Analyzing recursive probabilistic Programs - Informatik 2 · PDF file1 Introduction 2...

Analyzing recursive probabilisticProgramsJustin Winkens

30. June 2016

Justin Winkens | 29. Juni 2016 0/42Justin Winkens | 30. June 2016 1/42

1 Introduction

2 Preliminaries

3 Translating pGCL to pPDA

4 Analyzing Probabilistic Pushdown Automata

Table of contents

Justin Winkens | 30. June 2016 2/42

1 Introduction

2 Preliminaries

Introduction

Input ProbabilisticProgram

Output

Introduction

Probabilistic Programs

Output

Introduction

Output

TrueSkillTM

Introduction

Output

TrueSkillTM self-testing/correcting

Introduction

Output

biology

Introduction

Output

biology networks

Introduction

Output

biology networks Picture

Introduction

1 algorithm quicksort(A) is

2 list less , equal , greater

3 if length(A) <= 1

4 return A

5 pivot := A[length(A)]

6 for x in A

7 if x < pivot then append x to less

8 if x = pivot then append x to equal

9 if x > pivot then append x to greater

10 return concat(quicksort(less), equal ,quicksort(greater ))

Introduction

Recursive Probabilistic Programs

3 if length(A) <= 1

4 return A

6 for x in A

) Average case complexity O(n log n)

Introduction

3 if length(A) <= 1

4 return A

6 for x in A

) Average case complexity O(n log n)) Worst case complexity O(n2

Introduction

3 if length(A) <= 1

4 return A

5 pivot := uniformSel(A)

6 for x in A

Introduction

3 if length(A) <= 1

4 return A

5 pivot := uniformSel(A)

6 for x in A

) Worst case far less likely!

Introduction

I We wish to...

I ...reason about a programs behaviorI ...estimate expected outcomes and runtimesI ...quantify probabilities

I Recursive programs written in the probabilistic Guarded CommandLanguage

I Analysis by using probabilistic pushdown automata

Introduction

I We wish to...I ...reason about a programs behavior

I ...estimate expected outcomes and runtimesI ...quantify probabilities

Introduction

I We wish to...I ...reason about a programs behaviorI ...estimate expected outcomes and runtimes

I ...quantify probabilitiesI Recursive programs written in the probabilistic Guarded Command

LanguageI Analysis by using probabilistic pushdown automata

Introduction

I We wish to...I ...reason about a programs behaviorI ...estimate expected outcomes and runtimesI ...quantify probabilities

Introduction

1 Introduction

2 Preliminaries

Preliminaries

ControlUnit

... a b a d ...

Input tape

Preliminaries Probabilistic Pushdown Automata

Pushdown Automata

Definition: Probabilistic Pushdown Automaton (pPDA)

A probabilistic pushdown automaton (pPDA) is a tuple � = (Q,�, ,!,P)

defined as

I Q is a finite set of states.I

� is a finite stack alphabet.I ,!✓ (Q⇥ �)⇥ (Q⇥ �

⇤) is a set of transitions such that

I for every pX 2 Q⇥ � there is at least one transition of the formpX ,! q↵

I for every transition pX ,! q↵ we have that |↵| 2.

Probabilistic Pushdown Automata

defined asI Q is a finite set of states.

I� is a finite stack alphabet.

I ,!✓ (Q⇥ �)⇥ (Q⇥ �

defined asI Q is a finite set of states.I

� is a finite stack alphabet.

I ,!✓ (Q⇥ �)⇥ (Q⇥ �

defined asI P is a function which assigns probabilities to transitions

I P(pX ,! q↵) 2 [0, 1]

pX,!q↵

P(pX ,! q↵) = 1

I We write pXx

,! q↵ instead of P(pX ,! q↵) = x

I P(pX ,! q↵) 2 [0, 1]

pX,!q↵

P(pX ,! q↵) = 1

I We write pXx

I P(pX ,! q↵) 2 [0, 1]

pX,!q↵

P(pX ,! q↵) = 1

I We write pXx

I P(pX ,! q↵) 2 [0, 1]

pX,!q↵

P(pX ,! q↵) = 1

I We write pXx

Example pPDA

A pPDA � is given by:I Q = {q}I

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

What does this pPDA represent?

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

qII qIII qIIII ...

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

q" qI qII

qIII qIIII ...

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

q" qI qII qIII

qIIII ...

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

q" qI qII qIII qIIII ...

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

q" qI qII qIII qIIII ...

) Termination on empty stack!

Example pPDA

� = {I}I qI

0.5,! qII , qI

0.5,! q✏

What could an unbounded random walk look like?

Example pPDA

Definition: Regular Configurations

Let � = (Q,�, ,!,P) be a pPDA and C ✓ Q⇥�

⇤ a set of configurations.

We call C regular if p↵ 2 C iff the reversed stack of p↵ is accepted by adeterministic finite-state automaton (DFA) A.

We call C simple if there exists a set H ✓ Q⇥ � such that p↵ 2 C iff↵ 6= " and head(p↵) 2 H.

Definition: Paths and RunsLet S be a set of states and !✓ S⇥S a binary and total relation andP = (S,!).

I A path is a word w such that w(i) ! w(i+ 1) for everyi 2 {0, . . . , |w|� 1}

I A state t is reachable from s if there is a finite path w with w(0) = sand w(|w|� 1) = t.

I A run is an infinite path.I Runs(P,w) = set of all runs in P that start with w

Preliminaries Notions

Paths, Runs

Definition: Paths and RunsLet S be a set of states and !✓ S⇥S a binary and total relation andP = (S,!).I A path is a word w such that w(i) ! w(i+ 1) for every

i 2 {0, . . . , |w|� 1}

I A state t is reachable from s if there is a finite path w with w(0) = sand w(|w|� 1) = t.

Paths, Runs

i 2 {0, . . . , |w|� 1}I A state t is reachable from s if there is a finite path w with w(0) = s

and w(|w|� 1) = t.

Paths, Runs

and w(|w|� 1) = t.I A run is an infinite path.

I Runs(P,w) = set of all runs in P that start with w

Paths, Runs

and w(|w|� 1) = t.I A run is an infinite path.I Runs(P,w) = set of all runs in P that start with w

Paths, Runs

Definition: Reachability

Let � = (Q,�, ,!,P) be a pPDA, pX 2 Q⇥ � an initial configurationand C ✓ Q⇥ �

⇤ a set of target configurations.

I Reach(pX, C) set of all runs r 2 Runs(�, pX) with r(i) 2 C, i 2 N0

I P(Reach(pX, C))

Reachability

I P(Reach(pX, C))

Reachability

I P(Reach(pX, C))

Reachability

Definition: TerminationLet � = (Q,�, ,!,P) be a pPDA, pX 2 Q⇥ � an initial configurationand q 2 Q a (final) state.

I Reach(pX, {q"}) set of all runs r 2 Runs(�, pX) with r(i) = q",i 2 N0.

I P(Reach(pX, {q"})).

Termination

Definition: Qualitative/Quantitative Reachability/Termination

I Qualitative reachability/termination: P(Reach(pX,C)) = 1?I Quantitative reachability/termination: P(Reach(pX,C)) d for a

constant d 2 (0, 1)?

Qualitative/Quantitative Reachability/Termination

⇤ a set of target configurations.I Qualitative reachability/termination: P(Reach(pX,C)) = 1?

I Quantitative reachability/termination: P(Reach(pX,C)) d for aconstant d 2 (0, 1)?

⇤ a set of target configurations.I Qualitative reachability/termination: P(Reach(pX,C)) = 1?I Quantitative reachability/termination: P(Reach(pX,C)) d for a

skip | abort | var x | x:=E | Q;R | {Q}[ ]{R} | {Q}[p]{R}if(G){Q}else{R} | while(G){Q}

Preliminaries Probabilistic Guarded Command Language

Probabilistic Guarded Command Language

1 var flip := true;

2 flip (){

3 while(flip){

4 {skip ;}[0.5]{ flip = false;}

1 var flip := true;

2 flip (){

3 while(flip){

4 {skip ;}[0.5]{ flip = false;}

We allow function calls!

Dueling Cowboys

1 var cowboy := A;

2 var duel:= true;

4 {cowboy := A;}[p]{ cowboy := B;}

6 while(duel){

7 if(cowboy = A){

8 {duel := false ;}[a]{ cowboy := B;}

9 }else{

10 {duel := false ;}[b]{ cowboy := A;}

Dueling Cowboys (Orig.)

1 var player := A;

2 duel (){ //d03 {player :=A;}[p]{ player :=B;}

4 shoot (); //d15 }

1 kill (){ //k02

How do we analyze this?

1 shoot (){ //s02 if(player == A){ //s13 {

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

12 else if(player == B){//s213 {

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

Dueling Cowboys (Rec.)

1 var player := A;

4 shoot (); //d15 }

1 kill (){ //k02

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

1 var player := A;

4 shoot (); //d15 }

1 kill (){ //k02

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

1 var player := A;

4 shoot (); //d15 }

1 kill (){ //k02

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

1 Introduction

2 Preliminaries

Translating pGCL to pPDA

2 Steps:

1. Translate pGCL program to control flow graph2. Translate control flow graph to pPDA

2 Steps:1. Translate pGCL program to control flow graph

2. Translate control flow graph to pPDA

2 Steps:1. Translate pGCL program to control flow graph2. Translate control flow graph to pPDA

I Pick control points (k0,d0,d1,s0,...,s6) as nodes

I Connect nodes by edges if control flows from one node to the nextI Label edges with statements and transition probability (if 6= 1)

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

pGCL ! Control Flow Graph

I Pick control points (k0,d0,d1,s0,...,s6) as nodesI Connect nodes by edges if control flows from one node to the next

I Label edges with statements and transition probability (if 6= 1)

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

I Pick control points (k0,d0,d1,s0,...,s6) as nodesI Connect nodes by edges if control flows from one node to the nextI Label edges with statements and transition probability (if 6= 1)

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

1 var player := A;

4 shoot (); //d15 }

1� pplayer := B

shoot()

pplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

4 kill (); //s35 }

8 player := B;

9 shoot (); //s410 }

14 kill (); //s515 }

16 [t]

18 player := A;

19 shoot (); //s620 }

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

1 kill (){ //k02

Why do we have this method?I It yields an “easy” head (pk0) for analysisI All configuration with head pk0 terminate) Picking good control points and methods is key!

1 kill (){ //k02

Why do we have this method?

I It yields an “easy” head (pk0) for analysisI All configuration with head pk0 terminate) Picking good control points and methods is key!

1 kill (){ //k02

Why do we have this method?I It yields an “easy” head (pk0) for analysis

I All configuration with head pk0 terminate) Picking good control points and methods is key!

1 kill (){ //k02

Why do we have this method?I It yields an “easy” head (pk0) for analysisI All configuration with head pk0 terminate

) Picking good control points and methods is key!

1 kill (){ //k02

1� pplayer := B

shoot()

pplayer := A

shoot()

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

Reminder: pPDA

⇤) is a set of transitions

I P is a function which assigns probabilities to transitions

How does this relate to control flow graphs?

Control Flow Graph ! pPDA

Reminder: pPDA

⇤) is a set of transitions

I P is a function which assigns probabilities to transitions

How does this relate to control flow graphs?

I Q: Variable assignments

I�: Nodes in the graph

I P: Probabilities of the edgesI ,!: ???

I Q: Variable assignmentsI

�: Nodes in the graph

I P: Probabilities of the edgesI ,!: ???

�: Nodes in the graphI P: Probabilities of the edges

I ,!: ???

�: Nodes in the graphI P: Probabilities of the edgesI ,!: ???

I construct every pair Q⇥ �

I for each pair hq, �i 2 Q⇥ � consider edges of the control flow graph:I hq, �i ,! hq0, �0i if no function call between � and �0

I hq, �i ,! hq0,'�0i if function call between � and �0

I �0 return addressI ' entry point of function call

I hq, �i ,! hq0, "i if function call terminates

I for each pair hq, �i 2 Q⇥ � consider edges of the control flow graph:

I hq, �i ,! hq0, �0i if no function call between � and �0

hAk0i 1,! hA"i

hBk0i 1,! hB"i

k0 hAk0i 1,! hA"i

hBk0i 1,! hB"i

1� pplayer := B

shoot()

pplayer := A

shoot()

hAd0i p,! hAs0d1i

hAd0i 1�p,! hBs0d1i

hAd1i 1,! hA"i

hBd1i 1,! hB"i

1� pplayer := B

shoot()

pplayer := A

shoot()

hAd0i p,! hAs0d1i

hAd1i 1,! hA"i

hBd1i 1,! hB"i

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

hAs0i 1,! hAs1i

hBs0i 1,! hBs2i

hAs1i s,! hAk0s3i

hAs1i 1�s,! hBs0s4i

hAs3i 1,! hA"i

hAs4i 1,! hA"i

hBs4i 1,! hB"i

hBs2i t,! hBk0s5i

hBs2i 1�t,! hAs0s6i

hBs5i 1,! hB"i

hBs6i 1,! hB"i

hAs6i 1,! hA"i

s3 s4 s5 s6

if else if

skill()

1� splayer := B

shoot()

tkill()

1� tplayer := A

shoot()

hAs0i 1,! hAs1i

hBs0i 1,! hBs2i

hAs1i s,! hAk0s3i

hAs3i 1,! hA"i

hAs4i 1,! hA"i

hBs4i 1,! hB"i

hBs2i t,! hBk0s5i

hBs5i 1,! hB"i

hBs6i 1,! hB"i

hAs6i 1,! hA"i

hAd0i p,! hAs0d1i

hAd1i 1,! hA"i

hBd1i 1,! hB"i

hAs0i 1,! hAs1i

hBs0i 1,! hBs2i

hAs1i s,! hAk0s3i

hAs3i 1,! hA"i

hAs4i 1,! hA"i

hBs4i 1,! hB"i

hBs2i t,! hBk0s5i

hBs5i 1,! hB"i

hBs6i 1,! hB"i

hAs6i 1,! hA"i

hAk0i 1,! hA"i

hBk0i 1,! hB"i

1 Introduction

2 Preliminaries

Analyzing Probabilistic Pushdown Automata

Properties of pPDA

⇤ a set of target configurations.I Qualitative reachability/termination: P(Reach(pX,C)) = 1?I Quantitative reachability/termination: P(Reach(pX,C)) d for a

Analyzing Probabilistic Pushdown Automata

Reminder: Properties of pPDA

We solve this problem for a simple set of target configurations.

) Extensible to regular sets of target configurations

NotationLet � = (Q,�, ,!,P) be a pPDA, C ✓ Q⇥ �

⇤ a simple set of targetconfigurations and H be the associated set of heads.

I[pX•] = P(Reach(pX, C))

I[pXq] = probability of all r 2 Reach(pX, {q"}) that do not visit C

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

Analyzing Probabilistic Pushdown Automata Qualitative Reachability/Termination

Qualitative Reachability/Termination

We solve this problem for a simple set of target configurations.) Extensible to regular sets of target configurations

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

For pX 2 H:

I[pX•] = 1

I[pXq] = 0

For pX 2 H:I

[pX•] = 1

I[pXq] = 0

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

t2Qx·[rY t]·[tZ•]

[pXq] =X

x · [rY q] +X

,!rY Z

t2Qx · [rY t] · [tZq]

1. Construct equations for all [pX•] and [pXq] needed.2. Replace all [pX•] and [pXq] by fresh random variables hpX•i and

hpXqi3. Solve the system of recursive equations for least solution.

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

[pXq] =X

x · [rY q] +X

,!rY Z

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

[pXq] =X

x · [rY q] +X

,!rY Z

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

[pXq] =X

x · [rY q] +X

,!rY Z

1. Construct equations for all [pX•] and [pXq] needed.

2. Replace all [pX•] and [pXq] by fresh random variables hpX•i andhpXqi

3. Solve the system of recursive equations for least solution.

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

[pXq] =X

x · [rY q] +X

,!rY Z

3. Solve the system of recursive equations for least solution.

For pX /2 H:

[pX•] =X

x·[rY •]+X

,!rY Z

x·[rY •]+X

,!rY Z

[pXq] =X

x · [rY q] +X

,!rY Z

Let C be a set of target configurations with H = {Ak0, Bk0}

Qualitative Term./Reach. Dueling Cowboys

Let C be a set of target configurations with H = {Ak0, Bk0}) P(Reach(Ad0, C)) = [Ad0•] = 1 means qualitative termination

[Ad0•] =X

x·[rY •] +X

,!rY Z

x·[rY •] +X

,!rY Z

[Ad0•] =X

x·[rY •] +X

,!rY Z

x·[rY •] +X

,!rY Z

hAd0i p

,! hAs0d1ihAd0i 1�p

,! hBs0d1i

[Ad0•] =X

x·[rY •] +X

,!rY Z

x·[rY •] +X

,!rY Z

= p·[As0•]

hAd0i p

,! hBs0d1i

[Ad0•] =X

x·[rY •] +X

,!rY Z

x·[rY •] +X

,!rY Z

= p·[As0•]+(1�p)·[Bs0•]

hAd0i p

,! hBs0d1i

[Ad0•] =X

x·[rY •] +X

,!rY Z

x·[rY •] +X

,!rY Z

= p·[As0•] + (1�p)·[Bs0•]+

,!rY Z

(x·[rY A]·[AZ•] + x·[rY B]·[BZ•])

hAd0i p

,! hBs0d1i

[Ad0•] = p·[As0•] + (1�p)·[Bs0•]+ (p·[As0A]·[Ad1•] + p·[As0B]·[Bd1•])+ ((1�p)·[Bs0A]·[Ad1•] + (1�p)·[Bs0B]·[Bd1•])

hAd0i p

,! hBs0d1i

[Ad0•] = p·[As0•] + (1�p)·[Bs0•]+ (p·[As0A]·[Ad1•] + p·[As0B]·[Bd1•])+ ((1�p)·[Bs0A]·[Ad1•] + (1�p)·[Bs0B]·[Bd1•])

) Remove zero summands

hAd0i p

,! hBs0d1i

[Ad0•] = p·[As0•] + (1�p)·[Bs0•]

) We need to calculate [As0•] and [Bs0•]!

[Ad0•] = p·[As0•] + (1�p)·[Bs0•][As0•] = s+ (1�s)·[Bs0•][Bs0•] = t+ (1� t)·[As0•]

hAd0•i = p·hAs0•i+ (1�p)·hBs0•ihAs0•i = s+ (1�s)·hBs0•ihBs0•i = t+ (1� t)·hAs0•i

p = 1/2

s = 1/2

t = 1/3

hAd0•i = 1, hAs0•i = 1, hBs0•i = 1

p = 1/2

s = 1/2

t = 1/3

hAd0•i = 0, hAs0•i = 0, hBs0•i = 0

p = 1/2

I We know that the duel ends if both s, t > 0

I What is the probability that e.g. A wins the duel?I ) [Ad0A] =?, C = ;I Same equations apply!

Analyzing Probabilistic Pushdown Automata Quantitative Reachability/Termination

Quantitative Term./Reach Dueling Cowboys

I What is the probability that e.g. A wins the duel?

I ) [Ad0A] =?, C = ;I Same equations apply!

I What is the probability that e.g. A wins the duel?I ) [Ad0A] =?, C = ;

I Same equations apply!

[Ad0A] = p · [As0A] + (1� p)[Bs0A]

[As0A] = s+ (1� s)[Bs0A]

[Bs0A] = (1� t)[As0A]

hAd0Ai = p · hAs0Ai+ (1� p)hBs0AihAs0Ai = s+ (1� s)hBs0AihBs0Ai = (1� t)hAs0Ai

p = 1/2, s = 1/2, t = 1/2

hAd0Ai = 1/2

hAs0Ai = 2/3

hBs0Ai = 1/3

I pPDA nice way to model and analyze systems that use probability aswell as recursion

I 2 formulae enough to study quantitative and qualitative reachabilityI Can be done efficiently in polynomial spaceI Construction technique to pPBA allows analysis in polynomial time.

Conclusion

I 2 formulae enough to study quantitative and qualitative reachability

I Can be done efficiently in polynomial spaceI Construction technique to pPBA allows analysis in polynomial time.

Conclusion

I 2 formulae enough to study quantitative and qualitative reachabilityI Can be done efficiently in polynomial space

I Construction technique to pPBA allows analysis in polynomial time.

Conclusion

I 2 formulae enough to study quantitative and qualitative reachabilityI Can be done efficiently in polynomial spaceI Construction technique to pPBA allows analysis in polynomial time.

Conclusion

Definition: RuntimeLet � be a transition system with a set of configurations S and targetconfigurations T .

Let f : S ! R�0 be a function that assigns run times to states.For every run r 2 Runs(�) we define kr as the least n 2 N0 such thatr(n) 2 T .

T ime(�) =

f(r(i)) .

) kr depends on probability) E(T ime(�)), E(T ime(�)|Reach(pX, C))

Expected Runtime

Definition: RuntimeLet � be a transition system with a set of configurations S and targetconfigurations T .Let f : S ! R�0 be a function that assigns run times to states.

For every run r 2 Runs(�) we define kr as the least n 2 N0 such thatr(n) 2 T .

T ime(�) =

f(r(i)) .

Expected Runtime

Definition: RuntimeLet � be a transition system with a set of configurations S and targetconfigurations T .Let f : S ! R�0 be a function that assigns run times to states.For every run r 2 Runs(�) we define kr as the least n 2 N0 such thatr(n) 2 T .

T ime(�) =

f(r(i)) .

Expected Runtime

T ime(�) =

f(r(i)) .

Expected Runtime

T ime(�) =

f(r(i)) .

) kr depends on probability

) E(T ime(�)), E(T ime(�)|Reach(pX, C))

Expected Runtime

T ime(�) =

f(r(i)) .

Expected Runtime

Definition: Expected Runtime pPDA

Let � = (Q,�, ,!,P) be a pPDA.EpXq = E(T ime(�)|Reach(pX, {q"})) conditional expected runtime

hEpXqi = f(pX) +

x · [rY q]

[pXq]· hErY qi

,!rY Z

[pXq]· (hErY ti+ hEtZqi)

) Here: f(pX) = 1 for all pX 2 Q⇥ � ) discrete time

Expected Runtime

hEpXqi = f(pX) +

x · [rY q]

[pXq]· hErY qi

,!rY Z

Expected Runtime

hEpXqi = f(pX) +

x · [rY q]

[pXq]· hErY qi

,!rY Z

Expected Runtime

P(Reach(rY, {q"})|Reach(pX, {q"})) = P(Reach(rY, {q"}) \Reach(pX, {q"}))P(Reach(pX, {q"}))

P(Reach(rY, {q"}))P(Reach(pX, {q"}))

[rY q]

Expected Runtime

Expected runtime for all runs that start in Ad0 and end in A":

hEAs0Ai = 1 +

[As1A]

[As0A]

· hEAs1Ai

hEAs1Ai = 1 +

[As1A]

· (hEAk0Ai+ hEAs3Ai) + (1� s) · [Bs0A]

[As1A]

(hEBs0Ai)

hEBs0Ai = 1 +

[Bs2A]

[Bs0A]· hEBs2Ai

hEBs2Ai = 1 + (1� t) · [As0A]

[Bs2A]

· (hEAs0Ai+ hEAs6Ai)

hEAd1Ai = 1

hEAk0Ai = 1

hEAs3Ai = 1

hEAs6Ai = 1

Expected Runtime

Expected runtime for all runs that start in Ad0 and end in A":

hEAs0Ai = 1 +

[As1A]

[As0A]

· hEAs1Ai

hEAs1Ai = 1 +

[As1A]

· (hEAk0Ai+ hEAs3Ai) + (1� s) · [Bs0A]

[As1A]

(hEBs0Ai)

hEBs0Ai = 1 +

[Bs2A]

[Bs0A]· hEBs2Ai

hEBs2Ai = 1 + (1� t) · [As0A]

[Bs2A]

· (hEAs0Ai+ hEAs6Ai)

hEAd1Ai = 1

hEAk0Ai = 1

hEAs3Ai = 1

hEAs6Ai = 1

Expected Runtime

Let’s assume p = s = t = 1/2

hEAd0Ai = 2 +

· hAs0Ai+ 1

· hBs0AihEAs0Ai = 1 + hEAs1AihEAs1Ai =

hEBs0AihEBs0Ai = 1 + hEBs2AihEBs2Ai = 2 + hEAs0Ai

) hEAd0Ai = 26/3 = 8.66

Expected Runtime

hEAd0Ai = 2 +

· hAs0Ai+ 1

) hEAd0Ai = 26/3 = 8.66

Expected Runtime

hEAd0Ai = 2 +

· hAs0Ai+ 1

) hEAd0Ai = 26/3 = 8.66

Expected Runtime

hAd0Ai = �p� (1� p) · (1� t)

2(st� s� t)

hAs0Ai = �1

2(st� s� t)

hBs0Ai = �(1� t)

2(st� s� t)

Quantitative System Parametrized

Analyzing recursive probabilistic Programs - Informatik 2 · PDF file1 Introduction 2...

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