Cyclic Proofs for Logic-Based Program Verification

最近几年，无论是从学术角度还是从工业方面，对计算机程序的理想特性的数学验证都产生了越来越广泛的兴趣。例如，这些属性可以声明程序不会超出其内存边界（“内存安全”），或者它总是能够执行某些操作（“活动”）。因此，为程序和支持这些原则的证明系统和计算机工具开发适当的推理原则，目前在计算机科学研究界吸引了相当多的活动。尽管在开发各种形式的程序逻辑方面取得了进步，但程序属性的“证明”的基本概念几乎没有变化；默认情况下，证明是一棵有限树，它的构造尊重逻辑的特定推理规则（又称推导树），因此树的叶子是公理实例，树的根是要证明的定理，中间节点由逻辑推理联系起来。然而，最近，形式证明的另一种模式已被提出作为逻辑推理的范式，其特征是各种形式的递归，称为循环证明。一个循环证明本质上是通过在一个推导树中识别一些循环得到的，也就是说，一个循环证明实际上是一个正则的，无限的推导树，用循环图的形式表示。通常情况下，并不是每一个这样的证明结构都是可靠的，所以在循环证明上施加了一个额外的全局保护条件，以确保它们的可靠性。循环证明的现有研究，首先是在一阶逻辑和具有归纳定义的BI的纯逻辑设置中，然后是在证明简单程序终止的分离逻辑系统中，已经证明了它作为形式推理工具的可行性，以及它作为证明方法的潜在力量。此外，在具有归纳定义的一阶逻辑的情况下，一个相关无穷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上的工作已经基本上建立了循环证明领域，既是一个理论上自然的值得研究的领域，也是一个成熟的程序验证应用领域。该博士后的主要目标是建立在循环证明的理论基础上，特别是在其初步发展到基于分离逻辑的程序推理的基础上，以进一步开发应用方向的思想。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，并将循环证明概念扩展到混合归纳法和协归纳法。其次，我们希望研究循环证明作为自动定理证明工具的潜力。

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.