# Polynomial-Time Verification of PCTL Properties of MDPs with Convex Uncertainties

### Alberto Alessandro Angelo Puggelli, Wenchao Li, Alberto L. Sangiovanni-Vincentelli and Sanjit A. Seshia

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2013-24

April 3, 2013

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-24.pdf

We address the problem of verifying Probabilistic Computation Tree Logic (PCTL) properties of Markov Decision Processes (MDPs) whose state transition probabilities are only known to lie within uncertainty sets. We first introduce the model of Convex-MDPs (CMDPs), i.e., MDPs with convex uncertainty sets. CMDPs generalize Interval-MDPs (IMDPs) by allowing also more expressive (convex) descriptions of uncertainty. Using results on strong duality for convex programs, we then present a PCTL verification algorithm for CMDPs, and prove that it runs in time polynomial in the size of a CMDP for a rich subclass of convex uncertainty models. This result allows us to lower the previously known algorithmic complexity upper bound for IMDPs from co-NP to PTIME. Using the proposed approach, we verify a consensus protocol and a dynamic configuration protocol for IPv4 addresses.

BibTeX citation:

@techreport{Puggelli:EECS-2013-24, Author = {Puggelli, Alberto Alessandro Angelo and Li, Wenchao and Sangiovanni-Vincentelli, Alberto L. and Seshia, Sanjit A.}, Title = {Polynomial-Time Verification of PCTL Properties of MDPs with Convex Uncertainties}, Institution = {EECS Department, University of California, Berkeley}, Year = {2013}, Month = {Apr}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-24.html}, Number = {UCB/EECS-2013-24}, Abstract = {We address the problem of verifying Probabilistic Computation Tree Logic (PCTL) properties of Markov Decision Processes (MDPs) whose state transition probabilities are only known to lie within uncertainty sets. We first introduce the model of Convex-MDPs (CMDPs), i.e., MDPs with convex uncertainty sets. CMDPs generalize Interval-MDPs (IMDPs) by allowing also more expressive (convex) descriptions of uncertainty. Using results on strong duality for convex programs, we then present a PCTL verification algorithm for CMDPs, and prove that it runs in time polynomial in the size of a CMDP for a rich subclass of convex uncertainty models. This result allows us to lower the previously known algorithmic complexity upper bound for IMDPs from co-NP to PTIME. Using the proposed approach, we verify a consensus protocol and a dynamic configuration protocol for IPv4 addresses.} }

EndNote citation:

%0 Report %A Puggelli, Alberto Alessandro Angelo %A Li, Wenchao %A Sangiovanni-Vincentelli, Alberto L. %A Seshia, Sanjit A. %T Polynomial-Time Verification of PCTL Properties of MDPs with Convex Uncertainties %I EECS Department, University of California, Berkeley %D 2013 %8 April 3 %@ UCB/EECS-2013-24 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-24.html %F Puggelli:EECS-2013-24