Probabilistic Graph Transformation Systems Christian Krause, Holger - - PowerPoint PPT Presentation

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 Markov Decision Processes Probabilistic Graph Transformation Systems Tool Support and Analysis Conclusions and Future Work        background       

• ur contributions

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)

A rule p = L

ℓ

← − K

r

− → R is a pair of injective typed graph morphisms. L is called the left-hand side (LHS), and R the right-hand side (RHS) of the rule. L R K

Definition (Transformation)

Let p = L

ℓ

← − K

r

− → R be a rule and m : L → M by a typed graph morphism, called a match. A transformation M

p,m

= ⇒ N is defined by the right double pushout diagram. L

m

• (PO)

K

ℓ

• r
• (PO)

R

• M

C

• N

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:

TG

Node active:Bool id:ID Message has con

Figure : type graph

INIT

:Node :Node active=true id=2 con active=true id=1 :Message has :Node active=true id=3 con con

Figure : example of an instance graph

AP

:Node active=false id=x :Message has

Figure : atomic proposition received(x)

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:

RHS

[1]:Node

LHS

[1]:Node

NAC

[1]:Node [2]:Message :Message has has

RHS’

[1]:Node [2]:Message [3]:Node has con

LHS’

[1]:Node [2]:Message [3]:Node active=true has con :Message has [1]:Node active=false [2]:Message has [1]:Node active=true [2]:Message has

Figure : rule send1

NAC

[1]:Node [2]:Message :Message

RHS

[1]:Node

LHS

[1]:Node has has [1]:Node active=false [2]:Message has [1]:Node active=true [2]:Message has

Figure : rule send2

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:

RHS

[1]:Node

LHS

[1]:Node

RHS’

[1]:Node

LHS’

[1]:Node [1]:Node active=true [2]:Message has [3]:Message has [1]:Node [2]:Message has [3]:Message has :Message has

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:

p (1-p)

p

p (1-p) (1-p) 1 1 1 1 send send send reset reset send send send1 reset reset send2 send1 send1 send2 send2 send2 send2 send1 send1

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:

p (1-p)

p

p (1-p) (1-p) 1 1 1 1 send send send reset reset send send send1 reset reset send2 send1 send1 send2 send2 send2 send2 send1 send1

Figure : Labeled transition system (LTS)

⇒ message sending is completely nondeterministic here

Figure : Markov decision process (MDP)

⇒ probabilistic message sending 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, qinit, Steps) consists of a denumerable set of states Q, an initial state qinit ∈ Q and a probabilistic transition function Steps : Q → 2Dist(Q). A probabilistic transition q

µ

− → q′ is made from a state q ∈ Q by:

• 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

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic vs. Nondeterministic Behavior

Probabilistic Behavior

• Discrete probabilistic choices to quantify random and

unlikely behavior, e.g., the chance of a message loss.

Nondeterministic Behavior

• Nondeterministic behavior in the case of incomplete

knowledge, e.g., to ensure implementation freedom.

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 10 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic vs. Nondeterministic Behavior

Probabilistic Behavior

• Discrete probabilistic choices to quantify random and

unlikely behavior, e.g., the chance of a message loss. ⇒ predictable average or long-run behavior

Nondeterministic Behavior

• Nondeterministic behavior in the case of incomplete

knowledge, e.g., to ensure implementation freedom. ⇒ completely unpredictable behavior

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 10 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic vs. Nondeterministic Behavior

Probabilistic Behavior

• Discrete probabilistic choices to quantify random and

unlikely behavior, e.g., the chance of a message loss. ⇒ predictable average or long-run behavior ⇒ analysis by calculating probabilities and expected values

Nondeterministic Behavior

• Nondeterministic behavior in the case of incomplete

knowledge, e.g., to ensure implementation freedom. ⇒ completely unpredictable behavior ⇒ reasoning about worst case and best case scenarios

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 10 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic vs. Nondeterministic Behavior

Probabilistic Behavior

• Discrete probabilistic choices to quantify random and

unlikely behavior, e.g., the chance of a message loss. ⇒ predictable average or long-run behavior ⇒ analysis by calculating probabilities and expected values

Nondeterministic Behavior

• Nondeterministic behavior in the case of incomplete

knowledge, e.g., to ensure implementation freedom. ⇒ completely unpredictable behavior ⇒ reasoning about worst case and best case scenarios

Note: In combined probabilistic and nondeterministic models, e.g., in MDPs, it is only possibly to compute minimum and maximum probabilities / expected values.

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 10 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 10 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic Graph Transformation Systems

Motivation

Discrete probabilistic choices as required by the running example cannot be expressed in ordinary graph transformation systems. Solution: we introduce the concept of probabilistic rules.

Definition (Probabilistic Rule)

A probabilistic rule π = J, P, µ consists of a typed graph J, a finite non-empty set of rules P, s.t. J = L for all p = L

ℓ

← − K

r

− → R ∈ P, and a probability distribution µ ∈ Dist(P).

Informal Semantics

Find a match for J. Then randomly choose

• ne of the RHSs according to µ and

transform the host graph based on this RHS. J R1 Rn K1 Kn . . . µ(p1) µ(pn) . . .

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 11 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic Graph Transformation Systems

We use one probabilistic rule to correctly model the broadcasting:

NAC

[1]:Node [2]:Message :Message

RHS:p

[1]:Node

RHS:(1-p)

[1]:Node

LHS

[1]:Node has has

RHS’

[1]:Node [2]:Message [3]:Node has con

RHS’

[1]:Node [2]:Message [3]:Node has con

LHS’

[1]:Node [2]:Message [3]:Node active=true has con :Message has [1]:Node active=false [2]:Message has [1]:Node active=false [2]:Message has [1]:Node active=true [2]:Message has

/ /

Figure : probabilistic rule send with sending probability p

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 12 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic Graph Transformation Systems

Definition (Probabilistic Graph Transformation System)

A probabilistic graph transformation system (PGTS) is a tuple G = T, Ginit, Π consisting of a type graph T, an initial graph Ginit typed over T, and a set of probabilistic rules Π typed over T. A probabilistic transformation M

π,m,p

= ⇒ N is made by:

• 1. nondeterministically selecting an applicable rule π = J, P, µ ∈ Π,
• 2. nondeterministically selecting a match m : J → M,
• 3. making a probabilistic choice for a basic rule p ∈ P according to µ,
• 4. transforming M into N using the basic rule p and the match m.

Note: A probabilistic transformation M

π,m,p

= ⇒ N is a particular resolution of both the nondeterministic and the probabilistic choices in a PGTS. The choice for a rule and match is nondeterministic, whereas the effect of a rule is probabilistic.

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 13 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Probabilistic Graph Transformation Systems

Proposition (Induced Markov Decision Process)

A PGTS G = T, Ginit, Π induces the MDP MG = Q, qinit, Steps with:

• Q = {[G] | Ginit =

⇒∗

G G}, i.e., the set of isomorphism classes of

typed graphs reachable from Ginit,

• qinit = [Ginit],
• Steps([G]) = { ν | G

π,m

= ⇒ ν } where G

π,m

= ⇒ ν with π = J, P, µ ∈ Π denotes the fact that there exists p ∈ P and G′ ∈ Q such that G

π,m,p

= ⇒ G′, and where ν ∈ Dist(Q) is induced by µ as follows:a ν([G′]) =

• p∈P:G

π,m,p

= ⇒ G′

µ(p)

aWe use the convention ∅ = 0 for sums over empty sets.

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 14 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 14 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Tool Support

Henshin

We use Henshin to model PGTSs and to generate their state spaces in the form of MDPs.

PRISM

We use PRISM to model check the derived MDPs.

Figure : Schematic example network topology (left) and the derived state space (right)

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 15 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Quantitative Analysis

PCTL Model Checking

For the 3 × 3 grid network and a send probability of p = 0.7 we check:

• P>0.3(♦ received(8)): the probability that node 8

receives the message is greater than 0.3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) p=0.6 p=0.7 p=0.8 Nodes (and their distance to node 1 in hopes) Minimum / maximum probabilities

Figure : Minimum / maximum probabilities for message reception for each node in the 3 × 3 grid and send probabilities of p = 0.6, 0.7 and 0.8

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 16 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Quantitative Analysis

PCTL Model Checking

For the 3 × 3 grid network and a send probability of p = 0.7 we check:

• P>0.3(♦ received(8)): the probability that node 8

receives the message is greater than 0.3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) p=0.6 p=0.7 p=0.8 Nodes (and their distance to node 1 in hopes) Minimum / maximum probabilities

Figure : Minimum / maximum probabilities for message reception for each node in the 3 × 3 grid and send probabilities of p = 0.6, 0.7 and 0.8

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 16 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Quantitative Analysis

PCTL Model Checking

For the 3 × 3 grid network and a send probability of p = 0.7 we check:

• P>0.3(♦≤10 received(9)): the probability that node 9 receives the

message in at most 10 steps is greater than 0.3. χ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 Execution step n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Minimum / maximum probabilities Send probability p

(1) (2)

Figure : Minimum / maximum probabilities for message reception of node 9 in the 3 × 3 grid network for varying p (left) and fixed p = 0.7 (right)

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 17 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Quantitative Analysis

PCTL Model Checking

For the 3 × 3 grid network and a send probability of p = 0.7 we check:

• P>0.3(♦≤10 received(9)): the probability that node 9 receives the

message in at most 10 steps is greater than 0.3. χ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 Execution step n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Minimum / maximum probabilities Send probability p

(1) (2)

Figure : Minimum / maximum probabilities for message reception of node 9 in the 3 × 3 grid network for varying p (left) and fixed p = 0.7 (right)

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 17 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Quantitative Analysis

Analyzing different network topologies: xxxxxx

1 2 3 4 6 5 8 7 9 2 3 8 5 4 6 7 9

10

1 2 3 5 4 6 1 9 8 7

10 11

(a) (b) (c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (2) 7 (3) 8 (3) 9 (4) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (2) 5 (2) 6 (3) 7 (3) 8 (4) 9 (4) 10 (5) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 (1) 3 (1) 4 (1) 5 (1) 6 (2) 7 (3) 8 (3) 9 (4) 10 (4) 11 (4)

Network (b) Network (c) Network (d)

Minimum / maximum probabilities Nodes (and their distance to node 1 in hops)

Figure : Minimum / maximum probabilities for message reception for p = 0.7

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 18 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 18 of 19

Motivation Graph Transformation Markov Decision Processes Probabilistic Graph Transformation Systems Analysis Conclusions

Conclusions and Future Work

Conclusions

• PGTSs enable the modeling and analysis of combined

probabilistic and nondeterministic behavior in GTSs.

• Clear separation between the protocol and the network structure.
• We can easily analyze the protocol for different network

topologies by changing the initial graph.

• Dynamic changes (reconfigurations) in the network topology can

be modeled directly using transformation rules.

Future Work

• Study parallelism and concurrency for PGTSs.
• Exploit abstraction and compositionality for an efficient analysis.

Christian Krause, Holger Giese: Probabilistic Graph Transformation Systems, ICGT 2012 (Applications Track) Slide 19 of 19