efficient analysis of probabilistic systems that ... · the cost problem 3 10: 15 7 10: 20 2 10: 5...

Efficient Analysis of Probabilistic SystemsThat Accumulate Quantities

Christoph Haase Stefan Kiefer Markus Lohrey

Highlights 2018, Berlin20 September 2018

Stefan Kiefer Probabilistic Systems That Accumulate Quantities 1

The Cost Problem

310 : 15

710 : 20

210 : 5

810 : 10

What is the probability to reach the gate in 25–45min?After what time are half of the customers done?

Cost Problem :=

Input: Cost Markov chaincost formula ϕ

Output: Pr(ϕ)

25 ≤ cost ≤ 45


The Cost Problem

310 : 15

710 : 20

210 : 5

810 : 10

What is the probability to reach the gate in 25–45min?After what time are half of the customers done?

Cost Problem :=

Input: Cost Markov chaincost formula ϕthreshold τ ∈ [0,1]

Output: Is Pr(ϕ) ≥ τ ?

25 ≤ cost ≤ 45


Dynamic Programming

12 : 1

12 : 2

13 : 3

13 : 1

13 : 1

0 1 2 3 4

1

0

0

12

0

0

14

12

0

18+

16

12

12

12

12

12

12

12

Iterative Linear Programming [Baier, Daum, Dubslaff, Klein,Klüppelholz, NFM’14]

Computing expectations is much easier.


Dynamic Programming

12 : 1

12 : 2

13 : 3

13 : 1

13 : 1

0 1 2 3 4

1

0

0

12

0

0

14

12

0

18+

16

12

12

12

12

12

12

12

Iterative Linear Programming [Baier, Daum, Dubslaff, Klein,Klüppelholz, NFM’14]

Computing expectations is much easier.Stefan Kiefer Probabilistic Systems That Accumulate Quantities 3

Complexity of the Cost Problem

Theorem (Laroussinie, Sproston, FoSSaCS’05)The cost problem is in EXPTIME.The cost problem is NP-hard

NP-hard

.


by reduction from theK th largest subset problem

Theorem (HK, IPL’16)The K th largest subset problem is

PP

-complete.

CorollaryThe cost problem is PP-hard.

a superset of NP


Theorem (Laroussinie, Sproston, FoSSaCS’05)The cost problem is in EXPTIME.The cost problem is NP-hardNP-hard.



Theorem (HK, IPL’16)The K th largest subset problem is

PP

-complete.


a superset of NP


Theorem (Laroussinie, Sproston, FoSSaCS’05)The cost problem is in EXPTIME.The cost problem is NP-hardNP-hard.



Theorem (HK, IPL’16)The K th largest subset problem is PP-complete.


a superset of NP

The K th Largest Subset Problem is PP-Complete

K th Largest Subset Problem

Instance: A finite set X ⊆ N and K ,B ∈ N.Question: Are there at least K subsets of X whose elementssum up to at most B?

K th Largest-Area ConvexPolygon Problem

Instance: A finite setP ⊆ N× N and K ,B ∈ N.Question: Are there at least Kconvex polygons made from Pwhose area is at least B?

Vertex countingInstance: A matrix A, a vector b, and K ∈ N.Question: Does the polyhedron Ax ≤ b have ≤ K vertices?



EXPTIME

PSPACE

Cost

PP

NP

1 0

+ +

∗ −

∗

circuit value


Theorem (HK, ICALP’15)The cost problem is PosSLP-hard.

The cost problem is in PSPACE.

Theorem (HKL, LICS’17)The cost problem is in CH.


EXPTIME

PSPACE

Cost

PP PosSLP

NP

1 0

+ +

∗ −

∗

circuit value

:=Input: arithmetic circuitOutput: Is the value > 0 ?


Theorem (HK, ICALP’15)The cost problem is PosSLP-hard.

The cost problem is in PSPACE.



EXPTIME

PSPACE

Cost

PP PosSLP

NP

1 0

+ +

∗ −

∗

circuit value

:=Input: arithmetic circuitOutput: Is the value > 0 ?


Theorem (HK, ICALP’15)The cost problem is PosSLP-hard.The cost problem is in PSPACE.



EXPTIME

PSPACE

CH

Cost

PP PosSLP

NP

1 0

+ +

∗ −

∗

circuit value


Theorem (HK, ICALP’15)The cost problem is PosSLP-hard.The cost problem is in PSPACE.


Solving the Cost Problem

Cost Problem :=


Output: Pr(ϕ)

310 : 20

12 : 15

12 : 25

1 : 10

1 : 10

710 : 5

Enumerate the Parikh images.Problem: there might be multiple paths per Parikh image.



Cost Problem :=


Output: Pr(ϕ)

310 : 20

12 : 15

12 : 25

1 : 10

1 : 10

710 : 5

2

1

2

1

2

1




Cost Problem :=


Output: Pr(ϕ)

310 : 20

12 : 15

12 : 25

1 : 10

1 : 10

710 : 5

2

1

2

1

2

1

Enumerate the Parikh images.

Problem: there might be multiple paths per Parikh image.



Cost Problem :=


Output: Pr(ϕ)

310 : 20

12 : 15

12 : 25

1 : 10

1 : 10

710 : 5

2

1

2

1

2

1




Cost Problem :=


Output: Pr(ϕ)



The BEST Theorem

Theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte)The number of Eulerian cycles in an Eulerian graph G equals

t(G) ·∏v∈V

(d(v)− 1

)!

Theorem (Tutte’s matrix-tree theorem)

t(G) = det(L(G)1,1)


number of directedspanning trees in G

Laplacian of G


310 : 20

12 : 15

12 : 25

1 : 10

1 : 10

710 : 5

2

1

2

1

2

1

Enumerate the Parikh images. How?

There is a linear-sized existential Presburger formula thatencodes “valid” Parikh images (Euler-Hierholzer theorem)

Use an SMT solver to compute all valid Parikh images.


QUANT: A Tool for the Cost Problem

Theorem ([HK, ICALP’15], [HK, IPL’16], [HKL, LICS’17])The cost problem is hard for PP and PosSLP.The cost problem is in CH.




0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

What is Pr(cost ∈ [4,6]2)?


dimension d

time in sec

3 4 5 6 7 80

50

100

150

200

QUANT [8,10]dQUANT [13,15]dQUANT [18,20]d

PRISM [8,10]d

PRISM [13,15]d

PRISM [18,20]d


Counting Parikh ImagesΣ: finite alphabet

p ∈ NΣ: vector

A: language generator (DFA, NFA, CFG)

N(A,p): number of words accepted by A with Parikh image p

Example: for A = a∗ba∗ and p = (2,1): N(A,p) = 3

PosParikh :=

Input: Language generators A,Bvector p ∈ NΣ

Output: Is N(A,p) > N(B,p) ?

Different variants:language generator: DFA, NFA, CFGunary or binary encoding of pfixed or variable alphabet Σ


Complexity of PosParikh [HKL’17]

vector p size of Σ DFA NFA CFG

unaryencoding

unary in L NL-c. P-c.fixed PL-c.

variable PP-c.

binaryencoding

unary in L NL-c. DP-c.

fixedPosMatPow-hard,

in CHPSPACE-c. PEXP-c.

variable PosSLP-hard,in CH

BitParikh :=

Input: Language generator Avector p ∈ NΣ

number i ∈ N in binary

Output: Is the i-th bit of N(A,p) equal to 1 ?


PosMatPow

PosMatPow :=

Input: m×m integer matrix M (in binary)number n ∈ N (in binary)

Output: Is (Mn)1,m ≥ 0 ?

The entries of Mn are of exponential size.

Theorem (Galby, Ouaknine, Worrell, 2015)PosMatPow is in P for m = 2.PosMatPow is in P for m = 3 and M given in unary.

PosMatPow reduces to PosSLP.

Theorem (HKL, 2017)PosMatPow reduces toPosParikh with DFAs, binary encoding of p, even for |Σ| = 2.


From Matrix Powers to Graphs

G: multi-graph with edges labelled by multiplicities

u v w2 3

N(G,u, v ,n) : number of paths from u to v of length n

N(G,u,w ,2) = 6

LemmaGiven an integer matrix M and indices i , j , one can compute inlogspace a multi-graph G with vertices u, v+, v− such that

(Mn)i,j = N(G,u, v+,n)− N(G,u, v−,n) for all n ∈ N.


Getting Rid of Edge Weights

Replace u v21

by

10101 1010 101 10 1

u v

The edge weights are removed at the expense of longer paths.


From Graphs to DFAs

Replace v

v1

v2

v3

v4

by

v

v1

v2

v3

v4

a

b

b b b

ab b

ba

b

b

a


Complexity of PosParikh [HKL’17]

vector p size of Σ DFA NFA CFG

unaryencoding

unary in L NL-c. P-c.fixed PL-c.

variable PP-c.

binaryencoding

unary in L NL-c. DP-c.

fixedPosMatPow-hard,

in CHPSPACE-c. PEXP-c.

variable PosSLP-hard,in CH


Cost Markov Decision Processes

Theorem (HK, ICALP’15)The cost problem for MDPs is EXP-complete.

Theorem (HK, ICALP’15)The cost problem for acyclic MDPs is PSPACE-complete, evenfor atomic cost formulas.


Main Message

Study complexity!

The exponential-time complexity can be improved by:formal language theoryPresburger arithmeticthe BEST theoremSMT-solvers


efficient analysis of probabilistic systems that ... · the cost problem 3 10: 15 7 10: 20 2 10: 5...

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