Counting Problems for Parikh ImagesCH Cost PP PosSLP NP 1 0 + + circuit value Christoph Haase,...

Counting Problems for Parikh Images

Christoph Haase Stefan Kiefer Markus Lohrey

MFCS 2017, Aalborg25 August 2017

Christoph Haase, Stefan Kiefer, Markus Lohrey Counting Problems for Parikh Images 1

The Cost Problem

310 : 15

710 : 20

210 : 5

810 : 10

What is the probability to reach the gate in 25–45min?Quantiles?

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?Quantiles?

Cost Problem :=

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

Output: Is Pr(ϕ) ≥ τ ?

25 ≤ cost ≤ 45

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

-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.

-complete.

a superset of NP

-complete.

a superset of NP

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

a superset of NP

EXPTIME

PSPACE

∗ −

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

PP PosSLP

∗ −

circuit value

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

EXPTIME

PSPACE

PP PosSLP

∗ −

circuit value

EXPTIME

PSPACE

PP PosSLP

∗ −

circuit value

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

EXPTIME

PSPACE

PP PosSLP

∗ −

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

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

710 : 5

Cost Problem :=

Output: Pr(ϕ)

310 : 20

12 : 15

12 : 25

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

710 : 5

Cost Problem :=

Output: Pr(ϕ)

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)11)

number of directedspanning trees in G

Laplacian of G

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)?

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

0.45 : (1,0)

0.45 : (0,1)

0.1 : (0,0)

dimension d

time in sec

3 4 5 6 7 80

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

PRISM [8,10]d

PRISM [13,15]d

PRISM [18,20]d

dimension d

time in sec

3 4 5 6 7 80

PRISM [8,10]d

PRISM [13,15]d

PRISM [18,20]d

dimension d

time in sec

3 4 5 6 7 80

PRISM [8,10]d

PRISM [13,15]d

PRISM [18,20]d

Counting Parikh Images

Σ: 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

Results of This Paper: Complexity of PosParikh

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 ?

unaryencoding

variable PP-c.

binaryencoding

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

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

10101 1010 101 10 1

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

From Graphs to DFAs

Replace v

Summary

Counting Parikh images is closely related to performanceanalysis of probabilistic systems.

The complexity depends strongly on various parameters:language generator: DFA, NFA, CFGunary or binary encoding of pfixed or variable alphabet Σ

unaryencoding

variable PP-c.

binaryencoding

Counting Problems for Parikh ImagesCH Cost PP PosSLP NP 1 0 + + circuit value Christoph Haase,...

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