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Ordering Strict Partial Orders to Model Behavioral Refinement 对严格偏序进行排序以模拟行为细化
Pub Date : 2018-10-23 DOI: 10.4204/EPTCS.282.3
Mathieu Montin, M. Pantel
Software is now ubiquitous and involved in complex interactions with the human users and the physical world in so-called cyber-physical systems where the management of time is a major issue. Separation of concerns is a key asset in the development of these ever more complex systems. Two different kinds of separation exist: a first one corresponds to the different steps in a development leading from the abstract requirements to the system implementation and is qualified as vertical. It matches the commonly used notion of refinement. A second one corresponds to the various components in the system architecture at a given level of refinement and is called horizontal. Refinement has been studied thoroughly for the data, functional and concurrency concerns while our work focuses on the time modeling concern. This contribution aims at providing a formal construct for the verification of refinement in time models, through the definition of an order between strict partial orders used to relate the different instants in asynchronous systems. This relation allows the designer at the concrete level to distinguish events that are coincident at the abstract level while preserving the properties assessed at the abstract level. This work has been conducted using the proof assistant Agda and is connected to a previous work on the asynchronous language CCSL, which has also been modelled using the same tool.
软件现在无处不在,并且在所谓的网络-物理系统中与人类用户和物理世界进行复杂的交互,其中时间管理是一个主要问题。在开发这些越来越复杂的系统时,关注点分离是一项关键资产。存在两种不同类型的分离:第一种分离对应于从抽象需求到系统实现的开发中的不同步骤,并且被认为是垂直的。它符合常用的细化概念。第二个层次对应于给定精化层次上的系统架构中的各种组件,称为水平层次。当我们的工作集中在时间建模问题上时,我们已经对数据、功能和并发问题进行了深入的研究。该贡献旨在通过定义用于关联异步系统中不同瞬间的严格偏序之间的顺序,为时间模型的精化验证提供一个正式的构造。这种关系允许具体级别的设计师区分在抽象级别上一致的事件,同时保留在抽象级别上评估的属性。这项工作是使用证明助手Agda进行的,并与之前关于异步语言CCSL的工作相关联,后者也使用相同的工具进行了建模。
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引用次数: 0
A Theory of Lazy Imperative Timing 懒惰祈使句计时理论
Pub Date : 2018-10-23 DOI: 10.4204/EPTCS.282.1
E. Hehner
We present a theory of lazy imperative timing.
我们提出了一种懒惰祈使句计时理论。
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引用次数: 0
Fast Cut-Elimination using Proof Terms: An Empirical Study 使用证明项的快速删减:实证研究
Pub Date : 2018-10-17 DOI: 10.4204/EPTCS.281.3
Gabriel Ebner
Urban and Bierman introduced a calculus of proof terms for the sequent calculus LK with a strongly normalizing reduction relation. We extend this calculus to simply-typed higher-order logic with inferences for induction and equality, albeit without strong normalization. We implement thiscalculus in GAPT, our library for proof transformations. Evaluating the normalization on both artificial and real-world benchmarks, we show that this algorithm is typically several orders of magnitude faster than the existing Gentzen-like cut-reduction, and an order of magnitude faster than any other cut-elimination procedure implemented in GAPT.
Urban和Bierman为具有强归一化约化关系的序列微积分LK引入了证明项的演算。我们将这个演算扩展到具有归纳和相等推理的简单类型高阶逻辑,尽管没有强规范化。我们在GAPT中实现了这个演算,GAPT是我们的证明转换库。在人工和现实世界的基准上评估归一化,我们表明该算法通常比现有的genzen -like cut-reduction快几个数量级,比GAPT中实现的任何其他cut-elimination过程快一个数量级。
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引用次数: 0
Validating Back-links of FOLID Cyclic Pre-proofs FOLID循环预证明的反向链接验证
Pub Date : 2018-10-17 DOI: 10.4204/EPTCS.281.4
Sorin Stratulat
Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions, the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.
循环预证明可以表示为具有反向链接的有限树推导集。在具有归纳定义的一阶逻辑的框架中,树派生的节点用序列标记,并且反向链接将特定的终端节点(称为芽)连接到由相同序列标记的其他节点。然而,只有一些反向链接可以构成可靠的预证明。在此之前,我们已经证明了在有向图的最小环上定义的特殊的序性和可导性条件,对于验证反向链是足够的。在这种方法中,在处理不同的最小周期时可以多次检查相同的约束,因此可能需要额外的记录机制来避免冗余计算,以便将时间复杂度降低到多项式。我们提出了一种不需要处理最小周期的新方法。它基于一种标准形式,允许通过只考虑其有向图的非单例强连接组件的根-芽路径来定义验证条件。
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引用次数: 6
The Fundamental Theorem of Algebra in ACL2 ACL2中的代数基本定理
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.8
Ruben Gamboa, J. Cowles
We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a "context", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the "context" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r).
本文报道了ACL2(r)中代数基本定理的一个证明。证明由四个部分组成。首先,定义了复数的复值函数和实值函数的连续性,并证明了从复到实数的连续函数在一个封闭的平方区域上达到最小值。连续实值复函数的一个重要例子是取连续复函数的传统复范数。我们认为这些连续函数只有一个(复数)参数,但在ACL2(r)中,它们表现为两个参数的函数。额外的参数是一个“context”,它是未被解释的。例如,它可以是固定的其他参数,例如在具有底数和指数的指数函数中,它们中的任何一个都可以保持固定。其次,证明了复数多项式是连续的,因此复数多项式的范数是一个连续的实值函数,并且在以原点为中心的任意正方形区域上达到最小值。这部分证明得益于“上下文”论证的引入,它说明了一种创新,它简化了具有非绑定参数的经典性质的证明。第三,我们推导了离原点足够远的输入的非常数多项式范数的下界和上界。这意味着可以找到一个足够大的平方来保证它包含多项式范数的全局最小值。第四,证明了如果给定数不是一个非常多项式的根,那么它就不可能是全局最小值。最后,结合这些结果表明,全局最小值必须是多项式的一个根。这个结果是ACL2(r)中复数多项式形式化的更大努力的一部分。
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引用次数: 0
Real Vector Spaces and the Cauchy-Schwarz Inequality in ACL2(r) ACL2(r)中的实向量空间与Cauchy-Schwarz不等式
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.9
Carl Kwan, M. Greenstreet
We present a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of $mathbb{R}^n$ as an inner product space. We also provide an application of Cauchy-Schwarz by formalising $mathbb R^n$ as a metric space and exhibiting continuity for some simple functions $mathbb R^ntomathbb R$. The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another: [|langle u,vrangle| leq |u| |v|] with equality iff the vectors are linearly dependent. It finds frequent use in many branches of mathematics including linear algebra, real analysis, functional analysis, probability, etc. Indeed, the inequality is considered to be among "The Hundred Greatest Theorems" and is listed in the "Formalizing 100 Theorems" project. To the best of our knowledge, our formalisation is the first published proof using ACL2(r) or any other first-order theorem prover.
本文给出了ACL2(r)中Cauchy-Schwarz不等式的一个力学证明,并形式化了进行这种证明所必需的数学。这包括将$mathbb{R}^n$形式化为内部积空间。我们还通过将$mathbb R^n$形式化为度量空间并展示一些简单函数$mathbb R^ntomathbb R$的连续性来提供Cauchy-Schwarz的应用。Cauchy-Schwarz不等式将向量的大小与其投影(或内积)与另一个向量联系起来:[|langle u,vrangle| leq |u| |v|]如果向量是线性相关的,则相等。它在许多数学分支中经常使用,包括线性代数、实分析、泛函分析、概率论等。事实上,这个不等式被认为是“100个最伟大的定理”之一,并被列入“100个定理的形式化”项目。据我们所知,我们的形式化是第一个使用ACL2(r)或任何其他一阶定理证明的公开证明。
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引用次数: 4
DefunT: A Tool for Automating Termination Proofs by Using the Community Books (Extended Abstract) DefunT:一个使用社区图书自动终止证明的工具(扩展摘要)
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.12
Matt Kaufmann
We present a tool that automates termination proofs for recursive definitions by mining existing termination theorems.
我们提出了一个工具,通过挖掘现有的终止定理来自动化递归定义的终止证明。
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引用次数: 2
Convex Functions in ACL2(r) ACL2(r)中的凸函数
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.10
Carl Kwan, M. Greenstreet
This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space formalisation of R^n. Among the introduced theorems is a set of equivalent conditions for convex functions with Lipschitz continuous gradients from Yurii Nesterov's classic text on convex optimisation. To the best of our knowledge a full proof of the theorem has yet to be published in a single piece of literature. We also explore "proof engineering" issues, such as how to state Nesterov's theorem in a manner that is both clear and useful.
本文建立在我们先前在ACL2(R)中对R^n的形式化的基础上,提出了一组关于凸函数的推理定理。这是在度量空间形式化的R^n中可能的高维分析推理的演示。在介绍的定理中,有一组来自yuri Nesterov关于凸优化的经典文本的具有Lipschitz连续梯度的凸函数的等价条件。据我们所知,这个定理的完整证明还没有在一篇文献中发表过。我们还探讨了“证明工程”问题,例如如何以一种既清晰又有用的方式陈述Nesterov定理。
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引用次数: 4
Formalising Filesystems in the ACL2 Theorem Prover: an Application to FAT32 ACL2定理证明中的文件系统形式化:FAT32的应用
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.2
M. Mehta
In this work, we present an approach towards constructing executable specifications of existing filesystems and verifying their functional properties in a theorem proving environment. We detail an application of this approach to the FAT32 filesystem. We also detail the methodology used to build up this type of executable specification through a series of models which incrementally add features of the target filesystem. This methodology has the benefit of allowing the verification effort to start from simple models which encapsulate features common to many filesystems and which are thus suitable for reuse.
在这项工作中,我们提出了一种在定理证明环境中构建现有文件系统的可执行规范并验证其功能属性的方法。我们将详细介绍这种方法在FAT32文件系统中的应用。我们还详细介绍了通过一系列模型来构建这种类型的可执行规范的方法,这些模型增量地添加目标文件系统的特性。这种方法的好处是允许验证工作从简单的模型开始,这些模型封装了许多文件系统共有的特性,因此适合重用。
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引用次数: 2
Smtlink 2.0
Pub Date : 2018-10-10 DOI: 10.4204/EPTCS.280.11
Yan Peng, M. Greenstreet
Smtlink is an extension of ACL2 with Satisfiability Modulo Theories (SMT) solvers. We presented an earlier version at ACL2'2015. Smtlink 2.0 makes major improvements over the initial version with respect to soundness, extensibility, ease-of-use, and the range of types and associated theory-solvers supported. Most theorems that one would want to prove using an SMT solver must first be translated to use only the primitive operations supported by the SMT solver -- this translation includes function expansion and type inference. Smtlink 2.0 performs this translation using a sequence of steps performed by verified clause processors and computed hints. These steps are ensured to be sound. The final transliteration from ACL2 to Z3's Python interface requires a trusted clause processor. This is a great improvement in soundness and extensibility over the original Smtlink which was implemented as a single, monolithic, trusted clause processor. Smtlink 2.0 provides support for FTY defprod, deflist, defalist, and defoption types by using Z3's arrays and user-defined data types. We have identified common usage patterns and simplified the configuration and hint information needed to use Smtlink.
Smtlink是ACL2的扩展,具有可满足模理论(SMT)解算器。我们在ACL2'2015上展示了一个早期版本。Smtlink 2.0在可靠性、可扩展性、易用性以及支持的类型和相关理论求解器的范围等方面对初始版本进行了重大改进。想要使用SMT求解器证明的大多数定理必须首先转换为只使用SMT求解器支持的基本操作——这种转换包括函数展开和类型推断。Smtlink 2.0使用经过验证的子句处理器和计算提示执行的一系列步骤来执行此转换。确保这些步骤是合理的。从ACL2到Z3的Python接口的最终音译需要一个可信的子句处理器。与最初的Smtlink相比,这在可靠性和可扩展性方面有了很大的改进,原来的Smtlink是作为一个单一的、整体的、受信任的子句处理器实现的。Smtlink 2.0通过使用Z3的数组和用户定义的数据类型,提供了对ftydefprod、defist、defalist和defoption类型的支持。我们已经确定了常见的使用模式,并简化了使用Smtlink所需的配置和提示信息。
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引用次数: 7
期刊
CoRR
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