Donkó, István

Kaposi, Ambrus

Tóth, Melinda

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13th Joint Conference on Mathematics and Computer Science, October 1 – 3, 2020, ELTE, Hungary 45Formalizing a relational model of concurrent programs in adependently typed environment4István Donkó†, Ambrus Kaposi‡, Melinda Tóth§ELTE, Eötvös Loránd UniversityFaculty of InformaticsDepartment of Programming Languages and Compilers†isti115@inf.elte.hu ‡akaposi@inf.elte.hu §tothmelinda@elte.huSequential programming languages have already been formalized in dependently typedprogramming languages, such as for example Agda or Coq, but the computer based formalizationand verification of concurrent programs is still in its early days. The goal of our research is toformalize a relational model [4] that describes the behavior of distributed concurrent programsin a theorem prover system. Our long term goals include the formalization of the material of thesubject titled ”Specification and Implementation of Distributed Systems”, which serves as a corepart in the Computer Science education at Eötvös Loránd University in Budapest.Since our goal is not the introduction of a new way to address the problem of creatingcorrectness proofs for parallel programs, rather the adaptation of an existing system for formalizedimplementation, instead of discussing the theoretical methods, here we focus on a more practicalapproach and explore what others have achieved in the field of computer based verification.Motivation Software plays a critically important role in the life of modern societies. More or lesseverybody interacts with computer programs on countless occasions during our everyday lives, mostof the time probably not even noticing. For example, just paying with a credit card while shopping,being able to call someone with our mobile phones, or even as mundane tasks, as operating modernhome appliances, like washing machines or microwave ovens requires interaction with software.Problems occurring in these example situations mostly just cause inconveniences, but if we takethe more critical scenarios into account, such as for example software running on an airplane, orkeeping a nuclear power plant safe, we can see that programmers have an even bigger impact.Borrowing from Robert C. Martin we can define the beginning of programming aroundthe work of Alan Turing, since he was the first one, who wrote code for machines in the sense thatwe would recognize today. In his time, he described the future possibilities of his vision with thefollowing sentences in a lecture to the London Mathematical Society [6].“In order to supply the machine with these problems we shall need a great number ofmathematicians of ability. These mathematicians will be needed in order to do the preliminaryresearch on the problems, putting them into a form for computation.”He stated the need for mathematicians for the precise formalization of problems. As wecan see today, his approach is necessary for building critical systems, the correctness of which candecide between life and death, we need to have formal strategies to verify behaviors of programsunder all circumstances instead of just observing them for the most likely situations.Introduction There are lots of existing means for confirming the adherence of simple sequentialprograms to their specifications, ranging from formal verification procedures carried out on paperto contracts built into programming languages, that can be checked and enforced automatically,either via static code analysis, or during runtime by monitoring different values. Reasoning about4The research has been supported by the European Union, co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16- 2017-00002). The research is part of the ”Application Domain Specific Highly Reliable IT Solutions”project that has been implemented with the support provided from the National Research, Development and Inno-vation Fund of Hungary, financed under the Thematic Excellence Programme no. 2020-4.1.1.-TKP2020 (NationalChallenges Subprogramme) funding scheme.46 13th Joint Conference on Mathematics and Computer Science, October 1 – 3, 2020, ELTE, Hungaryparallel programs is a lot more complicated, but this complication also serves as an explanationfor the need to do so, since concurrency is often a result of multiple systems working together,in which case it is a lot easier to make mistakes because of the unpredictable order of executionof instructions. Several different ways are known to approach formal proofs of correctness forconcurrent programs. For example multiple specific methods can be seen in [2]. What we choseto base our research on is the material of the subject titled ”Specification and Implementation ofDistributed Systems”.Our formalization does not follow the material exactly, but we tried to stay as close tothe original notation as possible, as we also have intentions to later further expand this projectto cater for educational usage for example as part of the practical courses. We consider this to bea valuable opportunity for creating a teaching tool that can greatly aid the understanding of thesubject for students. The use of proof assistants has already been successfully introduced in severalother classes [5], which helps making this idea seem quite feasible.Background We used type theory (namely, the Agda [1] implementation of it) as the main toolfor creating and verifying our formalization, which – by being an expressive alternative foundationfor mathematics – enables the formalization of constructive proofs through the connections tointuitionistic logic given by the Brouwer–Heyting–Kolmogorov interpretation. After formalizing amodel by defining its types and their elements, one can express statements and theorems in formsof new types, the instances of which can be thought of as proofs for them. This is due to the socalled ”propositions-as-types” paradigm, formally known as the Curry–Howard isomorphism.Results The main results of our work include the fully formalized version of a big core partfrom the original model [3] we built our research around, which is precise enough for computerbased typechecking. This consists of a definition for a language which contains parallel conditionalassignments as well as several types of predicates and statements that can be used to describe spec-ifications for programs written in this language. On top of these foundational constructs we havedeveloped proofs for several generic lemmas and some bigger theorems. We have also formalized aparallelized version of the bubble sort algorithm and the verified some of its properties.References[1] Agda Development Team. Agda - https://github.com/agda/agda. Version 2.6.1. url: https://github.com/agda/agda.[2] Robin Candy. “Towards Concurrent Hoare Logic”. In: (Nov. 2012). url: https://ir.canterbury.ac.nz/handle/10092/14861.[3] István Donkó. Formalizing a relational model of concurrent programs in a dependently typedenvironment. Paper at the Student Association Conference, Faculty of Informatics, EötvösLoránd University, May 2020, Received 1st prize.[4] Ákos Fóthi and Zoltán Horváth. Párhuzamos és elosztott programozás (English: Parallel andDistributed Programming). ELTE, 2005.[5] Dániel Horpácsi et al. “Interactive Teaching of Programming Language Theory with a ProofAssistant”. In: Central-European Journal of New Technologies in Research, Education andPractice (2020).[6] Alan M Turing. “Lecture to the London Mathematical Society on 20 February 1947”. In: ().

Formalizing a relational model of concurrent programs in a dependently typed environment

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Formalizing a relational model of concurrent programs in a dependently typed environment

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