Expressiveness and Closure Properties for Quantitative Languages

Expressiveness and Closure Properties for

Quantitative Languages

Krishnendu Chatterjee, IST Austria

Laurent Doyen, ULB Belgium

Tom Henzinger, EPFL Switzerland

LICS 2009

Model-Checking

Property/ Specification

Yes / No

Satisfaction Relation

Program/ System

-perhaps a proof -perhaps some counterexamples

Model-Checking


Yes/No

Trace inclusion

Program/ System

Formula

Every request is followed by a grant

Finite automaton

Model-Checking


Yes/No

Trace inclusion

Program/ System

Finite automaton

Model-checking is boolean

Formula


- a trace is either good or bad

Quantitative Analysis


Value (R)


Program/ System

Finite automaton

Formula


-Measure of “fit” between system and spec

-e.g. average number of requests immediately granted

Distance (R)



Program/ System #1

Finite automaton

- Comparing two implementations

e.g. cost or quality measure

Program/ System #2


Quantitative Model-checking

Is there a Quantitative Framework with

- an appealing mathematical formulation, - useful expressive power, and

- good algorithmic properties ?

(Like the boolean theory of -regularity.)

Note: “Quantitative” is more than “timed” and “probabilistic”

A language is a boolean function:

Quantitative languages

A quantitative language is a function:

L(w) can be interpreted as:

• the amount of some resource needed by the system to produce w (power, energy, time consumption),

• a reliability measure (the average number of “faults” in w).

Outline

• Weighted automata

• Expressive power

• Closure properties

Weighted automata

Quantitative languages are generated by weighted automata.

Weight function

Weighted automata

Quantitative languages are generated by weighted automata.

Weight functionValue of a word w: max of {values of the runs r over w}Value of a run r: Val(r)

where is a value function

Some value functions

(reachability)

(Büchi)

(coBüchi)

(vi {0,1})

Outline




Reducibility

A class C of weighted automata can be reduced to a class C’ of weighted automata if

for all A C, there is A’ C’ such that LA = LA’.

Reducibility

A class C of weighted automata can be reduced to a class C’ of weighted automata if

for all A C, there is A’ C’ such that LA = LA’.

E.g. for boolean languages:

• Nondet. coBüchi can be reduced to nondet. Büchi

• Nondet. Büchi cannot be reduced to det. Büchi

(nondet. Büchi cannot be determinized)

cannot be determinized.

cannot be determinized.

Some known facts (CSL’08)

cannot be reduced to

cannot be reduced to

Reducibility relations

Cut-point languages

Words with value above some threshold:

ω-regular for Sup, LimSup, LimInf

can be non-ω-regular for LimAvg and Discounted

Cut-point languages

LimAvg:

«average number of a’s = 1»

is not ω-regular

A deterministic automaton for would accept (anb)ω for some n

Cut-point languages

Disc:

«disc. sum of a’s ≥ 1»

is not ω-regular

ambiguous word

1

p1 p2

Cut-point languages

Disc:

«disc. sum of a’s ≥ 1»

is not ω-regular

ambiguous word

1

From any two positions p1 and p2, there is a continuation accepted from p1 but not from p2

p1 p2

Cut-point Languages

Cut-point languages for deterministic LimAvg-automata are studied in [Alur/Degorre/Maler/Weiss’09]

Cut-point languages of LimAvg and Discounted can be non-ω-regular

Cut-point languages are not robust w.r.t. transition weights.

Cut-point Languages

isolated cut-point

Isolated cut-point languages are robust

Isolated cut-point languages are ω-regular(for deterministic automata)

Cut-point Languages

Each s.c.c. defines an interval of values.

Make accepting those s.c.c. with interval above

LimAvg:

s.c.c. decomposition

Either value is , then accept

or value is , the reject

Cut-point Languages

Disc:

after sufficiently long prefix, decision can be taken

is reducible to .

Expressive power of {0,1}-automata

is not reducible to .

is reducible to .


Store the value

A

B

is reducible to .


A

B

can take finitely many different values.

Store the value

is reducible to .


A

B

is reducible to .


B

A

Therefore for

is reducible to .


A B

for all

Outline




Operations

Operations on quantitative languages:

• shift(L1,c)(w) = L1(w) + c

• scale(L1,c)(w) = c·L1(w) (c>0)

Operations


• shift(L1,c)(w) = L1(w) + c

• scale(L1,c)(w) = c·L1(w) (c>0)

• max(L1,L2)(w) = max(L1(w),L2(w))

• min(L1,L2)(w) = min(L1(w),L2(w))

• complement(L1)(w) = 1-L1(w)

Operations


• shift(L1,c)(w) = L1(w) + c

• scale(L1,c)(w) = c·L1(w) (c>0)

• max(L1,L2)(w) = max(L1(w),L2(w))

• min(L1,L2)(w) = min(L1(w),L2(w))

• complement(L1)(w) = 1-L1(w)

• sum(L1,L2)(w) = L1(w) + L2(w)

Closure properties

All classes of weighted automata are closed under shift and scale.

All classes of nondeterministic weighted automata are closed under max.

Closure properties

Closure properties

Analogous results for boolean languages.

Closure properties

There is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).

Closure properties


Assume that L is definable by a LimAvg automaton C.

Closure properties



Then, some a-cycle or b-cycle in C has average weight >0.

(consider the word for large)

Closure properties


Then, some a-cycle or b-cycle in C has average weight >0.Then, some word gets value >0…


Closure properties


There is no nondeterministic Discounted automaton for the language Lm = min(La,Lb).

Proof: analogous argument.

Closure properties

Closure properties

min(L1,L2) = 1-max(1-L1,1-L2)

Closure properties

By analogous arguments (analysis of cycles).

Conclusion

• Quantitative generalization of languages to model programs/systems more accurately.

• Expressive power:

• Cut-point languages;

• {0,1} automata.

• Closure properties.

• Outlook: other/equivalent formalisms for quantitative specification ?

Thank you !

Questions ?

The end

Expressiveness and Closure Properties for Quantitative Languages

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