research-article

Open access

Modular Denotational Semantics for Effects with Guarded Interaction Trees

Authors:

Lars BirkedalAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 8, Issue POPL

Article No.: 12, Pages 332 - 361

https://doi.org/10.1145/3632854

Published: 05 January 2024 Publication History

Abstract

We present guarded interaction trees — a structure and a fully formalized framework for representing higher-order computations with higher-order effects in Coq, inspired by domain theory and the recently proposed interaction trees. We also present an accompanying separation logic for reasoning about guarded interaction trees. To demonstrate that guarded interaction trees provide a convenient domain for interpreting higher-order languages with effects, we define an interpretation of a PCF-like language with effects and show that this interpretation is sound and computationally adequate; we prove the latter using a logical relation defined using the separation logic. Guarded interaction trees also allow us to combine different effects and reason about them modularly. To illustrate this point, we give a modular proof of type soundness of cross-language interactions for safe interoperability of different higher-order languages with different effects. All results in the paper are formalized in Coq using the Iris logic over guarded type theory.

References

[1]

Casper Bach Poulsen and Cas van der Rest. 2023. Hefty Algebras: Modular Elaboration of Higher-Order Algebraic Effects. Proceedings of the ACM on Programming Languages, 7, POPL (2023), Jan., 62:1801–62:1831. https://doi.org/10.1145/3571255

[2]

Andrej Bauer and Matija Pretnar. 2015. Programming with Algebraic Effects and Handlers. Journal of Logical and Algebraic Methods in Programming, 84, 1 (2015), Jan., 108–123. issn:2352-2208 https://doi.org/10.1016/j.jlamp.2014.02.001

[3]

Lars Birkedal and Rasmus Ejlers Møgelberg. 2013. Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes. In 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, New Orleans, LA, USA, June 25-28, 2013. IEEE Computer Society, 213–222. https://doi.org/10.1109/LICS.2013.27

[4]

Lars Birkedal, Rasmus Ejlers Møgelberg, Jan Schwinghammer, and Kristian Støvring. 2012. First steps in synthetic guarded domain theory: step-indexing in the topos of trees. Log. Methods Comput. Sci., 8, 4 (2012), https://doi.org/10.2168/LMCS-8(4:1)2012

[5]

Lars Birkedal, Bernhard Reus, Jan Schwinghammer, and Hongseok Yang. 2008. A Simple Model of Separation Logic for Higher-Order Store. In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz (Eds.) (Lecture Notes in Computer Science, Vol. 5126). Springer, 348–360. https://doi.org/10.1007/978-3-540-70583-3_29

[6]

Lars Birkedal, Kristian Støvring, and Jacob Thamsborg. 2010. The Category-Theoretic Solution of Recursive Metric-Space Equations. Theoretical Computer Science, 411, 47 (2010), Oct., 4102–4122. issn:03043975 https://doi.org/10.1016/j.tcs.2010.07.010

[7]

Lars Birkedal and Hongseok Yang. 2008. Relational Parametricity and Separation Logic. Log. Methods Comput. Sci., 4, 2 (2008), https://doi.org/10.2168/LMCS-4(2:6)2008

[8]

Ales Bizjak, Hans Bugge Grathwohl, Ranald Clouston, Rasmus Ejlers Møgelberg, and Lars Birkedal. 2016. Guarded Dependent Type Theory with Coinductive Types. CoRR, abs/1601.01586 (2016), arXiv:1601.01586. arxiv:1601.01586

[9]

Paulo Emílio de Vilhena and François Pottier. 2021. A Separation Logic for Effect Handlers. Proceedings of the ACM on Programming Languages, 5, POPL (2021), Jan., 33:1–33:28. https://doi.org/10.1145/3434314

[10]

Matthias Felleisen and Robert Hieb. 1992. The revised report on the syntactic theories of sequential control and state. Theoretical Computer Science, 103, 2 (1992), 235–271. issn:0304-3975 https://doi.org/10.1016/0304-3975(92)90014-7

[11]

Peter Freyd. 1991. Algebraically Complete Categories. In Category Theory, Aurelio Carboni, Maria Cristina Pedicchio, and Guiseppe Rosolini (Eds.) (Lecture Notes in Mathematics). Springer, Berlin, Heidelberg. 95–104. isbn:978-3-540-46435-8 https://doi.org/10.1007/BFb0084215

[12]

Naohiko Hoshino. 2012. Step Indexed Realizability Semantics for a Call-by-Value Language Based on Basic Combinatorial Objects. In 2012 27th Annual IEEE Symposium on Logic in Computer Science. 385–394. issn:1043-6871 https://doi.org/10.1109/LICS.2012.74

[13]

Patricia Johann, Alex Simpson, and Janis Voigtländer. 2010. A Generic Operational Metatheory for Algebraic Effects. In Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, United Kingdom. IEEE Computer Society, 209–218. https://doi.org/10.1109/LICS.2010.29

[14]

Ralf Jung, Robbert Krebbers, Lars Birkedal, and Derek Dreyer. 2016. Higher-order ghost state. In Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming, ICFP 2016, Nara, Japan, September 18-22, 2016, Jacques Garrigue, Gabriele Keller, and Eijiro Sumii (Eds.). ACM, 256–269. https://doi.org/10.1145/2951913.2951943

[15]

Ralf Jung, Robbert Krebbers, Jacques-Henri Jourdan, Ales Bizjak, Lars Birkedal, and Derek Dreyer. 2018. Iris from the ground up: A modular foundation for higher-order concurrent separation logic. J. Funct. Program., 28 (2018), e20. https://doi.org/10.1017/S0956796818000151

[16]

Ralf Jung, David Swasey, Filip Sieczkowski, Kasper Svendsen, Aaron Turon, Lars Birkedal, and Derek Dreyer. 2015. Iris: Monoids and Invariants as an Orthogonal Basis for Concurrent Reasoning. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2015, Mumbai, India, January 15-17, 2015, Sriram K. Rajamani and David Walker (Eds.). ACM, 637–650. https://doi.org/10.1145/2676726.2676980

[17]

Nicolas Koh, Yao Li, Yishuai Li, Li-yao Xia, Lennart Beringer, Wolf Honoré, William Mansky, Benjamin C. Pierce, and Steve Zdancewic. 2019. From C to Interaction Trees: Specifying, Verifying, and Testing a Networked Server. In Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2019). Association for Computing Machinery, New York, NY, USA. 234–248. isbn:978-1-4503-6222-1 https://doi.org/10.1145/3293880.3294106

[18]

Robbert Krebbers, Ralf Jung, Ales Bizjak, Jacques-Henri Jourdan, Derek Dreyer, and Lars Birkedal. 2017. The Essence of Higher-Order Concurrent Separation Logic. In Programming Languages and Systems - 26th European Symposium on Programming, ESOP 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017, Uppsala, Sweden, April 22-29, 2017, Proceedings, Hongseok Yang (Ed.) (Lecture Notes in Computer Science, Vol. 10201). Springer, 696–723. https://doi.org/10.1007/978-3-662-54434-1_26

[19]

Robbert Krebbers, Amin Timany, and Lars Birkedal. 2017. Interactive proofs in higher-order concurrent separation logic. In Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages, POPL 2017, Paris, France, January 18-20, 2017, Giuseppe Castagna and Andrew D. Gordon (Eds.). ACM, 205–217. https://doi.org/10.1145/3009837.3009855

[20]

Mohsen Lesani, Li-yao Xia, Anders Kaseorg, Christian J. Bell, Adam Chlipala, Benjamin C. Pierce, and Steve Zdancewic. 2022. C4: Verified Transactional Objects. Proceedings of the ACM on Programming Languages, 6, OOPSLA1 (2022), April, 80:1–80:31. https://doi.org/10.1145/3527324

[21]

Jacob Matthews and Robert Bruce Findler. 2007. Operational Semantics for Multi-Language Programs. ACM SIGPLAN Notices, 42, 1 (2007), Jan., 3–10. issn:0362-1340 https://doi.org/10.1145/1190215.1190220

[22]

Rasmus E. Møgelberg and Marco Paviotti. 2019. Denotational Semantics of Recursive Types in Synthetic Guarded Domain Theory. Mathematical Structures in Computer Science, 29, 3 (2019), March, 465–510. issn:0960-1295, 1469-8072

[23]

Rasmus Ejlers Møgelberg and Andrea Vezzosi. 2021. Two Guarded Recursive Powerdomains for Applicative Simulation. Electronic Proceedings in Theoretical Computer Science, 351 (2021), Dec., 200–217. issn:2075-2180 arxiv:2112.14056.

[24]

Daniel Patterson, Noble Mushtak, Andrew Wagner, and Amal Ahmed. 2022. Semantic Soundness for Language Interoperability. In Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation (PLDI 2022). Association for Computing Machinery, New York, NY, USA. 609–624. isbn:978-1-4503-9265-5 https://doi.org/10.1145/3519939.3523703

[25]

Marco Paviotti, Rasmus Ejlers Møgelberg, and Lars Birkedal. 2015. A Model of PCF in Guarded Type Theory. Electronic Notes in Theoretical Computer Science, 319 (2015), Dec., 333–349. issn:1571-0661

[26]

Andrew M. Pitts. 1996. Relational Properties of Domains. Information and Computation, 127, 2 (1996), June, 66–90. issn:0890-5401 https://doi.org/10.1006/inco.1996.0052

[27]

G. D. Plotkin. 1977. LCF Considered as a Programming Language. Theoretical Computer Science, 5, 3 (1977), Dec., 223–255. issn:0304-3975 https://doi.org/10.1016/0304-3975(77)90044-5

[28]

Gordon D. Plotkin and Matija Pretnar. 2013. Handling Algebraic Effects. Logical Methods in Computer Science, Volume 9, Issue 4 (2013), Dec., issn:1860-5974 https://doi.org/10.2168/LMCS-9(4:23)2013

[29]

Dana Scott. 1976. Data Types as Lattices. SIAM J. Comput., 5, 3 (1976), Sept., 522–587. issn:0097-5397 https://doi.org/10.1137/0205037

[30]

Lucas Silver, Paul He, Ethan Cecchetti, Andrew K Hirsch, and Steve Zdancewic. 2023. Semantics for Noninterference with Interaction Trees.

[31]

M. B. Smyth and G. D. Plotkin. 1982. The Category-Theoretic Solution of Recursive Domain Equations. SIAM J. Comput., 11, 4 (1982), Nov., 761–783. issn:0097-5397 https://doi.org/10.1137/0211062

[32]

Birthe van den Berg, Tom Schrijvers, Casper Bach Poulsen, and Nicolas Wu. 2021. Latent Effects for Reusable Language Components. In Programming Languages and Systems, Hakjoo Oh (Ed.) (Lecture Notes in Computer Science). Springer International Publishing, Cham. 182–201. isbn:978-3-030-89051-3 https://doi.org/10.1007/978-3-030-89051-3_11

[33]

Nicolas Wu, Tom Schrijvers, and Ralf Hinze. 2014. Effect Handlers in Scope. In Proceedings of the 2014 ACM SIGPLAN Symposium on Haskell (Haskell ’14). Association for Computing Machinery, New York, NY, USA. 1–12. isbn:978-1-4503-3041-1 https://doi.org/10.1145/2633357.2633358

[34]

Li-yao Xia, Yannick Zakowski, Paul He, Chung-Kil Hur, Gregory Malecha, Benjamin C. Pierce, and Steve Zdancewic. 2019. Interaction Trees: Representing Recursive and Impure Programs in Coq. Proceedings of the ACM on Programming Languages, 4, POPL (2019), Dec., 51:1–51:32. https://doi.org/10.1145/3371119

[35]

Zhixuan Yang and Nicolas Wu. 2021. Reasoning about Effect Interaction by Fusion. Proceedings of the ACM on Programming Languages, 5, ICFP (2021), Aug., 73:1–73:29. https://doi.org/10.1145/3473578

[36]

Kangfeng Ye, Simon Foster, and Jim Woodcock. 2022. Formally Verified Animation for RoboChart Using Interaction Trees. In Formal Methods and Software Engineering, Adrian Riesco and Min Zhang (Eds.) (Lecture Notes in Computer Science). Springer International Publishing, Cham. 404–420. isbn:978-3-031-17244-1 https://doi.org/10.1007/978-3-031-17244-1_24

[37]

Yannick Zakowski, Calvin Beck, Irene Yoon, Ilia Zaichuk, Vadim Zaliva, and Steve Zdancewic. 2021. Modular, Compositional, and Executable Formal Semantics for LLVM IR. Proceedings of the ACM on Programming Languages, 5, ICFP (2021), Aug., 67:1–67:30. https://doi.org/10.1145/3473572

[38]

Hengchu Zhang, Wolf Honoré, Nicolas Koh, Yao Li, Yishuai Li, Li-Yao Xia, Lennart Beringer, William Mansky, Benjamin Pierce, and Steve Zdancewic. 2021. Verifying an HTTP Key-Value Server with Interaction Trees and VST. In 12th International Conference on Interactive Theorem Proving (ITP 2021), Liron Cohen and Cezary Kaliszyk (Eds.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 193). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany. 32:1–32:19. isbn:978-3-95977-188-7 issn:1868-8969 https://doi.org/10.4230/LIPIcs.ITP.2021.32

Cited By

Stassen PMøgelberg RZwart MAguirre ABirkedal L(2025)Modelling Recursion and Probabilistic Choice in Guarded Type TheoryProceedings of the ACM on Programming Languages10.1145/37048849:POPL(1417-1445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704884
Giovannini EDing TNew M(2025)Denotational Semantics of Gradual Typing using Synthetic Guarded Domain TheoryProceedings of the ACM on Programming Languages10.1145/37048639:POPL(772-801)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704863
Vistrup MSammler MJung R(2025)Program Logics à la CarteProceedings of the ACM on Programming Languages10.1145/37048479:POPL(300-331)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704847
Show More Cited By

Index Terms

Modular Denotational Semantics for Effects with Guarded Interaction Trees
1. Software and its engineering
  1. Software notations and tools
    1. Software libraries and repositories
2. Theory of computation
  1. Logic
    1. Logic and verification
  2. Semantics and reasoning
    1. Program semantics

Recommendations

Call-by-push-value in Coq: operational, equational, and denotational theory
CPP 2019: Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs

Call-by-push-value (CBPV) is an idealised calculus for functional and imperative programming, introduced as a subsuming paradigm for both call-by-value (CBV) and call-by-name (CBN). We formalise weak and strong operational semantics for (effect-free) ...
Interactive proofs in higher-order concurrent separation logic
POPL '17

When using a proof assistant to reason in an embedded logic -- like separation logic -- one cannot benefit from the proof contexts and basic tactics of the proof assistant. This results in proofs that are at a too low level of abstraction because they ...
Interactive proofs in higher-order concurrent separation logic
POPL '17: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages

When using a proof assistant to reason in an embedded logic -- like separation logic -- one cannot benefit from the proof contexts and basic tactics of the proof assistant. This results in proofs that are at a too low level of abstraction because they ...

Comments

Information & Contributors

Information

Published In

cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 8, Issue POPL

January 2024

2820 pages

EISSN:2475-1421

DOI:10.1145/3554315

Editor:
Michael Hicks
Amazon, USA

Issue’s Table of Contents

Copyright © 2024 Owner/Author.

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 January 2024

Published in PACMPL Volume 8, Issue POPL

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article

Funding Sources

Villum Fonden

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
516
Total Downloads

Downloads (Last 12 months)448
Downloads (Last 6 weeks)57

Reflects downloads up to 02 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Stassen PMøgelberg RZwart MAguirre ABirkedal L(2025)Modelling Recursion and Probabilistic Choice in Guarded Type TheoryProceedings of the ACM on Programming Languages10.1145/37048849:POPL(1417-1445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704884
Giovannini EDing TNew M(2025)Denotational Semantics of Gradual Typing using Synthetic Guarded Domain TheoryProceedings of the ACM on Programming Languages10.1145/37048639:POPL(772-801)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704863
Vistrup MSammler MJung R(2025)Program Logics à la CarteProceedings of the ACM on Programming Languages10.1145/37048479:POPL(300-331)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704847
Zhang CTang XPeng Y(2024)Enhancing Regular Expression Processing through Field-Programmable Gate Array-Based Multi-Character Non-Deterministic Finite AutomataElectronics10.3390/electronics1309163513:9(1635)Online publication date: 24-Apr-2024
https://doi.org/10.3390/electronics13091635
Timany AKrebbers RDreyer DBirkedal L(2024)A Logical Approach to Type SoundnessJournal of the ACM10.1145/367695471:6(1-75)Online publication date: 11-Nov-2024
https://dl.acm.org/doi/10.1145/3676954

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Figures

Tables

Media

View Issue’s Table of Contents