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 Property/ Specification Yes/No Trace inclusion Program/ System Formula Every request is followed by a grant Finite automaton

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

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

Distance (R) 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

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 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 (reachability) (Büchi) (coBüchi) (v i  {0,1})

Some value functions (reachability) (Büchi) (coBüchi) (v i  {0,1})

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 L A = L A’.

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 L A = L A’. 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. Some known facts (CSL’08) 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 (a n b) ω for some n

Cut-point languages Disc: «disc. sum of a’s ≥ 1» is not ω-regular ambiguous word 1 p1p1 p2p2

Cut-point languages Disc: «disc. sum of a’s ≥ 1» is not ω-regular ambiguous word 1 From any two positions p 1 and p 2, there is a continuation accepted from p 1 but not from p 2 p1p1 p2p2

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. Expressive power of {0,1}-automata Store the value A B

is reducible to. Expressive power of {0,1}-automata A B can take finitely many different values. Store the value

is reducible to. Expressive power of {0,1}-automata A B

is reducible to. Expressive power of {0,1}-automata A B

is reducible to. Expressive power of {0,1}-automata A B

is reducible to. Expressive power of {0,1}-automata B A

Therefore for is reducible to. Expressive power of {0,1}-automata AB for all

Outline Weighted automata Expressive power Closure properties

Operations Operations on quantitative languages: shift(L 1,c)(w) = L 1 (w) + c scale(L 1,c)(w) = c·L 1 (w) (c>0)

Operations Operations on quantitative languages: shift(L 1,c)(w) = L 1 (w) + c scale(L 1,c)(w) = c·L 1 (w) (c>0) max(L 1,L 2 )(w) = max(L 1 (w),L 2 (w)) min(L 1,L 2 )(w) = min(L 1 (w),L 2 (w)) complement(L 1 )(w) = 1-L 1 (w)

Operations Operations on quantitative languages: shift(L 1,c)(w) = L 1 (w) + c scale(L 1,c)(w) = c·L 1 (w) (c>0) max(L 1,L 2 )(w) = max(L 1 (w),L 2 (w)) min(L 1,L 2 )(w) = min(L 1 (w),L 2 (w)) complement(L 1 )(w) = 1-L 1 (w) sum(L 1,L 2 )(w) = L 1 (w) + L 2 (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

Analogous results for boolean languages.

Closure properties There is no nondeterministic LimAvg automaton for the language L m = min(L a,L b ).

Closure properties There is no nondeterministic LimAvg automaton for the language L m = min(L a,L b ). Assume that L is definable by a LimAvg automaton C.

Closure properties There is no nondeterministic LimAvg automaton for the language L m = min(L a,L b ). 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 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 L m = min(L a,L b ).

Closure properties There is no nondeterministic LimAvg automaton for the language L m = min(L a,L b ). There is no nondeterministic Discounted automaton for the language L m = min(L a,L b ). Proof: analogous argument.

Closure properties

min(L 1,L 2 ) = 1-max(1-L 1,1-L 2 )

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