On Delay and Regret Determinization of Max-Plus
Automata
Guillermo A. Pérez
joint work with E. Filiot, I. Jecker, N. Lhote, and J.-F. Raskin
from Université libre de Bruxelles, Belgium
Verification Seminar @ Oxford
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q,
q0
Guillermo A. Pérez (ULB)
q1
On Delay & Regret Deteterminization
q2
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q, initial states I,
q0
Guillermo A. Pérez (ULB)
q1
On Delay & Regret Deteterminization
q2
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q, initial states I, final states F
q0
Guillermo A. Pérez (ULB)
q1
On Delay & Regret Deteterminization
q2
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q, initial states I, final states F
an alphabet A = {a, b, . . . }, and transition function ∆ ⊆ Q × A × Q
A
q0
A
q1
a
q2
b
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q, initial states I, final states F
an alphabet A = {a, b, . . . }, and transition function ∆ ⊆ Q × A × Q
a weight function w : ∆ → Z
A, 0
q0
A, 0
q1
a, 0
q2
b, 1
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
(Weighted) automata
A (non-deterministic weighted finite) automaton N = (Q, I, A, ∆, w , F )
states Q, initial states I, final states F
an alphabet A = {a, b, . . . }, and transition function ∆ ⊆ Q × A × Q
a weight function w : ∆ → Z
A, 0
q0
A, 0
q1
a, 0
q2
b, 1
it is deterministic if ‘∆ is functional’ and I is a singleton {qI }
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
2 / 23
Max-Plus Automata
A, −2
q1
a, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
Runs, acceptance, and value
A run (on input word) is a state-action sequence, e.g. q0 , b, q1 , a, q3 ; it
can can be initial, final, or accepting; it has a value e.g.
w (q0 , b, q1 , a, q3 ) = w (q0 , b, q1 ) + w (q1 , a, q3 ).
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
3 / 23
Max-Plus Automata
A, −2
q1
a, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
Language, function, maximal runs
A (finite) word α is accepted if the automaton has an accepting run
on α; the set LN of all such words is the language of the automaton.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
3 / 23
Max-Plus Automata
A, −2
q1
a, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
Language, function, maximal runs
A (finite) word α is accepted if the automaton has an accepting run
on α; the set LN of all such words is the language of the automaton.
The automaton induces a function JN K : LN → Z by mapping words
α to the maximal value of an accepting run on α.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
3 / 23
Max-Plus Automata
A, −2
q1
a, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
Language, function, maximal runs
A (finite) word α is accepted if the automaton has an accepting run
on α; the set LN of all such words is the language of the automaton.
The automaton induces a function JN K : LN → Z by mapping words
α to the maximal value of an accepting run on α.
An accepting run ρ on α is maximal if w (ρ) = JN K(ρ).
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
3 / 23
Outline/Summary
1
Determinizability
always possible for Boolean automata
deterministic max-plus automata strictly less expressive
deciding determinizability is open for general max-plus automata
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
4 / 23
Outline/Summary
1
Determinizability
always possible for Boolean automata
deterministic max-plus automata strictly less expressive
deciding determinizability is open for general max-plus automata
2
Bounded-delay determinizability
delay is popular in transducer theory
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
4 / 23
Outline/Summary
1
Determinizability
always possible for Boolean automata
deterministic max-plus automata strictly less expressive
deciding determinizability is open for general max-plus automata
2
Bounded-delay determinizability
delay is popular in transducer theory
3
Bounded-regret determinizability
quantitative generalization of good-for-games
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
4 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
D
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
D
≤k
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
Properties:
D
≤k
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
Properties:
D
≤k
k-delay determinization is decidable;
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization I
Bounded delay: stronger notion of determinization
Properties:
D
≤k
k-delay determinization is decidable;
A
Guillermo A. Pérez (ULB)
A determinizable ⇒ ∃k s.t. A k-delay
determinizable;
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
5 / 23
Bounded-delay determinization II
Ai
a,0
b,0
a,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A0 :
A1 :
Ai :
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
A0 :
q0
a,0
q1
q2
c,0
a,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
Ai :
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
Ai :
r = r0
r1
ri
abs val diff
a,0
a
b
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
Ai :
lag(r , r0 ) = 0,
b
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
b
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
lag(r , ri ) = i.
b
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
b
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
lag(r , ri ) = i.
lag(D, A) = min{k|∀ run r of D, ∃ run r 0 of A s.t. lag(r , r 0 ) ≤ k}
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
b
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
lag(r , ri ) = i.
lag(D, A0 ) = 0,
lag(D, A) = min{k|∀ run r of D, ∃ run r 0 of A s.t. lag(r , r 0 ) ≤ k}
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
b
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
lag(r , ri ) = i.
lag(D, A0 ) = 0,
lag(D, A1 ) = 1,
lag(D, A) = min{k|∀ run r of D, ∃ run r 0 of A s.t. lag(r , r 0 ) ≤ k}
Bounded-delay determinization II
Ai
Deterministic automaton D
b,0
a,0
b,0
q0
A0 :
a,0
q1
q2
c,0
c,0
a,1
b,-1
a,-1
c,1
a,i
b,-i
a,-i
c,i
A1 :
r = r0
r1
ri
abs val diff
a,0
a
b
Ai :
lag(r , r0 ) = 0,
lag(r , r1 ) = 1,
lag(r , ri ) = i.
lag(D, A0 ) = 0,
lag(D, A1 ) = 1,
lag(D, Ai ) = i.
lag(D, A) = min{k|∀ run r of D, ∃ run r 0 of A s.t. lag(r , r 0 ) ≤ k}
Bounded-delay determinization III
k-Delay Determinization Problem
Given A, does there exist D deterministic such that
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
7 / 23
Bounded-delay determinization III
k-Delay Determinization Problem
Given A, does there exist D deterministic such that
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
7 / 23
Bounded-delay determinization III
k-Delay Determinization Problem
Given A, does there exist D deterministic such that
JDK = JAK;
A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
7 / 23
Bounded-delay determinization III
k-Delay Determinization Problem
Given A, does there exist D deterministic such that
JDK = JAK;
D
Guillermo A. Pérez (ULB)
A
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
7 / 23
Bounded-delay determinization III
k-Delay Determinization Problem
Given A, does there exist D deterministic such that
JDK = JAK;
lag(D, A) ≤ k.
D
≤k
Guillermo A. Pérez (ULB)
A
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
7 / 23
Bounded-regret determinization I
0-regret: stronger notion of determinization
A
{B s.t. JBK = JAK}
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
8 / 23
Bounded-regret determinization I
0-regret: stronger notion of determinization
h-morphism
A
{B s.t. JBK = JAK}
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
8 / 23
Bounded-regret determinization I
0-regret: stronger notion of determinization
h-morphism
D
A
{B s.t. JBK = JAK}
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
8 / 23
Bounded-regret determinization I
0-regret: stronger notion of determinization
h-morphism
D
A
{B s.t. JBK = JAK}
Previous work
Decidable in polynomial time [Aminof, Kupferman, Lampert 2010].
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
8 / 23
Bounded-regret determinization II
r -regret: approximate determinization under a structural restriction
A
{B s.t. JBK = JAK}
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
9 / 23
Bounded-regret determinization II
r -regret: approximate determinization under a structural restriction
A
{B s.t. |JBK − JAK| ≤ r }
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
9 / 23
Bounded-regret determinization II
r -regret: approximate determinization under a structural restriction
h-morphism
A
{B s.t. |JBK − JAK| ≤ r }
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
9 / 23
Bounded-regret determinization II
r -regret: approximate determinization under a structural restriction
h-morphism
A
D
{B s.t. |JBK − JAK| ≤ r }
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
9 / 23
Bounded-regret determinization III: a game
A, −2
A
q1
A, 2
q2
q4
a, 0
Guillermo A. Pérez (ULB)
A, 0
b, −1
b, 0
q0
A, 2
q3
On Delay & Regret Deteterminization
A, 0
Verification Seminar @ Oxford
10 / 23
Bounded-regret determinization III: a game
A, −2
A
q1
A, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
Strategies
A strategy to resolve non-determinisim on the fly:
σ(q0 , a) = q1 , for all a ∈ A
σ(q0 . . . q1 , a) = q3 , for all a ∈ A
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
10 / 23
Bounded-regret determinization III: a game
A, −2
q1
A, 2
A, 2
A, 0
b, −1
b, 0
q0
A
q3
q2
q4
a, 0
A, 0
Strategies
A strategy to resolve non-determinisim on the fly:
σ(q0 , a) = q1 , for all a ∈ A
σ(q0 . . . q1 , a) = q3 , for all a ∈ A
JAK(bb) = 1, Val(σ(bb)) = 0
=⇒ regσ (bb) = 1
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
10 / 23
Bounded-regret determinization IV
r-Regret Determinization Problem
Given A, does there exist σ such that
σ(α) is accepting whenever α ∈ LA ;
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
11 / 23
Bounded-regret determinization IV
r-Regret Determinization Problem
Given A, does there exist σ such that
σ(α) is accepting whenever α ∈ LA ;
supα JAK(α) − Val(σ(α)) ≤ r
the LHS of the inequality is the regret of σ
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
11 / 23
Bounded-regret determinization IV
r-Regret Determinization Problem
Given A, does there exist σ such that
σ(α) is accepting whenever α ∈ LA ;
supα JAK(α) − Val(σ(α)) ≤ r
the LHS of the inequality is the regret of σ
Related problems
Good-for-games automata
Determinization by pruning
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
11 / 23
Example
a, k − 1
b, k
Guillermo A. Pérez (ULB)
a, −k
a, k
a, −k
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
12 / 23
Example
a, k − 1
b, k
a, −k
a, k
a, −k
For all k ∈ N,
1-regret and k-delay determinizable,
for all 0 ≤ j < k it is NOT j-delay determinizable.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
12 / 23
Algorithms
Bounded delay
Bounded regret
the 0 case
EXP-complete
polynomial time
general k/r
EXP-complete
EXP-complete
Determinization problems
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
13 / 23
Classical constructions for delay
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
15 / 23
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q0
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q0
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q0
a,0
q1 q2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q0
a,0
q1 q2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
b,2
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
q3
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
b,2
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
q3
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
b,2
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
q3
c,2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
b,2
q3
a,0
q2
c,2
q4
a,1
q0
a,0
q1 q2
q3
c,2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
a,0
q2
c,2
b,2
b,0
q3
q5
q4
a,1
q0
a,0
q1 q2
q3
c,2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
c,0
a,1
q1
b,2
q0
a,0
q2
c,2
b,2
b,0
q3
q5
q4
a,1
q0
a,0
q1 q2
q3
c,2
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
a,1
c,0
Guillermo A. Pérez (ULB)
q1
b,2
q0
a,0
q2
c,2
b,2
b,0
q3
q5
q4
c,0
On Delay & Regret Deteterminization
a,1
q0
a,0
q1 q2
q3
c,2
Verification Seminar @ Oxford
15 / 23
Subset construction
Theorem
A is 0-delay determinizable ⇔ P(A) is 0-regret determinizable.
a,0
b,0
q5
q4
a,1
c,0
0-del
Guillermo A. Pérez (ULB)
q1
b,2
q0
a,0
q2
c,2
b,2
b,0
q3
q5
a,1
q4
c,0
q0
0-reg
On Delay & Regret Deteterminization
a,0
q1 q2
q3
c,2
Verification Seminar @ Oxford
15 / 23
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
16 / 23
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
(q1 , 0)
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
(q0 , −1)
(q1 , 1)
(q0 , 0)
(q0 , 1)
(q3 , 0)
(q3 , 0)
(q2 , −1)
(q2 , 0)
(q2 , 1)
(q3 , 0)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
(q1 , 0)
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
(q1 , 1)
(q0 , 0)
(q3 , 0)
(q2 , −1)
(q2 , 0)
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
(q1 , 0)
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
(q1 , 1)
(q0 , 0)
(q3 , 0)
(q2 , −1)
(q2 , 0)
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
a,0
(q1 , 1)
(q0 , 0)
(q3 , 0)
(q2 , −1)
(q2 , 0)
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
c,1
a,0
(q1 , 1)
(q0 , 0)
(q3 , 0)
(q2 , −1)
(q2 , 0)
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
(q0 , 0)
(q3 , 0)
a,0 (q2 , −1)
c,1
a,-1
(q2 , 0)
a,-2
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
(q0 , 0)
(q3 , 0)
a,0 (q2 , −1)
c,1
a,-1
(q2 , 0)
a,-2
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
b,-2
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
(q0 , 0)
b,0
(q3 , 0)
a,0 (q2 , −1)
c,1
b,-1
a,-1
(q2 , 0)
a,-2
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
b,-2
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
(q0 , 0)
b,0
(q3 , 0)
a,0 (q2 , −1)
c,1
b,-1
a,-1
(q2 , 0)
a,-2
(q2 , 1)
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
b,-2
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
b,-1
b,0
(q0 , 0)
(q3 , 0)
a,0 (q2 , −1) c,0
c,1
a,-1
(q2 , 0)
a,-2
c,1
c,2
(q2 , 1)
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
16 / 23
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
b,-2
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
b,-1
b,0
(q0 , 0)
(q3 , 0)
a,0 (q2 , −1) c,0
c,1
a,-1
(q2 , 0)
a,-2
c,1
c,2
(q2 , 1)
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
16 / 23
Delay construction
Theorem
A is k-delay determinizable ⇔ δ k (A) is 0-delay determinizable.
(q1 , −1)
b,-2
a,2
(q1 , 0)
a,1
a,1
q1
b,-1
q0
a,-1
q3
q2
a,0
(q1 , 1)
b,-1
b,0
(q0 , 0)
(q3 , 0)
a,0 (q2 , −1) c,0
c,1
a,-1
(q2 , 0)
a,-2
c,1
c,2
(q2 , 1)
Theorem
k-delay determinizability is in EXP
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
16 / 23
Regret games between ∃ve and ∀dam
PO energy games
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
18 / 23
PO energy games
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
JAK(bbaaa) = 5
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
18 / 23
PO energy games
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
JAK(bbaaa) = 5
JAK(bbaaabbbb) = 6
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
18 / 23
PO energy games
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
JAK(bbaaa) = 5
JAK(bbaaabbbb) = 6
Theorem
The r -regret determinizability problem reduces to a PO energy game with
fixed initial credit.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
18 / 23
PO energy games II
The game
EGs are infinite-horizon quantitative games played by ∃ve against ∀dam.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
19 / 23
PO energy games II
The game
EGs are infinite-horizon quantitative games played by ∃ve against ∀dam.
Weighted automaton with observations
The arena in which an EG is played is a tuple (Q, qI , A, ∆, w , Obs).
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
19 / 23
PO energy games II
The game
EGs are infinite-horizon quantitative games played by ∃ve against ∀dam.
Weighted automaton with observations
The arena in which an EG is played is a tuple (Q, qI , A, ∆, w , Obs).
A,-1
b,-1
a,-1
A,+1
b,-1
a,-1
A,-1
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
19 / 23
PO energy games II
The game
EGs are infinite-horizon quantitative games played by ∃ve against ∀dam.
Weighted automaton with observations
The arena in which an EG is played is a tuple (Q, qI , A, ∆, w , Obs).
A,-1
b,-1
a,-1
A,+1
b,-1
a,-1
A,-1
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
19 / 23
PO energy games II
The game
EGs are infinite-horizon quantitative games played by ∃ve against ∀dam.
Weighted automaton with observations
The arena in which an EG is played is a tuple (Q, qI , A, ∆, w , Obs).
A,-1
b,-1
a,-1
A,+1
b,-1
a,-1
A,-1
∃ve wants to keep the running sum of weights non-negative,
∀dam wants the opposite.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
19 / 23
Quantitative Joker game
a, 1
b, 0
a, 1
b, 0
s0
a, 0
b, 1
s1
s2
a, 0
b, 1
Regret as an energy game
r becomes the initial credit
after each chosen transition, the energy is changed by difference of
the transitions’ weights
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
20 / 23
Quantitative Joker game
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
Q. Joker game (i.e. an EG with resets)
∀dam chooses a letter, ∃ve chooses a transition/state, ∀dam chooses his
transition
The weight of a turn: difference of weights
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
20 / 23
Quantitative Joker game
a, 1
b, 0
s0
a, 1
b, 0
a, 0
b, 1
s1
s2
a, 0
b, 1
Q. Joker game (i.e. an EG with resets)
∀dam chooses a letter, ∃ve chooses a transition/state, ∀dam chooses his
transition or plays joker
The weight of a turn: difference of weights
joker: reset energy level
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
20 / 23
Using the joker game
Losing
If ∀dam wins the Joker game, he wins the PO game, and the automaton is
not r -regret determinizable.
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
21 / 23
Using the joker game
Losing
If ∀dam wins the Joker game, he wins the PO game, and the automaton is
not r -regret determinizable.
Winning
If ∃ve wins the Joker game, her strategy is a witness of the automaton
being B-bounded:
∀max acc run prefix r , ∀run r 0 : w (r 0 ) − w (r ) ≤ B.
where r 0 and r are the same length
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
21 / 23
A second energy game: Example
A, −2
A
q1
A, 2
q2
q4
a, 0
Guillermo A. Pérez (ULB)
A, 0
b, −1
b, 0
q0
A, 2
q3
On Delay & Regret Deteterminization
A, 0
Verification Seminar @ Oxford
22 / 23
A second energy game: Example
A, −2
A
q1
A, 2
q2
q4
a, 0
Guillermo A. Pérez (ULB)
A, 0
b, −1
b, 0
q0
A, 2
q3
A, 0
D
A, 0
a, 0
b, 1
A, 0
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
22 / 23
A second energy game: Example
A, −2
A
q1
A, 2
A, 0
b, −1
b, 0
q0
A, 2
q3
q2
q4
a, 0
A, 0
D
A, 0
a, 0
b, 1
A, 0
(No Joker) energy game
∀dam chooses a letter, ∃ve chooses a transition/state, ∀dam chooses his
transition
The weight of a turn: difference of weights
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
22 / 23
Conclusion
Bounded delay
Bounded regret
Guillermo A. Pérez (ULB)
the 0 case
EXP-complete
polynomial time
On Delay & Regret Deteterminization
general k/r
EXP-complete
EXP-complete
Verification Seminar @ Oxford
23 / 23
Conclusion
Bounded delay
Bounded regret
the 0 case
EXP-complete
polynomial time
general k/r
EXP-complete
EXP-complete
Are the regret and Joker games equivalent?
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
23 / 23
Conclusion
Bounded delay
Bounded regret
the 0 case
EXP-complete
polynomial time
general k/r
EXP-complete
EXP-complete
Are the regret and Joker games equivalent?
What is the complexity of deciding if ∃r s.t. an automaton is r -regret
determinizable?
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
23 / 23
Conclusion
Bounded delay
Bounded regret
the 0 case
EXP-complete
polynomial time
general k/r
EXP-complete
EXP-complete
Are the regret and Joker games equivalent?
What is the complexity of deciding if ∃r s.t. an automaton is r -regret
determinizable?
What happens when delay and regret are defined with respect to ratio
instead of difference?
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
23 / 23
Conclusion
Bounded delay
Bounded regret
the 0 case
EXP-complete
polynomial time
general k/r
EXP-complete
EXP-complete
Are the regret and Joker games equivalent?
What is the complexity of deciding if ∃r s.t. an automaton is r -regret
determinizable?
What happens when delay and regret are defined with respect to ratio
instead of difference?
Is general determinizability decidable?
Guillermo A. Pérez (ULB)
On Delay & Regret Deteterminization
Verification Seminar @ Oxford
23 / 23