Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Probabilistic Graph Transformation Systems Christian Krause, Holger Giese Hasso Plattner Institute (HPI), University of Potsdam, Germany ICGT 2012, Applications Track, 25 September 2012 Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 1 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Outline Motivation Typed Graph Transformation background Markov Decision Processes Probabilistic Graph Transformation Systems our contributions Tool Support and Analysis Conclusions and Future Work Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 1 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Outline Motivation Typed Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Tool Support and Analysis Conclusions and Future Work Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 1 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Motivation Graph Transformation • Graph transformation systems (GTSs) are a powerful formalism for modeling and analyzing functional and structural aspects of distributed and mobile systems. Probabilistic Behavior • Probabilistic behavior is needed to quantify unlikely behavior and random failures, e.g., message losses in unreliable media. • Many protocols in distributed / mobile systems use randomization to ensure functional properties or to optimize QoS properties. • Some examples: • Randomized distributed algorithms: randomized leader election protocol, randomized Byzantine agreement protocol • Communication and multimedia protocols: Bluetooth device discovery, IEEE 802.11 Wireless LAN, IPv4 Zeroconf protocol Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 2 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Motivation Why Probabilistic Graph Transformation? • The required concept of discrete probabilistic choices are not supported by any of the existing GTS variants. • Observation: often a combination of probabilistic and nondeterministic behavior is needed. Running Example • We model probabilistic broadcasting in wireless sensor networks. • Gossiping protocol : every node decides with a given probability whether it forwards a received message to its neighbors or not ⇒ reduction of communication costs. • Assumption: decision whether to forward a message is probabilistic; the order of the message sending is nondeterministic. Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 3 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Outline Motivation Typed Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Tool Support and Analysis Conclusions and Future Work Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 3 of 19
� � � � � � Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Modeling Definition (Rule) ℓ r A rule p = � L ← − K − → R � is a pair of injective typed L K R graph morphisms. L is called the left-hand side (LHS), and R the right-hand side (RHS) of the rule. Definition (Transformation) ℓ r ℓ r L K R Let p = � L ← − K − → R � be a rule and m : L → M by a typed graph morphism, m ( PO ) ( PO ) p , m = ⇒ N is called a match . A transformation M � N M C defined by the right double pushout diagram. The applicability of rules can be further restricted using negative application conditions (NACs). Matching and transformation of an unbounded number of substructures is achieved using nested rules . Attributes over finite domains can be encoded in typed graphs. Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 4 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Modeling Modeling wireless sensor networks as graphs: INIT :Node con active=true TG AP id=2 con :Node :Node Node active=true con has has active=false id=1 active:Bool :Message Message :Node id=x id:ID has con active=true :Message id=3 Figure : atomic Figure : type graph Figure : example of an proposition received ( x ) instance graph Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 5 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Modeling Modeling nondeterministic message broadcasting using nested graph transformation rules with NACs: NAC NAC has has has has [2]:Message [1]:Node :Message [2]:Message [1]:Node :Message LHS RHS LHS RHS [1]:Node [1]:Node [1]:Node [1]:Node [1]:Node [1]:Node [1]:Node [1]:Node active=true active=false active=true active=false has has has has [2]:Message [2]:Message [2]:Message [2]:Message Figure : rule send 2 LHS’ RHS’ con con [1]:Node [3]:Node [1]:Node [3]:Node has has has active=true [2]:Message [2]:Message :Message Figure : rule send 1 Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 6 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Modeling Resetting nodes in the case of message collisions: LHS RHS [1]:Node [1]:Node [1]:Node [1]:Node active=true has has [2]:Message [3]:Message LHS’ RHS’ [1]:Node [1]:Node has has has [2]:Message [3]:Message :Message Figure : rule reset Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 7 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Semantics Derived semantics for initial graph with topology consisting of 3 nodes: reset reset send 1 p 1 send 1 send send send 2 (1-p) 1 send 2 p send 2 send 2 (1-p) 1 send 1 send send 1 send send 1 p send 1 send 2 reset reset (1-p) Figure : Labeled transition system (LTS) ⇒ message sending is completely nondeterministic here Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 8 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Typed Graph Transformation: Semantics Derived semantics for initial graph with topology consisting of 3 nodes: reset reset send 1 p 1 send 1 send send send 2 (1-p) 1 send 2 p send 2 send 2 (1-p) 1 send 1 send send 1 send send 1 p send 1 send 2 reset reset (1-p) Figure : Labeled transition system (LTS) Figure : Markov decision process (MDP) ⇒ message sending is completely ⇒ probabilistic message sending nondeterministic here can be modeled in an MDP Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 8 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Outline Motivation Typed Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Tool Support and Analysis Conclusions and Future Work Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 8 of 19
Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions Markov Decision Processes Definition (Discrete Probability Distribution) For a denumerable set Q , a discrete probability distribution is a function µ : Q → [ 0 , 1 ] with � q ∈ Q µ ( q ) = 1. We use Dist ( Q ) to denote the set of all discrete probability distributions over Q . Definition (Markov Decision Process) A Markov decision process (MDP) M = ( Q , q init , Steps ) consists of a denumerable set of states Q , an initial state q init ∈ Q and a probabilistic transition function Steps : Q → 2 Dist ( Q ) . → q ′ is made from a state q ∈ Q by: µ A probabilistic transition q − 1. nondeterministically selecting a distribution µ ∈ Steps ( q ) , and 2. probabilistically choosing a target state q ′ according to µ . Quantitative properties for MDPs can be specified as PCTL formulae and verified using probabilistic model checking. Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 9 of 19
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