Cyclic Proofs for Logic-Based Program Verification

在过去的几年里，无论是从学术的角度还是从工业界的角度来看，对计算机程序所需性质的数学验证都越来越引起广泛的兴趣。例如，这样的属性可能声明程序没有超过其内存限制(“内存安全”)，或者它总是能够执行特定的操作(“活跃性”)。因此，程序和证明系统以及支持这些原则的计算机工具的适当归因原则的发展正在计算机科学研究界中吸引着相当多的活动。尽管在开发各种形式的程序逻辑方面取得了进展，但程序性质的*证明*的基本概念几乎没有改变；默认情况下，证明是一棵有限树，其构造尊重逻辑的特定推理规则(又名。派生树)，所以树的叶子是公理实例，树的根是要证明的定理，中间节点通过逻辑推理联系在一起。然而，最近，另一种形式证明模式被讨论为逻辑中的一种推理范例，其特点是各种形式的递归，称为“循环证明”。循环证明实质上是通过识别派生树中的一些圈来获得的，即循环证明实际上是以循环图的形式表示的正则无限派生树。通常，并不是每个这样的证明结构都代表声音证明，所以在循环证明上附加一个全局保护条件来确保它们的可靠性。现有的循环证明研究，首先是在一阶逻辑和具有归纳定义的BI的纯逻辑环境下，然后是在用于证明简单程序的终止性的分离逻辑系统中，已经证明了它作为形式推理工具的可行性和作为证明方法的潜在能力。此外，在具有归纳定义的一阶逻辑的情况下，相关无限证明系统的完备性结果建立了该方法的语义自然性。我们认为，我们以前在这些方向的工作基本上建立了循环证明领域，这在理论上是一个值得研究的自然领域，也是一个在程序验证中应用的成熟领域。拟议的博士后联谊会的广泛目标是建立在循环证明的发展理论基础上，特别是它最初发展成基于分离逻辑的程序推理，以进一步开发应用方向的想法。首先，我们希望建立和分析基于分离逻辑的程序验证循环证明系统，并将循环证明的概念扩展到混合归纳和协归纳。其次，我们希望研究循环证明作为自动化定理证明的工具的潜力。

The last few years have seen a increasingly widespreadinterest, both from the academic perspective and from elementsof industry, in the mathematical verification of desirableproperties of computer programs. Such properties might state,for example, that a program does not exceed its memory bounds(``memory safety'') or that it is always able to perform acertain action (``liveness''). The development of appropriatereasoning principles for programs and of proof systems andcomputer tools supporting those principles is thus currentlyattracting considerable activity amongst the computer scienceresearch community.Despite the advances in developing various forms of programlogic, the underlying notion of a *proof* of a programproperty has changed very little; by default, a proof is afinite tree whose construction respects the particularinference rules of the logic (a.k.a. derivation tree ), sothat the leaves of the tree are axiom instances and the root ofthe tree is the theorem to be proven, with intermediate nodesrelated by logical inferences. Recently, however, analternative mode of formal proof has been mooted as a paradigmfor reasoning in logics that feature various forms ofrecursion, known as *cyclic proof*. A cyclic proof isessentially obtained by identifying some cycles in a derivationtree, i.e., a cyclic proof is really a regular, infinitederivation tree, represented in cyclic graph form. Typically,not every such proof structure represents a sound proof, so anadditional, global guardedness condition is imposed oncyclic proofs that ensures their soundness. The existing investigations of cyclic proof, first in the purelogical settings of first-order logic and BI with inductivedefinitions, and subsequently in a separation logic systemfor proving termination of simple programs, have demonstratedits viability as a tool for formal reasoning, and its potentialpower as a proof method. Moreover, in the case of first-orderlogic with inductive definitions, a completeness result foran associated infinitary proof system establishes the semanticnaturality of the approach. We view our previous work in thesedirections as having substantially established the area ofcyclic proof as both a theoretically natural one worthy ofstudy, and ripe for applications in program verification.The broad aim of the proposed Postdoctoral Fellowship is tobuild on the developed theoretical foundations of cyclic proof,and especially its initial development into separationlogic-based reasoning about programs, in order to furtherexploit the ideas in the direction of applications. First, we wish to formulate and analyse cyclic proof systems for program verification based on separation logic, and to extend the cyclic proof concept to mixed induction and coinduction. Second, we wish to investigate the potential of cyclic proof as a vehicle for automated theorem proving.