expressiveness and closure properties for quantitative languages krishnendu chatterjee, ist austria...
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![Page 1: Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST Austria Laurent Doyen, ULB Belgium Tom Henzinger, EPFL Switzerland](https://reader035.vdocuments.us/reader035/viewer/2022062517/56649f135503460f94c275d0/html5/thumbnails/1.jpg)
Expressiveness and Closure Properties for
Quantitative Languages
Krishnendu Chatterjee, IST Austria
Laurent Doyen, ULB Belgium
Tom Henzinger, EPFL Switzerland
LICS 2009
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Model-Checking
Property/ Specification
Yes / No
Satisfaction Relation
Program/ System
-perhaps a proof -perhaps some counterexamples
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Model-Checking
Property/ Specification
Yes/No
Trace inclusion
Program/ System
Formula
Every request is followed by a grant
Finite automaton
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Model-Checking
Property/ Specification
Yes/No
Trace inclusion
Program/ System
Finite automaton
Model-checking is boolean
Formula
Every request is followed by a grant
- a trace is either good or bad
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Quantitative Analysis
Property/ Specification
Value (R)
Quantitative Analysis
Program/ System
Finite automaton
Formula
Every request is followed by a grant
-Measure of “fit” between system and spec
-e.g. average number of requests immediately granted
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Distance (R)
Quantitative Analysis
Quantitative Analysis
Program/ System #1
Finite automaton
- Comparing two implementations
e.g. cost or quality measure
Program/ System #2
Every request is followed by a grant
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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”
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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).
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Outline
• Weighted automata
• Expressive power
• Closure properties
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Weighted automata
Quantitative languages are generated by weighted automata.
Weight function
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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
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Some value functions
(reachability)
(Büchi)
(coBüchi)
(vi {0,1})
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Some value functions
(reachability)
(Büchi)
(coBüchi)
(vi {0,1})
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Outline
• Weighted automata
• Expressive power
• Closure properties
![Page 15: Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST Austria Laurent Doyen, ULB Belgium Tom Henzinger, EPFL Switzerland](https://reader035.vdocuments.us/reader035/viewer/2022062517/56649f135503460f94c275d0/html5/thumbnails/15.jpg)
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’.
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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)
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cannot be determinized.
cannot be determinized.
Some known facts (CSL’08)
cannot be reduced to
cannot be reduced to
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Reducibility relations
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Cut-point languages
Words with value above some threshold:
ω-regular for Sup, LimSup, LimInf
can be non-ω-regular for LimAvg and Discounted
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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
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Cut-point languages
Disc:
«disc. sum of a’s ≥ 1»
is not ω-regular
ambiguous word
1
p1 p2
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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
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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.
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Cut-point Languages
isolated cut-point
Isolated cut-point languages are robust
Isolated cut-point languages are ω-regular(for deterministic automata)
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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
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Either value is , then accept
or value is , the reject
Cut-point Languages
Disc:
after sufficiently long prefix, decision can be taken
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is reducible to .
Expressive power of {0,1}-automata
is not reducible to .
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is reducible to .
Expressive power of {0,1}-automata
Store the value
A
B
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is reducible to .
Expressive power of {0,1}-automata
A
B
can take finitely many different values.
Store the value
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is reducible to .
Expressive power of {0,1}-automata
A
B
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is reducible to .
Expressive power of {0,1}-automata
A
B
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is reducible to .
Expressive power of {0,1}-automata
A
B
![Page 33: Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST Austria Laurent Doyen, ULB Belgium Tom Henzinger, EPFL Switzerland](https://reader035.vdocuments.us/reader035/viewer/2022062517/56649f135503460f94c275d0/html5/thumbnails/33.jpg)
is reducible to .
Expressive power of {0,1}-automata
B
A
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Therefore for
is reducible to .
Expressive power of {0,1}-automata
A B
for all
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Outline
• Weighted automata
• Expressive power
• Closure properties
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Operations
Operations on quantitative languages:
• shift(L1,c)(w) = L1(w) + c
• scale(L1,c)(w) = c·L1(w) (c>0)
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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)
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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)
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Closure properties
All classes of weighted automata are closed under shift and scale.
All classes of nondeterministic weighted automata are closed under max.
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Closure properties
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Closure properties
Analogous results for boolean languages.
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Closure properties
There is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).
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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.
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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)
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Closure properties
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…
There is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).
![Page 46: Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST Austria Laurent Doyen, ULB Belgium Tom Henzinger, EPFL Switzerland](https://reader035.vdocuments.us/reader035/viewer/2022062517/56649f135503460f94c275d0/html5/thumbnails/46.jpg)
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.
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Closure properties
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Closure properties
min(L1,L2) = 1-max(1-L1,1-L2)
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Closure properties
By analogous arguments (analysis of cycles).
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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 ?
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Thank you !
Questions ?
The end
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