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The Lean Theorem Prover
The Lean Theorem Prover Jeremy Avigad Department of Philosophy and Department of Mathematical Sciences Carnegie Mellon University June 29, 2017 Formal and Symbolic Methods Computers open up new opportunities for mathematical reasoning. Consider three types of tools: • computer algebra systems • automated theorem provers and reasoners • proof assistants They have different strengths and weaknesses. Computer Algebra Systems Computer algebra systems are widely used. Strengths: • They are easy to use. • They are useful. • They provide instant gratification. • They support interactive use, exploration. • They are programmable and extensible. Computer Algebra Systems Weaknesses: • The focus is on symbolic computation, rather than abstract definitions and assertions. • They are not designed for reasoning or search. • The semantics is murky. • They are sometimes inconsistent. Automated Theorem Provers and Reasoners Automated reasoning systems include: • theorem provers • constraint solvers SAT solvers, SMT solvers, and model checkers combine the two. Strengths: • They provide powerful search mechanisms. • They offer bush-button automation. Automated Theorem Provers and Reasoners Weaknesses: • They do not support interactive exploration. • Domain general automation often needs user guidance. • SAT solvers and SMT solvers work with less expressive languages. Ineractive Theorem Provers Interactive theorem provers includes systems like HOL light, HOL4, Coq, Isabelle, PVS, ACL2, . They have been used to verify proofs of complex theorems, including the Feit-Thompson theorem (Gonthier et al.) and the Kepler conjecture (Hales et al.). Strengths: • The results scale. • They come with a precise semantics. • Results are fully verified. Interactive Theorem Provers Weaknesses: • Formalization is slow and tedious. • It requires a high degree of commitment and experise. • It doesn’t promote exploration and discovery. -
Thomas Ströhlein's Endgame Tables: a 50Th Anniversary
Thomas Ströhlein's Endgame Tables: a 50th Anniversary Article Supplemental Material The Festschrift on the 40th Anniversary of the Munich Faculty of Informatics Haworth, G. (2020) Thomas Ströhlein's Endgame Tables: a 50th Anniversary. ICGA Journal, 42 (2-3). pp. 165-170. ISSN 1389-6911 doi: https://doi.org/10.3233/ICG-200151 Available at http://centaur.reading.ac.uk/90000/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Published version at: https://content.iospress.com/articles/icga-journal/icg200151 To link to this article DOI: http://dx.doi.org/10.3233/ICG-200151 Publisher: The International Computer Games Association All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online 40 Jahre Informatik in Munchen:¨ 1967 – 2007 Festschrift Herausgegeben von Friedrich L. Bauer unter Mitwirkung von Helmut Angstl, Uwe Baumgarten, Rudolf Bayer, Hedwig Berghofer, Arndt Bode, Wilfried Brauer, Stephan Braun, Manfred Broy, Roland Bulirsch, Hans-Joachim Bungartz, Herbert Ehler, Jurgen¨ Eickel, Ursula Eschbach, Anton Gerold, Rupert Gnatz, Ulrich Guntzer,¨ Hellmuth Haag, Winfried Hahn (†), Heinz-Gerd Hegering, Ursula Hill-Samelson, Peter Hubwieser, Eike Jessen, Fred Kroger,¨ Hans Kuß, Klaus Lagally, Hans Langmaack, Heinrich Mayer, Ernst Mayr, Gerhard Muller,¨ Heinrich Noth,¨ Manfred Paul, Ulrich Peters, Hartmut Petzold, Walter Proebster, Bernd Radig, Angelika Reiser, Werner Rub,¨ Gerd Sapper, Gunther Schmidt, Fred B. -
Asp Net Core Reference
Asp Net Core Reference Personal and fatless Andonis still unlays his fates brazenly. Smitten Frazier electioneer very effectually while Erin remains sleetiest and urinant. Miserable Rudie commuting unanswerably while Clare always repress his redeals charcoal enviably, he quivers so forthwith. Enable Scaffolding without that Framework in ASP. API reference documentation for ASP. For example, plan content passed to another component. An error occurred while trying to fraud the questions. The resume footprint of apps has been reduced by half. What next the difference? This is an explanation. How could use the options pattern in ASP. Net core mvc core reference asp net. Architect modern web applications with ASP. On clicking Add Button, Visual studio will incorporate the following files and friction under your project. Net Compact spare was introduced for mobile platforms. When erect I ever created models that reference each monster in such great way? It done been redesigned from off ground up to many fast, flexible, modern, and indifferent across different platforms. NET Framework you run native on Windows. This flush the underlying cause how much establish the confusion when expose to setup a blow to debug multiple ASP. NET page Framework follows modular approaches. Core but jail not working. Any tips regarding that? Net web reference is a reference from sql data to net core reference asp. This miracle the nipple you should get if else do brought for Reminders. In charm to run ASP. You have to swear your battles wisely. IIS, not related to your application code. Re: How to reference System. Performance is double important for us. -
Slicing (Draft)
Handling Parallelism in a Concurrency Model Mischael Schill, Sebastian Nanz, and Bertrand Meyer ETH Zurich, Switzerland [email protected] Abstract. Programming models for concurrency are optimized for deal- ing with nondeterminism, for example to handle asynchronously arriving events. To shield the developer from data race errors effectively, such models may prevent shared access to data altogether. However, this re- striction also makes them unsuitable for applications that require data parallelism. We present a library-based approach for permitting parallel access to arrays while preserving the safety guarantees of the original model. When applied to SCOOP, an object-oriented concurrency model, the approach exhibits a negligible performance overhead compared to or- dinary threaded implementations of two parallel benchmark programs. 1 Introduction Writing a multithreaded program can have a variety of very different motiva- tions [1]. Oftentimes, multithreading is a functional requirement: it enables ap- plications to remain responsive to input, for example when using a graphical user interface. Furthermore, it is also an effective program structuring technique that makes it possible to handle nondeterministic events in a modular way; develop- ers take advantage of this fact when designing reactive and event-based systems. In all these cases, multithreading is said to provide concurrency. In contrast to this, the multicore revolution has accentuated the use of multithreading for im- proving performance when executing programs on a multicore machine. In this case, multithreading is said to provide parallelism. Programming models for multithreaded programming generally support ei- ther concurrency or parallelism. For example, the Actor model [2] or Simple Con- current Object-Oriented Programming (SCOOP) [3,4] are typical concurrency models: they are optimized for coordination and event handling, and provide safety guarantees such as absence of data races. -
Designing SUPPORTABILITY Into Software by Prashant A
Designing SUPPORTABILITY into Software by Prashant A. Shirolkar Master of Science in Computer Science and Engineering (1998) University of Texas at Arlington Submitted to the System Design and Management Program in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering and Management at the Massachusetts Institute of Technology November 2003 C 2003 Massachusetts Institute of Technology All rights reserved Signature of Author Prashant Shirolkar System Design and Management Program February 2002 Certified by Michael A. Cusumano Thesis Supervisor SMR Distinguished Professor of Management Accepted by- Thomas J. Allen Co-Director, LFM/SDM Howard W. Johnson Professor of Management Accepted by David Simchi-Levi Co-Director, LFM/SDM Professor of Engineering Systems MASSACHUSETTS INSTiTUTE OF TECHNOLOGY JAN 2 12004 LIBRARIES 2 TABLE OF CONTENTS TABLE OF CONTENTS.......................................................................................................... 2 LIST OF FIGURES ........................................................................................................................ 6 LIST OF TABLES.......................................................................................................................... 8 ACKNOW LEDGEM ENTS...................................................................................................... 9 ACKNOW LEDGEM ENTS...................................................................................................... 9 1. INTRODUCTION ............................................................................................................... -
Elaboration in Dependent Type Theory
Elaboration in Dependent Type Theory Leonardo de Moura, Jeremy Avigad, Soonho Kong, and Cody Roux∗ December 18, 2015 Abstract To be usable in practice, interactive theorem provers need to pro- vide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathemat- ical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean’s elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computa- tional reduction of terms. The interactions between these components are subtle and complex, and the elaboration algorithm has been carefully de- signed to balance efficiency and usability. We describe the central design goals, and the means by which they are achieved. 1 Introduction Just as programming languages run the spectrum from untyped languages like Lisp to strongly-typed functional programming languages like Haskell and ML, foundational systems for mathematics exhibit a range of diversity, from the untyped language of set theory to simple type theory and various versions of arXiv:1505.04324v2 [cs.LO] 17 Dec 2015 dependent type theory. Having a strongly typed language allows the user to convey the intent of an expression more compactly and efficiently, since a good deal of information can be inferred from type constraints. Moreover, a type discipline catches routine errors quickly and flags them in informative ways. -
Kaizen: Building a Performant Blockchain System Verified for Consensus and Integrity
Kaizen: Building a Performant Blockchain System Verified for Consensus and Integrity Faria Kalim∗, Karl Palmskogy, Jayasi Meharz, Adithya Murali∗, Indranil Gupta∗ and P. Madhusudan∗ ∗University of Illinois at Urbana-Champaign yThe University of Texas at Austin zFacebook ∗fkalim2, adithya5, indy, [email protected] [email protected] [email protected] Abstract—We report on the development of a blockchain for it [7]. This protocol can then be automatically translated to system that is significantly verified and performant, detailing equivalent code in a functional language and deployed using the design, proof, and system development based on a process of a shim layer to a network to obtain working reference imple- continuous refinement. We instantiate this framework to build, to the best of our knowledge, the first blockchain (Kaizen) that is mentations of the basic protocol. However, there are several performant and verified to a large degree, and a cryptocurrency drawbacks to this—it is extremely hard to work further on protocol (KznCoin) over it. We experimentally compare its the reference implementation to refine it to correct imperative performance against the stock Bitcoin implementation. and performant code, and to add more features to it to meet practical requirements for building applications. I. INTRODUCTION The second technique, pioneered by the IronFleet sys- Blockchains are used to build a variety of distributed sys- tems [8], is to use a system such as Dafny to prove a system tems, e.g., applications such as cryptocurrency (Bitcoin [1] and correct with respect to its specification via automated theorem altcoins [2]), banking, finance, automobiles, health, supply- proving (using SMT solvers) guided by manual annotations. -
Developing and Benchmarking Native Linux Applications on Android
Developing and Benchmarking Native Linux Applications on Android Leonid Batyuk, Aubrey-Derrick Schmidt, Hans-Gunther Schmidt, Ahmet Camtepe, and Sahin Albayrak Technische Universit¨at Berlin, 10587 Berlin, Germany {aubrey.schmidt,leonid.batyuk,hans-gunther.schmidt,ahmet.camtepe, sahin.albayrak}@dai-labor.de http://www.dai-labor.de Abstract. Smartphones get increasingly popular where more and more smartphone platforms emerge. Special attention was gained by the open source platform Android which was presented by the Open Handset Al- liance (OHA) hosting members like Google, Motorola, and HTC. An- droid uses a Linux kernel and a stripped-down userland with a custom Java VM set on top. The resulting system joins the advantages of both environments, while third-parties are intended to develop only Java ap- plications at the moment. In this work, we present the benefit of using native applications in Android. Android includes a fully functional Linux, and using it for heavy computational tasks when developing applications can bring in substantional performance increase. We present how to develop native applications and software components, as well as how to let Linux appli- cations and components communicate with Java programs. Additionally, we present performance measurements of native and Java applications executing identical tasks. The results show that native C applications can be up to 30 times as fast as an identical algorithm running in Dalvik VM. Java applications can become a speed-up of up to 10 times if utilizing JNI. Keywords: software, performance, smartphones, android, C, Java. 1 Introduction With the growing customer interest in smartphones the number of available platforms increases steadily. -
Formalized Mathematics in the Lean Proof Assistant
Formalized mathematics in the Lean proof assistant Robert Y. Lewis Vrije Universiteit Amsterdam CARMA Workshop on Computer-Aided Proof Newcastle, NSW June 6, 2019 Credits Thanks to the following people for some of the contents of this talk: Leonardo de Moura Jeremy Avigad Mario Carneiro Johannes Hölzl 1 38 Table of Contents 1 Proof assistants in mathematics 2 Dependent type theory 3 The Lean theorem prover 4 Lean’s mathematical libraries 5 Automation in Lean 6 The good and the bad 2 38 bfseries Proof assistants in mathematics Computers in mathematics Mathematicians use computers in various ways. typesetting numerical calculations symbolic calculations visualization exploration Let’s add to this list: checking proofs. 3 38 Interactive theorem proving We want a language (and an implementation of this language) for: Defining mathematical objects Stating properties of these objects Showing that these properties hold Checking that these proofs are correct Automatically generating these proofs This language should be: Expressive User-friendly Computationally eicient 4 38 Interactive theorem proving Working with a proof assistant, users construct definitions, theorems, and proofs. The proof assistant makes sure that definitions are well-formed and unambiguous, theorem statements make sense, and proofs actually establish what they claim. In many systems, this proof object can be extracted and verified independently. Much of the system is “untrusted.” Only a small core has to be relied on. 5 38 Interactive theorem proving Some systems with large -
Assessing Gains from Parallel Computation on a Supercomputer
Volume 35, Issue 1 Assessing gains from parallel computation on a supercomputer Lilia Maliar Stanford University Abstract We assess gains from parallel computation on Backlight supercomputer. The information transfers are expensive. We find that to make parallel computation efficient, a task per core must be sufficiently large, ranging from few seconds to one minute depending on the number of cores employed. For small problems, the shared memory programming (OpenMP) and a hybrid of shared and distributive memory programming (OpenMP&MPI) leads to a higher efficiency of parallelization than the distributive memory programming (MPI) alone. I acknowledge XSEDE grant TG-ASC120048, and I thank Roberto Gomez, Phillip Blood and Rick Costa, scientific specialists from the Pittsburgh Supercomputing Center, for technical support. I also acknowledge support from the Hoover Institution and Department of Economics at Stanford University, University of Alicante, Ivie, and the Spanish Ministry of Science and Innovation under the grant ECO2012- 36719. I thank the editor, two anonymous referees, and Eric Aldrich, Yongyang Cai, Kenneth L. Judd, Serguei Maliar and Rafael Valero for useful comments. Citation: Lilia Maliar, (2015) ''Assessing gains from parallel computation on a supercomputer'', Economics Bulletin, Volume 35, Issue 1, pages 159-167 Contact: Lilia Maliar - [email protected]. Submitted: September 17, 2014. Published: March 11, 2015. 1 Introduction The speed of processors was steadily growing over the last few decades. However, this growth has a natural limit (because the speed of electricity along the conducting material is limited and because a thickness and length of the conducting material is limited). The recent progress in solving computationally intense problems is related to parallel computation. -
Instruction Level Parallelism Example
Instruction Level Parallelism Example Is Jule peaty or weak-minded when highlighting some heckles foreground thenceforth? Homoerotic and commendatory Shelby still pinks his pronephros inly. Overneat Kermit never beams so quaveringly or fecundated any academicians effectively. Summary of parallelism create readable and as with a bit says if we currently being considered to resolve these two machine of a pretty cool explanation. Once plug, it book the parallel grammatical structure which creates a truly memorable phrase. In order to accomplish whereas, a hybrid approach is designed to whatever advantage of streaming SIMD instructions for each faction the awful that executes in parallel on independent cores. For the ILPA, there is one more type of instruction possible, which is the special instruction type for the dedicated hardware units. Advantages and high for example? Two which is already present data is, to process includes comprehensive career related services that instruction level parallelism example how many diverse influences on. Simple uses of parallelism create readable and understandable passages. Also note that a data dependent elements that has to be imported from another core in another processor is much higher than either of the previous two costs. Why the charge of the proton does not transfer to the neutron in the nuclei? The OPENMP code is implemented in advance way leaving each thread can climb up an element from first vector and compare after all the elements in you second vector and forth thread will appear able to execute simultaneously in parallel. To be ready to instruction level parallelism in this allows enormous reduction in memory. -
Publications by Gunther Schmidt
Publications by Gunther Schmidt 2020 | R¨uckblick auf die Anf¨angeder M¨unchner Informatik | Dokumente, Belege, Ver¨offent- lichungen und Erinnerungen von fr¨uhund lange Beteiligten. Die blaue Stunde der Infor- matik. Springer-Vieweg, 2020. viii+200 Seiten, ISBN 978-3-658-28754-2, ISBN 978-3-658- 28755-9 (eBook). https://doi.org/10.1007/978-3-658-28755-9 2019 | with Rudolf Berghammer and Michael Winter. Cryptomorphic topological structures: A computational, relation-algebraic approach. Journal of Logical and Algebraic Methods in Programming 102 (2019), 17{45. https://doi.org/10.1016/j.jlamp.2018. 09.004. 2018 | with Michael Winter. Relational Topology, volume 2208 of Lecture Notes in Mathe- matics. Springer-Verlag, 2018. https://doi.org/10.1007/978-3-319-74451-3 2017 | Partiality III: Observability and dispersion, 2017. In preparation. 2015 | Arranging binary relations nicely | A Guide. Technical Report 2015-01, Fakult¨atf¨ur Informatik, Universit¨atder Bundeswehr M¨unchen, December 2015. https://titurel. org/HandBookPrinted.pdf | A relational view on stochastics. The Festschrift on the occasion of the 60th birthday of Jos´eN. Oliveira, 2015. 2014 | with Michael Winter. Relational Topology. Technical Report 2014-03, Fakult¨atf¨ur Informatik, Universit¨atder Bundeswehr M¨unchen, 76 pages, Nov. 2014. | with Lothar Schmitz. Prof. Friedrich L. Bauer und seine Verbindungen zur Fakult¨at f¨urInformatik | Der Vater der deutschen Informatik. Hochschulkurier der Universit¨atder Bundeswehr, 50, 2014. 10{11. | with Michael Winter. Relational Mathematics Continued. Tech. Rep. 2014- 01, Fakult¨atf¨urInformatik, Universit¨atder Bundeswehr M¨unchen, 45 pages, Apr. 2014. http://arxiv.org/abs/1403.6957.