent},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {13--26},
doi = {10.1145/3497775.3503686},
year = {2022},
}
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Verbatim++: Verified, Optimized, and Semantically Rich Lexing with Derivatives
Derek Egolf, Sam Lasser, and Kathleen Fisher
(Northeastern University, USA; Tufts University, USA)
Lexers and parsers are attractive targets for attackers because they often sit at the boundary between a software system's internals and the outside world. Formally verified lexers can reduce the attack surface of these systems, thus making them more secure.
One recent step in this direction is the development of Verbatim, a verified lexer based on the concept of Brzozowski derivatives. Two limitations restrict the tool's usefulness. First, its running time is quadratic in the length of its input string. Second, the lexer produces tokens with a simple "tag and string" representation, which limits the tool's ability to integrate with parsers that operate on more expressive token representations.
In this work, we present a suite of extensions to Verbatim that overcomes these limitations while preserving the tool's original correctness guarantees. The lexer achieves effectively linear performance on a JSON benchmark through a combination of optimizations that, to our knowledge, has not been previously verified. The enhanced version of Verbatim also enables users to augment their lexical specifications with custom semantic actions, and it uses these actions to produce semantically rich tokens---i.e., tokens that carry values with arbitrary, user-defined types. All extensions were implemented and verified with the Coq Proof Assistant.
@InProceedings{CPP22p27,
author = {Derek Egolf and Sam Lasser and Kathleen Fisher},
title = {Verbatim++: Verified, Optimized, and Semantically Rich Lexing with Derivatives},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {27--39},
doi = {10.1145/3497775.3503694},
year = {2022},
}
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Formally Verified Superblock Scheduling
Cyril Six, Léo Gourdin,
Sylvain Boulmé ,
David Monniaux , Justus Fasse, and Nicolas Nardino
(Kalray, France; Grenoble Alps University, France; CNRS, France; Grenoble INP, France; Verimag, France; ENS Lyon, France)
On in-order processors, without dynamic instruction scheduling,
program running times may be significantly reduced by compile-time
instruction scheduling. We present here the first effective certified
instruction scheduler that operates over superblocks (it may move
instructions across branches), along with its performance
evaluation. It is integrated within the CompCert C compiler, providing
a complete machine-checked proof of semantic preservation from C to
assembly.
Our optimizer composes several passes designed by translation
validation: program transformations are proposed by untrusted oracles,
which are then validated by certified and scalable checkers. Our main
checker is an architecture-independent simulation-test over
superblocks modulo register liveness, which relies on hash-consed
symbolic execution.
@InProceedings{CPP22p40,
author = {Cyril Six and Léo Gourdin and Sylvain Boulmé and David Monniaux and Justus Fasse and Nicolas Nardino},
title = {Formally Verified Superblock Scheduling},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {40--54},
doi = {10.1145/3497775.3503679},
year = {2022},
}
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Semantics
Certified Abstract Machines for Skeletal Semantics
Guillaume Ambal, Sergueï Lenglet, and Alan Schmitt
(University of Rennes, France; University of Lorraine, France; Inria, France)
Skeletal semantics is a framework to describe semantics of programming
languages. We propose an automatic generation of a certified OCaml interpreter
for any language written in skeletal semantics. To this end, we introduce two
new interpretations, i.e., formal meanings, of skeletal semantics, in the form
of non-deterministic and deterministic abstract machines. These machines are
derived from the usual big-step interpretation of skeletal semantics using
functional correspondence, a standard transformation from big-step
evaluators to abstract machines. All these interpretations are
formalized in the Coq proof assistant and we certify their soundness. We finally
use the extraction from Coq to OCaml to obtain the certified interpreter.
@InProceedings{CPP22p55,
author = {Guillaume Ambal and Sergueï Lenglet and Alan Schmitt},
title = {Certified Abstract Machines for Skeletal Semantics},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {55--67},
doi = {10.1145/3497775.3503676},
year = {2022},
}
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A Compositional Proof Framework for FRETish Requirements
Esther Conrad
,
Laura Titolo , Dimitra Giannakopoulou, Thomas Pressburger, and Aaron Dutle
(NASA Langley Research Center, USA; National Institute of Aerospace, USA; NASA Ames Research Center, USA)
Structured natural languages provide a trade space between ambiguous natural languages that make up most written requirements, and mathematical formal specifications such as Linear Temporal Logic. FRETish is a structured natural language for the elicitation of system requirements developed at NASA. The related open-source tool Fret provides support for translating FRETish requirements into temporal logic formulas that can be input to several verification and analysis tools. In the context of safety-critical systems, it is crucial to ensure that a generated formula captures the semantics of the corresponding FRETish requirement precisely. This paper presents a rigorous formalization of the FRETish language including a new denotational semantics and a proof of semantic equivalence between FRETish specifications and their temporal logic counterparts computed by Fret. The complete formalization and the proof have been developed in the Prototype Verification System (PVS) theorem prover.
@InProceedings{CPP22p68,
author = {Esther Conrad and Laura Titolo and Dimitra Giannakopoulou and Thomas Pressburger and Aaron Dutle},
title = {A Compositional Proof Framework for FRETish Requirements},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {68--81},
doi = {10.1145/3497775.3503685},
year = {2022},
}
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Verified Data Structures
Specification and Verification of a Transient Stack
Alexandre Moine
,
Arthur Charguéraud, and
François Pottier
(Inria, France; University of Strasbourg, France; CNRS, France)
A transient data structure is a package of an ephemeral data structure, a
persistent data structure, and fast conversions between them. We describe the
specification and proof of a transient stack and its iterators. This data
structure is a scaled-down version of the general-purpose transient sequence
data structure implemented in the OCaml library Sek. Internally, it relies on
fixed-capacity arrays, or chunks, which can be shared between several
ephemeral and persistent stacks. Dynamic tests are used to determine whether a
chunk can be updated in place or must be copied: a chunk can be updated if it
is uniquely owned or if the update is monotonic. Using CFML, which implements
Separation Logic with Time Credits inside Coq, we verify the functional
correctness and the amortized time complexity of this data structure. Our
verification effort covers iterators, which involve direct pointers to
internal chunks. The specification of iterators describes what the operations
on iterators do, how much they cost, and under what circumstances an iterator
is invalidated.
@InProceedings{CPP22p82,
author = {Alexandre Moine and Arthur Charguéraud and François Pottier},
title = {Specification and Verification of a Transient Stack},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {82--99},
doi = {10.1145/3497775.3503677},
year = {2022},
}
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Mechanized Verification of a Fine-Grained Concurrent Queue from Meta’s Folly Library
Simon Friis Vindum , Dan Frumin
, and
Lars Birkedal
(Aarhus University, Denmark; University of Groningen, Netherlands)
We present the first formal specification and verification of the fine-grained concurrent multi-producer-multi-consumer queue algorithm from Meta’s C++ library Folly of core infrastructure components. The queue is highly optimized, practical, and used by Meta in production where it scales to thousands of consumer and producer threads. We present an implementation of the algorithm in an ML-like language and formally prove that it is a contextual refinement of a simple coarse-grained queue (a property that implies that the MPMC queue is linearizable). We use the ReLoC relational logic and the Iris program logic to carry out the proof and to mechanize it in the Coq proof assistant. The MPMC queue is implemented using three modules, and our proof is similarly modular. By using ReLoC and Iris’s support for modular reasoning we verify each module in isolation and compose these together. A key challenge of the MPMC queue is that it has a so-called external linearization point, which ReLoC has no support for reasoning about. Thus we extend ReLoC, both on paper and in Coq, with novel support for reasoning about external linearization points.
@InProceedings{CPP22p100,
author = {Simon Friis Vindum and Dan Frumin and Lars Birkedal},
title = {Mechanized Verification of a Fine-Grained Concurrent Queue from Meta’s Folly Library},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {100--115},
doi = {10.1145/3497775.3503689},
year = {2022},
}
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Applying Formal Verification to Microkernel IPC at Meta
Quentin Carbonneaux, Noam Zilberstein, Christoph Klee, Peter W. O'Hearn
, and Francesco Zappa Nardelli
(Meta, France; Meta, USA; Cornell University, USA; Meta, UK; University College London, UK)
We use Iris, an implementation of concurrent separation logic in the Coq proof assistant, to verify two queue data structures used for inter-process communication in an operating system under development. Our motivations are twofold. First, we wish to leverage formal verification to boost confidence in a delicate piece of industrial code that was subject to numerous revisions. Second, we aim to gain information on the cost-benefit tradeoff of applying a state-of-the-art formal verification tool in our industrial setting. On both fronts, our endeavor has been a success. The verification effort proved that the queue algorithms are correct and uncovered four algorithmic simplifications as well as bugs in client code. The simplifications involve the removal of two memory barriers, one atomic load, and one boolean check, all in a performance-sensitive part of the OS. Removing the redundant boolean check revealed unintended uses of uninitialized memory in multiple device drivers, which were fixed. The proof work was completed in person months, not years, by engineers with no prior familiarity with Iris. These findings are spurring further use of verification at Meta.
@InProceedings{CPP22p116,
author = {Quentin Carbonneaux and Noam Zilberstein and Christoph Klee and Peter W. O'Hearn and Francesco Zappa Nardelli},
title = {Applying Formal Verification to Microkernel IPC at Meta},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {116--129},
doi = {10.1145/3497775.3503681},
year = {2022},
}
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Distributed Systems and Concurrency
Forward Build Systems, Formally
Sarah Spall, Neil Mitchell, and Sam Tobin-Hochstadt
(Indiana University, USA; Meta, UK)
Build systems are a fundamental part of software construction, but
their correctness has received comparatively little attention,
relative to more prominent parts of the toolchain. In this paper, we
address the correctness of forward build systems, which
automatically determine the dependency structure of the build, rather
than having it specified by the programmer.
We first define what it means for a forward build system to be
correct---it must behave identically to simply executing the
programmer-specified commands in order. Of course, realistic build
systems avoid repeated work, stop early when possible, and run
commands in parallel, and we prove that these optimizations, as
embodied in the recent forward build system Rattle, preserve
our definition of correctness. Along the way, we show that other
forward build systems, such as Fabricate and Memoize,
are also correct.
We carry out all of our work in Agda, and describe in detail the
assumptions underlying both Rattle itself and our modeling of it.
@InProceedings{CPP22p130,
author = {Sarah Spall and Neil Mitchell and Sam Tobin-Hochstadt},
title = {Forward Build Systems, Formally},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {130--142},
doi = {10.1145/3497775.3503687},
year = {2022},
}
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Formal Verification of a Distributed Dynamic Reconfiguration Protocol
William Schultz, Ian Dardik, and Stavros Tripakis
(Northeastern University, USA)
We present a formal, machine checked TLA+ safety proof of MongoRaftReconfig, a distributed dynamic reconfiguration protocol. MongoRaftReconfig was designed for and implemented in MongoDB, a distributed database whose replication protocol is derived from the Raft consensus algorithm. We present an inductive invariant for MongoRaftReconfig that is formalized in TLA+ and formally proved using the TLA+ proof system (TLAPS). We also present a formal TLAPS proof of two key safety properties of MongoRaftReconfig, LeaderCompleteness and StateMachineSafety. To our knowledge, these are the first machine checked inductive invariant and safety proof of a dynamic reconfiguration protocol for a Raft based replication system.
@InProceedings{CPP22p143,
author = {William Schultz and Ian Dardik and Stavros Tripakis},
title = {Formal Verification of a Distributed Dynamic Reconfiguration Protocol},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {143--152},
doi = {10.1145/3497775.3503688},
year = {2022},
}
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Blockchains and Cryptography
A Verified Algebraic Representation of Cairo Program Execution
Jeremy Avigad , Lior Goldberg, David Levit, Yoav Seginer, and Alon Titelman
(Carnegie Mellon University, USA; Starkware Industries, Israel)
Cryptographic interactive proof systems provide an efficient and scalable means of verifying the results of computation on blockchain. A prover constructs a proof, off-chain, that the execution of a program on a given input terminates with a certain result. The prover then publishes a certificate that can be verified efficiently and reliably modulo commonly accepted cryptographic assumptions. The method relies on an algebraic encoding of execution traces of programs. Here we report on a verification of the correctness of such an encoding of the Cairo model of computation with respect to the STARK interactive proof system, using the Lean 3 proof assistant.
@InProceedings{CPP22p153,
author = {Jeremy Avigad and Lior Goldberg and David Levit and Yoav Seginer and Alon Titelman},
title = {A Verified Algebraic Representation of Cairo Program Execution},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {153--165},
doi = {10.1145/3497775.3503675},
year = {2022},
}
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Reflection, Rewinding, and Coin-Toss in EasyCrypt
Denis Firsov
and Dominique Unruh
(Tallinn University of Technology, Estonia; Guardtime, Estonia; University of Tartu, Estonia)
In this paper we derive a suite of lemmas which allows users to
internally reflect EasyCrypt programs into distributions which
correspond to their denotational semantics (probabilistic
reflection). Based on this we develop techniques for reasoning about
rewinding of adversaries in EasyCrypt. (A widely used technique in
cryptology.) We use our reflection and rewindability results to prove
the security of a coin-toss protocol.
@InProceedings{CPP22p166,
author = {Denis Firsov and Dominique Unruh},
title = {Reflection, Rewinding, and Coin-Toss in EasyCrypt},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {166--179},
doi = {10.1145/3497775.3503693},
year = {2022},
}
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Proof Infrastructure
An Extension of the Framework Types-To-Sets for Isabelle/HOL
Mihails Milehins
In their article titled From Types to Sets by Local Type Definitions in Higher-Order Logic and published in the proceedings of the conference Interactive Theorem Proving in 2016, Ondřej Kunčar and Andrei Popescu propose an extension of the logic Isabelle/HOL and an associated algorithm for the relativization of the type-based theorems to more flexible set-based theorems, collectively referred to as Types-To-Sets. One of the aims of their work was to open an opportunity for the development of a software tool for applied relativization in the implementation of the logic Isabelle/HOL of the proof assistant Isabelle. In this article, we provide a description of a software framework for the interactive automated relativization of definitions and theorems in Isabelle/HOL, developed as an extension of the proof language Isabelle/Isar, building upon some of the ideas for further work expressed in the original article on Types-To-Sets by Ondřej Kunčar and Andrei Popescu and the subsequent article Smooth Manifolds and Types to Sets for Linear Algebra in Isabelle/HOL, which was written by Fabian Immler and Bohua Zhan and published in the proceedings of the International Conference on Certified Programs and Proofs in 2019.
@InProceedings{CPP22p180,
author = {Mihails Milehins},
title = {An Extension of the Framework Types-To-Sets for Isabelle/HOL},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {180--196},
doi = {10.1145/3497775.3503674},
year = {2022},
}
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A Drag-and-Drop Proof Tactic
Pablo Donato
, Pierre-Yves Strub, and Benjamin Werner
(École Polytechnique, France)
We explore the features of a user interface where formal proofs can
be built through gestural actions. In particular, we show how proof
construction steps can be associated to drag-and-drop actions. We
argue that this can provide quick and intuitive proof construction
steps. This work builds on theoretical tools coming from deep
inference. It also resumes and integrates some ideas of the former
proof-by-pointing project.
@InProceedings{CPP22p197,
author = {Pablo Donato and Pierre-Yves Strub and Benjamin Werner},
title = {A Drag-and-Drop Proof Tactic},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {197--209},
doi = {10.1145/3497775.3503692},
year = {2022},
}
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Rewriting and Automated Reasoning
CertiStr: A Certified String Solver
Shuanglong Kan, Anthony Widjaja Lin,
Philipp Rümmer, and Micha Schrader
(TU Kaiserslautern, Germany; MPI-SWS, Germany; Uppsala University, Sweden)
Theories over strings are among the most heavily researched logical theories in the SMT community in the past decade, owing to the error-prone nature of string manipulations, which often leads to security vulnerabilities (e.g. cross-site scripting and code injection). The majority of the existing decision procedures and solvers for these theories are themselves intricate; they are complicated algorithmically, and also have to deal with a very rich vocabulary of operations. This has led to a plethora of bugs in implementation, which have for instance been discovered through fuzzing.
In this paper, we present CertiStr, a certified implementation of a string constraint solver for the theory of strings with concatenation and regular constraints. CertiStr aims to solve string constraints using a forward-propagation algorithm based on symbolic representations of regular constraints as symbolic automata, which returns three results: sat, unsat, and unknown, and is guaranteed to terminate for the string constraints whose concatenation dependencies are acyclic. The implementation has been developed and proven correct in Isabelle/HOL, through which an effective solver in OCaml was generated. We demonstrate the effectiveness and efficiency of CertiStr against the standard Kaluza benchmark, in which 80.4% tests are in the string constraint fragment of CertiStr. Of these 80.4% tests, CertiStr can solve 83.5% (i.e. CertiStr returns sat or unsat) within 60s.
@InProceedings{CPP22p210,
author = {Shuanglong Kan and Anthony Widjaja Lin and Philipp Rümmer and Micha Schrader},
title = {CertiStr: A Certified String Solver},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {210--224},
doi = {10.1145/3497775.3503691},
year = {2022},
}
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Safe, Fast, Concurrent Proof Checking for the lambda-Pi Calculus Modulo Rewriting
Michael Färber
(University of Innsbruck, Austria)
Several proof assistants, such as Isabelle or Coq,
can concurrently check multiple proofs.
In contrast, the vast majority of today's small proof checkers
either does not support concurrency at all or only limited forms thereof,
restricting the efficiency of proof checking on multi-core processors.
This work shows the design of a small, memory- and thread-safe kernel that
efficiently checks proofs both concurrently and sequentially.
This design is implemented in a new proof checker called Kontroli for
the lambda-Pi calculus modulo rewriting, which is
an established framework to uniformly express a multitude of logical systems.
Kontroli is faster than
the reference proof checker for this calculus, Dedukti,
on all of five evaluated datasets
obtained from proof assistants and interactive theorem provers.
Furthermore, Kontroli reduces the time of
the most time-consuming part of proof checking
using eight threads by up to 6.6x.
@InProceedings{CPP22p225,
author = {Michael Färber},
title = {Safe, Fast, Concurrent Proof Checking for the lambda-Pi Calculus Modulo Rewriting},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {225--238},
doi = {10.1145/3497775.3503683},
year = {2022},
}
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Formalized Mathematics
Formalising Lie Algebras
Oliver Nash
(Imperial College London, UK)
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none is assumed; the intention is that the overall themes will be accessible even to readers unfamiliar with Lie theory.
Particular attention is paid to the construction of the classical and exceptional Lie algebras. Thanks to these constructions, it is possible to state the classification theorem for finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero.
In addition to the focus on Lie theory, we also aim to highlight the unity of Mathlib. To this end, we include examples of achievements made possible only by leaning on several branches of the library simultaneously.
@InProceedings{CPP22p239,
author = {Oliver Nash},
title = {Formalising Lie Algebras},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {239--250},
doi = {10.1145/3497775.3503672},
year = {2022},
}
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Windmills of the Minds: An Algorithm for Fermat’s Two Squares Theorem
Hing Lun Chan
(Australian National University, Australia)
The two squares theorem of Fermat is a gem in number theory,
with a spectacular one-sentence "proof from the Book".
Here is a formalisation of this proof, with an interpretation using windmill patterns.
The theory behind involves involutions on a finite set,
especially the parity of the number of fixed points in the involutions.
Starting as an existence proof that is non-constructive, there is an ingenious way to turn it into a constructive one. This gives an algorithm to compute the two squares by iterating the two involutions alternatively from a known fixed point.
@InProceedings{CPP22p251,
author = {Hing Lun Chan},
title = {Windmills of the Minds: An Algorithm for Fermat’s Two Squares Theorem},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {251--264},
doi = {10.1145/3497775.3503673},
year = {2022},
}
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A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem
Ariel Kellison
(Cornell University, USA)
A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.
@InProceedings{CPP22p265,
author = {Ariel Kellison},
title = {A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {265--273},
doi = {10.1145/3497775.3503682},
year = {2022},
}
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Formalization of Logic
Undecidability, Incompleteness, and Completeness of Second-Order Logic in Coq
Mark Koch and Dominik Kirst
(Saarland University, Germany)
We mechanise central metatheoretic results about second-order logic (SOL) using the Coq proof assistant. Concretely, we consider undecidability via many-one reduction from Diophantine equations (Hilbert's tenth problem), incompleteness regarding full semantics via categoricity of second-order Peano arithmetic, and completeness regarding Henkin semantics via translation to mono-sorted first-order logic (FOL). Moreover, this translation is used to transport further characteristic properties of FOL to SOL, namely the compactness and Löwenheim-Skolem theorems.
@InProceedings{CPP22p274,
author = {Mark Koch and Dominik Kirst},
title = {Undecidability, Incompleteness, and Completeness of Second-Order Logic in Coq},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {274--290},
doi = {10.1145/3497775.3503684},
year = {2022},
}
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Semantic Cut Elimination for the Logic of Bunched Implications, Formalized in Coq
Dan Frumin
(University of Groningen, Netherlands)
The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs. Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient. In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus.
The presented proof is semantic, in the sense that is obtained by interpreting sequents in a particular “universal” model. This results in a more modular and elegant proof than a standard Gentzen-style cut elimination argument, which can be subtle and error-prone in manual proofs for BI. In particular, our semantic approach avoids unnecessary inversions on proof derivations, or the uses of cut reductions and the multi-cut rule.
Besides modular, our approach is also robust: we demonstrate how our method scales, with minor modifications, to (i) an extension of BI with an arbitrary set of simple structural rules, and (ii) an extension with an S4-like □ modality.
@InProceedings{CPP22p291,
author = {Dan Frumin},
title = {Semantic Cut Elimination for the Logic of Bunched Implications, Formalized in Coq},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {291--306},
doi = {10.1145/3497775.3503690},
year = {2022},
}
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Category Theory and HoTT
Implementing a Category-Theoretic Framework for Typed Abstract Syntax
Benedikt Ahrens ,
Ralph Matthes , and
Anders Mörtberg
(TU Delft, Netherlands; University of Birmingham, UK; IRIT, France; Université de Toulouse, France; CNRS, France; Toulouse INP, France; UT3, France; Stockholm University, Sweden)
In previous work ("From signatures to monads in UniMath"),we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant.
In the present work, we describe what was necessary to generalize that work to account for simply-typed languages. First, some definitions had to be generalized to account for the natural appearance of non-endofunctors in the simply-typed case. As it turns out, in many cases our mechanized results carried over to the generalized definitions without any code change. Second, an existing mechanized library on 𝜔-cocontinuous functors had to be extended by constructions and theorems necessary for constructing multi-sorted syntax. Third, the theoretical framework for the semantical signatures had to be generalized from a monoidal to a bicategorical setting, again to account for non-endofunctors arising in the typed case. This uses actions of endofunctors on functors with given source, and the corresponding notion of strong functors between actions, all formalized in UniMath using a recently developed library of bicategory theory. We explain what needed to be done to plug all of these ingredients together, modularly.
The main result of our work is a general construction that, when fed with a signature for a simply-typed language, returns an implementation of that language together with suitable boilerplate code, in particular, a certified monadic substitution operation.
@InProceedings{CPP22p307,
author = {Benedikt Ahrens and Ralph Matthes and Anders Mörtberg},
title = {Implementing a Category-Theoretic Framework for Typed Abstract Syntax},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {307--323},
doi = {10.1145/3497775.3503678},
year = {2022},
}
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(Deep) Induction Rules for GADTs
Patricia Johann and Enrico Ghiorzi
(Appalachian State University, USA)
Deep data types are those that are constructed from other data types, including, possibly, themselves. In this case, they are said to be truly nested. Deep induction is an extension of structural induction that traverses all of the structure in a deep data type, propagating predicates on its primitive data throughout the entire structure. Deep induction can be used to prove properties of nested types, including truly nested types, that cannot be proved via structural induction. In this paper we show how to extend deep induction to GADTs that are not truly nested GADTs. This opens the way to incorporating automatic generation of (deep) induction rules for them into proof assistants. We also show that the techniques developed in this paper do not suffice for extending deep induction to truly nested GADTs, so more sophisticated techniques are needed to derive deep induction rules for them.
@InProceedings{CPP22p324,
author = {Patricia Johann and Enrico Ghiorzi},
title = {(Deep) Induction Rules for GADTs},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {324--337},
doi = {10.1145/3497775.3503680},
year = {2022},
}
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On Homotopy of Walks and Spherical Maps in Homotopy Type Theory
Jonathan Prieto-Cubides
(University of Bergen, Norway)
We work with combinatorial maps
to represent graph embeddings into surfaces up to isotopy. The surface
in which the graph is embedded is left implicit in this approach. The
constructions herein are proof-relevant and stated with a
subset of the language of homotopy type theory.
This article presents a refinement of one characterisation of
embeddings in the sphere, called spherical maps, of connected
and directed multigraphs with discrete node sets. A combinatorial
notion of homotopy for walks and the normal form of walks under a
reduction relation is introduced. The first characterisation of
spherical maps states that a graph can be embedded in the sphere if
any pair of walks with the same endpoints are merely walk-homotopic.
The refinement of this definition filters out any walk with inner
cycles. As we prove in one of the lemmas, if a spherical map is
given for a graph with a discrete node set, then any walk in the graph
is merely walk-homotopic to a normal form.
The proof assistant Agda contributed to formalising the results
recorded in this article.
@InProceedings{CPP22p338,
author = {Jonathan Prieto-Cubides},
title = {On Homotopy of Walks and Spherical Maps in Homotopy Type Theory},
booktitle = {Proc.\ CPP},
publisher = {ACM},
pages = {338--351},
doi = {10.1145/3497775.3503671},
year = {2022},
}
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