Expressiveness and
Closure Properties for
Quantitative Languages
Krishnendu Chatterjee, IST Austria
Laurent Doyen, ULB Belgium
Tom Henzinger, EPFL Switzerland
LICS 2009
Model-Checking
Program/
Property/
System Specification
Satisfaction
Relation
Yes / No
-perhaps a proof
-perhaps some counterexamples
Model-Checking
Finite
automaton
Program/
Property/
System Specification
Trace
inclusion
Yes/No
Formula
Every request is
followed by a grant
Model-Checking
Finite
automaton
Program/
Property/
System Specification
Formula
Every request is
followed by a grant
Trace
inclusion
Yes/No
- a trace is either good or bad
Model-checking is boolean
Quantitative Analysis
Finite
automaton
Program/
Property/
System Specification
Formula
Every request is
followed by a grant
Quantitative
Analysis
Value (R)
-Measure of “fit” between system and spec
-e.g. average number of requests
immediately granted
Quantitative Analysis
Finite
automaton
Program/
System #1
Program/
System #2
Every request is
followed by a grant
Quantitative
Analysis
Distance (R)
- Comparing two implementations
e.g. cost or quality measure
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”
Quantitative languages
A language is a boolean function:
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 function
Value 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
(vi  {0,1})
(reachability)
(Büchi)
(coBüchi)
Some value functions
(vi  {0,1})
(reachability)
(Büchi)
(coBüchi)
Outline
• Weighted automata
• Expressive power
• Closure properties
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)
Some known facts (CSL’08)
cannot be reduced to
cannot be reduced to
cannot be determinized.
cannot be determinized.
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
p1
p2
From any two positions p1 and p2, there is a
continuation accepted from p1 but not from p2
Cut-point Languages
Cut-point languages of LimAvg and Discounted
can be non-ω-regular
Cut-point languages for deterministic LimAvg-automata
are studied in [Alur/Degorre/Maler/Weiss’09]
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
LimAvg:
s.c.c. decomposition
Each s.c.c. defines an interval of values.
Make accepting those s.c.c. with interval above
Cut-point Languages
Disc:
after sufficiently long prefix, decision can be taken
Either value is
or value is
, then accept
, the reject
Expressive power of {0,1}-automata
is reducible to
is not reducible to
.
.
Expressive power of {0,1}-automata
is reducible to
.
B
A
Store the value
Expressive power of {0,1}-automata
is reducible to
.
B
A
Store the value
can take finitely many different values.
Expressive power of {0,1}-automata
is reducible to
.
B
A
Expressive power of {0,1}-automata
is reducible to
.
B
A
Expressive power of {0,1}-automata
is reducible to
.
B
A
Expressive power of {0,1}-automata
is reducible to
.
B
A
Expressive power of {0,1}-automata
is reducible to
A
B
for all
Therefore
.
for
Outline
• Weighted automata
• Expressive power
• Closure properties
Operations
Operations on quantitative languages:
• shift(L1,c)(w) = L1(w) + c
• scale(L1,c)(w) = c·L1(w)
(c>0)
Operations
Operations on quantitative languages:
• 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
Operations on quantitative languages:
• 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
There is no nondeterministic LimAvg automaton for the
language Lm = min(La,Lb).
Assume that L is definable by a LimAvg automaton C.
Closure properties
There is no nondeterministic LimAvg automaton for the
language Lm = min(La,Lb).
Assume that L is definable by a LimAvg automaton C.
Then, some a-cycle or b-cycle in C has average weight >0.
(consider the word
for
large)
Closure properties
There is no nondeterministic LimAvg automaton for the
language Lm = min(La,Lb).
Assume that L is definable by a LimAvg automaton C.
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 LimAvg automaton for the
language Lm = min(La,Lb).
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 ?
The end
Thank you !
Questions ?