# Technical Reports

## The report FIMU-RS-2007-03

### Strategy Synthesis for Markov Decision Processes and Branching-Time Logics

by *
Tomáš Brázdil,
Vojtěch Forejt,
* A full version of the paper presented at CONCUR 2007 July 2007, 28 pages.

**FIMU-RS-2007-03.**
Available as *Postscript*,
**PDF**.

#### Abstract:

We consider a class of finite
$1\frac{1}{2}$-player games (Markov decision processes) where the
winning objectives are specified in
the branching-time temporal logic qPECTL$^*$
(an extension of the qualitative PCTL$^*$).
We study decidability and complexity of existence of a
winning strategy in these games.
%The logic is more expressive than the qualitative fragment of PCTL$^*$.
We identify a fragment of qPECTL$^*$ called detPECTL$^*$ for which the
existence of a winning strategy is decidable in exponential time, and
also the winning strategy can be computed in exponential time (if it exists).
Consequently we show that every formula of qPECTL$^*$ can be translated
to a formula of detPECTL$^*$ (in exponential time) so that the resulting formula
is equivalent to the original one
over finite Markov chains. From this we obtain that for the whole qPECTL$^*$,
the existence of a winning finite-memory strategy is decidable in
double exponential time. An immediate
consequence is that the existence of a
winning finite-memory strategy is decidable for the qualitative fragment of
PCTL$^*$ in triple exponential time.
We also obtain a single exponential upper bound on the same problem for
the qualitative PCTL.
Finally, we study the power of finite-memory strategies with
respect to objectives described in the qualitative PCTL.