 
Summary: DECISIVE MARKOV CHAINS
PAROSH ABDULLA, NOOMENE BEN HENDA, AND RICHARD MAYR
Uppsala University, Department of Information Technology, Box 337, SE751 05 Uppsala, Sweden
email address: parosh@it.uu.se
Uppsala University, Department of Information Technology, Box 337, SE751 05 Uppsala, Sweden
email address: noomene.benhenda@it.uu.se
North Carolina State University, Department of Computer Science, Campus Box 8206, Raleigh, NC 27695,
USA
email address: mayr@csc.ncsu.edu
ABSTRACT. We consider qualitative and quantitative verification problems for infinitestate 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,
all Markov chains induced by probabilistic lossy channel systems (PLCS) contain a finite attractor
and are thus decisive. Furthermore, all globally coarse Markov chains are decisive. The class of
globally coarse Markov chains includes, e.g., those induced by probabilistic vector addition systems
(PVASS) with upwardclosed sets F , and all Markov chains induced by probabilistic noisy Turing
machines (PNTM) (a generalization of the noisy Turing machines (NTM) of Asarin and Collins).
