抽象循环证明

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2024-04-19 DOI:10.1017/s0960129524000070
Bahareh Afshari, Dominik Wehr
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引用次数: 0

摘要

与传统的推导树不同,循环证明系统允许有限图的推导。这种证明的健全性是通过对推导施加健全性条件来保证的。最常见的此类条件是全局踪迹条件(GTC),这是一个关于通过推导图的无限路径的条件。为了统一处理这种循环证明系统,Brotherston 提出了一个抽象的轨迹概念。我们将兄弟斯顿的方法扩展到循环推导的范畴论演绎中,并从两个方面推进了这一框架:首先,我们引入了激活代数,使现有循环证明系统中的迹条件形式化得更自然。其次,考虑到踪迹信息的构成,我们可以推导出关于循环证明的新结果,比如引入拉姆齐式的踪迹条件。此外,我们将我们的轨迹概念与自动机理论联系起来,并证明验证具有特定轨迹条件的抽象循环证明的 GTC 是 PSPACE-complete的。
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Abstract cyclic proofs
Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the global trace condition (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston’s approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce activation algebras which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete.
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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