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Decisive Markov Chains
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Datorteknik.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Datorteknik.
2007 (Engelska)Ingår i: Logical Methods in Computer Science, ISSN 1860-5974, Vol. 3, nr 4, 1-32 s.Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We consider qualitative and quantitative verification problems for infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm solves the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM.

Ort, förlag, år, upplaga, sidor
2007. Vol. 3, nr 4, 1-32 s.
Nyckelord [en]
Infinite Markov Chains, Verification, Model Checking
Nationell ämneskategori
Data- och informationsvetenskap
Identifikatorer
URN: urn:nbn:se:uu:diva-10734DOI: 10.2168/LMCS-3(4:7)2007ISI: 000262642600007OAI: oai:DiVA.org:uu-10734DiVA: diva2:38502
Tillgänglig från: 2007-05-21 Skapad: 2007-05-21 Senast uppdaterad: 2010-12-15Bibliografiskt granskad

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