Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online

Reducing the Number of Annotations in a Verification-oriented Imperative Language

Automated software verification is a very active field of research which has made enormous progress both in theoretical and practical aspects. Recently, an important amount of research effort has been put into applying these techniques on top of mainstream programming languages. These languages typically provide powerful features such as reflection, aliasing and polymorphism which are handy for practitioners but, in contrast, make verification a real challenge. In this work we present Pest, a simple experimental, while-style, multiprocedural, imperative programming language which was conceived with verifiability as one of its main goals. This language forces developers to concurrently think about both the statements needed to implement an algorithm and the assertions required to prove its correctness. In order to aid programmers, we propose several techniques to reduce the number and complexity of annotations required to successfully verify their programs. In particular, we show that high-level iteration constructs may alleviate the need for providing complex loop annotations.Comment: 15 pages, 8 figure

Applying Formal Methods to Networking: Theory, Techniques and Applications

Despite its great importance, modern network infrastructure is remarkable for the lack of rigor in its engineering. The Internet which began as a research experiment was never designed to handle the users and applications it hosts today. The lack of formalization of the Internet architecture meant limited abstractions and modularity, especially for the control and management planes, thus requiring for every new need a new protocol built from scratch. This led to an unwieldy ossified Internet architecture resistant to any attempts at formal verification, and an Internet culture where expediency and pragmatism are favored over formal correctness. Fortunately, recent work in the space of clean slate Internet design---especially, the software defined networking (SDN) paradigm---offers the Internet community another chance to develop the right kind of architecture and abstractions. This has also led to a great resurgence in interest of applying formal methods to specification, verification, and synthesis of networking protocols and applications. In this paper, we present a self-contained tutorial of the formidable amount of work that has been done in formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial

Formally Verified Verifiable Electronic Voting Scheme

Since the introduction of secret ballots in Victoria, Australia in 1855, paper (ballots) are widely used around the world to record the preferences of eligible voters. Paper ballots provide three important ingredients: correctness, privacy, and verifiability. However, the paper ballot election brings various other challenges, e.g. it is slow for large democracies like India, error prone for complex voting method like single transferable vote, and poses operational challenges for large countries like Australia. In order to solve these problems and various others, many countries are adopting electronic voting. However, electronic voting has a whole new set of problems. In most cases, the software programs used to conduct the election have numerous problems, including, but not limited to, counting bugs, ballot identification, etc. Moreover, these software programs are treated as commercial in confidence and are not allowed to be inspected by members of the public. As a consequence, the result produced by these software programs can not be substantiated. In this thesis, we address the three main concerns posed by electronic voting, i.e. correctness, privacy, and verifiability. We address the correctness concern by using theorem prover to implement the vote counting algorithm, privacy concern by using cryptography, and verifiability concern by generating a independently checkable scrutiny sheet (certificate). Our work has been carried out in the Coq theorem prover

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers

Principles of Security and Trust

This open access book constitutes the proceedings of the 8th International Conference on Principles of Security and Trust, POST 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 10 papers presented in this volume were carefully reviewed and selected from 27 submissions. They deal with theoretical and foundational aspects of security and trust, including on new theoretical results, practical applications of existing foundational ideas, and innovative approaches stimulated by pressing practical problems

Die Integration von Verifikation und Test in Übersetzungssysteme

In dieser Arbeit wird die Architektur für einen Compiler vorgestellt, der die Korrektheit der übersetzten Quellen als Teil des Übersetzungsvorgangs überprüfen kann. Dabei soll es möglich sein, verschiedene Methoden, wie etwa formaler Test und formaler Beweis, einzusetzen, um die Korrektheit nachzuweisen. Ein vollautomatischer Nachweis ist sehr aufwendig und häufig auch gar nicht möglich. Es ist also nicht praktikabel, aus Spezifikation und Programm die Korrektheit automatisch abzuleiten. Wir erweitern daher die Sprache um Korrektheitsnachweise (justifications), die der Benutzer in den Quelltext einfügen muß ("literate justification"). Je nach gewählter Methode muß der Benutzer den Korrektheitsnachweis mehr oder weniger genau ausführen. Durch die Einführung der Korrektheitsnachweise muß der Übersetzer Beweise nur noch überprüfen anstatt sie automatisch abzuleiten. Die Überprüfung der Korrektheitsnachweise kann in den Übersetzer integriert werden oder an ein externes Werkzeug delegiert werden. Ein externes Werkzeug erlaubt die Einbindung bereits existierender Werkzeuge, aber auch eine Neuentwicklung eigener Werkzeuge ist möglich. Wir zeigen am Beispiel eines taktischen Theorembeweisers, daß eine Eigenentwicklung nicht unbedingt aufwendiger ist als die Anpassung eines vorhandenen Werkzeugs. Um Tests während der Übersetzung durchführen zu können, muß ein Interpreter zur Verfügung stehen. Die Ausführung ungetesteten Codes birgt allerdings auch Sicherheitsprobleme. Wir diskutieren verschiedene Möglichkeiten, mit diesem Problem umzugehen. Der Korrektheit einer Übersetzungseinheit entspricht in der Semantik die Konsistenz einer algebraischen Spezifikation. Wir betrachten zwei Beweismethoden: zum einen durch Konstruktion eines Modells und zum andern durch Nachweis einer korrektheitserhaltenden Relation. Die Beweisverpflichtungen ergeben sich zunächst aus der Beweismethode, außerdem werden Beweisverpflichtungen eingeführt, um die Korrektheit von zusammengesetzten (modularen) Programmen zuzusichern. Die in dieser Arbeit beschriebene Architektur ist prototypisch implementiert worden. Dazu wurde das Opal-System um Elemente zur Spezifikation und zur Beschreibung von Korektheitsnachweisen erweitert. In der Arbeit werden einige kurze Beispiele vorgeführt. Der Opal/J-Prototyp ist seit Version 2.3e Teil der Opal-Distribution.In this thesis we present a compiler architecture that enables the compiler to check the correctness of the source code as part of the compilation process. It allows to perform these correctness checks with different methods, in particular formal testing and formal proof. A fully automated check is very expensive and often impossible. Hence, it is not feasible to check correctness automatically with the help of specification and implementation. We extend the programming language by (correctness) justifications that the user must insert into the source code ("literate justification"). Depending on the chosen justification method the user must work out the justification in more or less detail. The introduction of justifications changes the compiler's task from deriving a correctness proof by itself to checking a correctness proof provided by the user. The correctness check for justifications can be integrated into the compiler or delegated to an external tool. An external tool allows the integration of existing tools but the development of specialized tools is also possible. The example development of a specialized tactical theorem-prover shows that the development of a specialized tool is not necessarily more expensive than the adaptation of an existing tool. For test execution during the compilation an interpreter must be available. The execution of untested code causes security risks. We discuss different possibilities to deal with this problem. The correctnessof a compilation unit corresponds to the consistency of an algebraic specification. We study two proof methods: either by construction of a model or by establishing a correctness-preserving relation. The proofo bligations result from the proof method, in addition proof obligations arei ntroduced to ensure the correctness of modular programs. The compiler architecture described in this thesis has been prototypically implemented. The Opal system has been extended with language elements to denote specifications and (correctness) justifications. The thesis presents some short examples. The Opal/J prototype is part of the Opal distribution since version 2.3e