Game Theory in Formal Verification Scribe Notes for Lecture 8 Markov Decision Processes
نویسنده
چکیده
Simple Stochastic Game nondeterminism nondeterminism probability probability nondeterminism nondeterminism probability 1 Definitions Markov Chain (MC) Definition 1. A Markov Chain ((S, E), δ) is a graph (S, E) with a function δ : S → D(S) that maps every state to a probability distribution of successor states. There is an edge between two states s, t in S iff the probability of going from s to t is nonzero. More formally, E = {(s, t) | δ(s)(t) > 0}. where the states a partitioned into player 1 states S 1 and probabilistic states
منابع مشابه
Scribe Notes: Game Theory in Formal Verification, Krishnendu Chatterjee Lecture 2
Game Graphs The model is based on a graph G = (S,E), consisting of a set S of states and a set E of directed edges between those states, the state transitions. We assume that every state has at least one outgoing edge. Figure 1(a) contains an example of such a graph representing a closed system where all indeterminism comes from the system itself (rather than from interaction with an outside en...
متن کاملGame Theory in Formal Verification Scribe notes for Lecture 6 Algorithms for Parity Games
In this lecture we will consider two algorithms for games with parity objectives. As before, game graph G = ((S,E), (S1, S2)) is a graph (S,E) with the finite set of states S, partitioned into Player 1 states S1 and Player 2 states S2, and edges E ∈ S × S. The set of all infinite paths in the game graph is denoted by Π. Given a path π = π0π1π2 . . ., we denote by Inf(π) the set of states occurr...
متن کاملA Proof System for Compositional Verification of Probabilistic Concurrent Processes
We present a formal proof system for compositional verification of probabilistic concurrent processes. Processes are specified using an SOS-style process algebra with probabilistic operators. Properties are expressed using a probabilistic modal μ-calculus. And the proof system is formulated as a sequent calculus in which sequents are given a quantitative interpretation. A key feature is that th...
متن کاملUbiquitous Abstraction: A New Approach for Mechanized Formal Verification
ion. Submitted for publication, Sept. 1998. Available at http://www.csl.sri.com/ ̃singermn/ integration.html.[8] H. Saı̈di and S. Graf. Construction of abstract state graphs with PVS. In O. Grumberg, editor, Computer-Aided Verification, CAV ’97, volume 1254 of Lecture Notes in Computer Science, pages 72–83, Haifa, Israel, June 1997. Springer-
متن کاملCMSC 858 F : Algorithmic Game Theory Fall 2010 Market Clearing with Applications
We will look at MARKET CLEARING or MARKET EQUILIBRIUM prices. We will consider two models with linear utilities : FISHER model and ARROWDEBREU model. See [1] and chapter 5 of [2] for more details. Finally we see an application of the FISHER model to wireless networks. I have also used as reference the scribe notes for this same lecture when this course was taught by Prof. Hajiaghayi at Rutgers ...
متن کامل