Rabin's theorem in the concurrency setting: A conjecture

ISCAS OpenIR

	Rabin's theorem in the concurrency setting: A conjecture
	Thiagarajan, P.S. (1); Yang, Shaofa (2); Thiagarajan, P.S.
	2014
发表期刊	Theoretical Computer Science
ISSN	3043975
卷号	546 页码:225-236
摘要	Rabin's theorem says that the monadic second order theory of the infinite binary tree is decidable. This result has had a far reaching influence in the theory of branching time temporal logics. A simple consequence of Rabin's theorem is that for every finite state transition system, the monadic second order theory of its computation tree is decidable. Concurrency theory strongly suggests that finite 1-safe Petri nets (or simply, net systems) are a natural generalization of the notion of a finite state transition system while labelled event structures arising as the unfoldings of net systems are the proper counterparts to the computation trees obtained by unwinding finite (sequential) transition systems. It is easy to define the monadic second order theory of such event structures. It turns out that unlike the sequential case, not every net system (i.e. its event structure unfolding) has a decidable monadic second order theory. This gives rise to the question: Which net systems admit a decidable monadic second order theory? Here we present a conjecture based on a property called grid-freeness. Our conjecture is that a net system has a decidable monadic second order theory iff its event structure unfolding is grid-free. We show that it is decidable whether a net system has this property. We also prove that the monadic second order theory of a net system is undecidable if its event structure unfolding is not grid-free. In addition we show that our conjecture can be effectively reduced to the sub-class of free choice net systems. Finally we point out how the positive resolution of our conjecture will settle the decidability of a range of distributed controller synthesis problems.; Rabin's theorem says that the monadic second order theory of the infinite binary tree is decidable. This result has had a far reaching influence in the theory of branching time temporal logics. A simple consequence of Rabin's theorem is that for every finite state transition system, the monadic second order theory of its computation tree is decidable. Concurrency theory strongly suggests that finite 1-safe Petri nets (or simply, net systems) are a natural generalization of the notion of a finite state transition system while labelled event structures arising as the unfoldings of net systems are the proper counterparts to the computation trees obtained by unwinding finite (sequential) transition systems. It is easy to define the monadic second order theory of such event structures. It turns out that unlike the sequential case, not every net system (i.e. its event structure unfolding) has a decidable monadic second order theory. This gives rise to the question: Which net systems admit a decidable monadic second order theory? Here we present a conjecture based on a property called grid-freeness. Our conjecture is that a net system has a decidable monadic second order theory iff its event structure unfolding is grid-free. We show that it is decidable whether a net system has this property. We also prove that the monadic second order theory of a net system is undecidable if its event structure unfolding is not grid-free. In addition we show that our conjecture can be effectively reduced to the sub-class of free choice net systems. Finally we point out how the positive resolution of our conjecture will settle the decidability of a range of distributed controller synthesis problems.
收录类别	SCI ; EI
关键词	Rabin's Theorem Monadic Second Order Logic Event Structure Petri Net Grid-free
部门归属	(1) School of Computing, National University of Singapore, Singapore; (2) State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, China
语种	英语
WOS记录号	WOS:000340691500018
引用统计
内容类型	期刊论文
URI标识	http://ir.iscas.ac.cn/handle/311060/16836
专题	中国科学院软件研究所
通讯作者	Thiagarajan, P.S.
推荐引用方式 GB/T 7714	Thiagarajan, P.S. ,Yang, Shaofa ,Thiagarajan, P.S.. Rabin's theorem in the concurrency setting: A conjecture[J]. Theoretical Computer Science,2014,546:225-236.
APA	Thiagarajan, P.S. ,Yang, Shaofa ,&Thiagarajan, P.S..(2014).Rabin's theorem in the concurrency setting: A conjecture.Theoretical Computer Science,546,225-236.
MLA	Thiagarajan, P.S. ,et al."Rabin's theorem in the concurrency setting: A conjecture".Theoretical Computer Science 546(2014):225-236.