LTL with the Freeze Quantifier
and Register Automata
Stéphane Demri1
1 LSV,
Ranko Lazić2
CNRS & ENS Cachan & INRIA Futurs
2 University
of Warwick
LICS, August 2006, Seattle
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Overview
LTL with freeze
Models
Syntax
Example
Register automata
Introduction
Alternation
Expressiveness
FO over data words
Register automata
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Models
Syntax
Example
Data words
Finite alphabet & infinite domain:
Infinite-state computations, XML documents, Timed words, . . .
a
a
b
b
a
b
URL1 URL2 URL1 URL2 URL3 URL3
a a b b a b
3 2.5 3 2.5 4 4
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Models
Syntax
Example
Data words
Finite alphabet & infinite domain:
Infinite-state computations, XML documents, Timed words, . . .
a
a
b
b
a
b
URL1 URL2 URL1 URL2 URL3 URL3
a a b b a b
3 2.5 3 2.5 4 4
a a b b a b
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Models
Syntax
Example
LTL↓n
φ ::= > | a | ↑r ∼ |
¬φ | φ ∧ φ |
O(φ, . . . , φ) |
↓r φ
r
∈ {1, . . . , n}
O ∈ {X, X−1 , F, F−1 , U, U−1 , . . .}
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Models
Syntax
Example
Freeze quantifier
Timed logics
Hybrid logics
Modal logics
x · φ(x)
↓p φ(p)
hλx · φ(x)i(c)
Stéphane Demri, Ranko Lazić
[Alur & Henzinger, JACM ’94]
[Goranko, JoLLI ’96]
[Fitting, JLC ’02]
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Models
Syntax
Example
Example
F(a ∧ ↓1 XF(a ∧ ↑1 ∼))
a a b b a b
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Introduction
Alternation
Register Automata
3
a ∧ ↓1
1
a ∧ ↑1 ∼
>
1-way or 2-way, Nondeterministic or Alternating, n registers
Over infinite data words: weak parity acceptance.
Special case of Büchi and co-Büchi acceptance.
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Register
Pebble
Timed
Data
automata
automata
automata
automata
Introduction
Alternation
[Kaminski & Francez, TCS ’94]
[Neven, Schwentick & Vianu, ToCL ’04]
[Alur & Dill, TCS ’94]
[Bouyer, Petit & Thérien, I & C ’03]
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Introduction
Alternation
a a b b a b
G (a ⇒ ↓1 X((a ⇒ ¬↑1 ∼) U (b ∧ ↑1 ∼)))
a ⇒ ↓1
4
end∨
Stéphane Demri, Ranko Lazić
b ∧ ↑1 ∼
3
>
a ⇒ ↑1 6∼
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Introduction
Alternation
[Kaminski & Francez, TCS ’94],
[Neven, Schwentick & Vianu, ToCL ’04]
1ARA
1NRA
1URA
1DRA
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
Introduction
Alternation
[Kaminski & Francez, TCS ’94],
[Neven, Schwentick & Vianu, ToCL ’04]
2ARA
2NRA
2URA
2DRA
Provided LogSpace ⊂ NLogSpace ⊂ PTime.
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
FO over data words
Register automata
FO(∼, <, +1)
Theorem
LogSpace
Simple LTL↓1 (X, X−1 , F, F−1 )
−→
PSpace
←−
FO2 (∼, <, +1)
[Etessami, Vardi & Wilke, I & C ’02]
LTL(X, X−1 , F, F−1 ) is equivalent to FO2 (<, +1).
[Bojańczyk et al., LICS ’06]
FO2 (∼, <, +1) SAT is as hard as Petri net Reachability.
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
FO over data words
Register automata
Simple LTL↓1 (X, X−1, F, F−1)
F(a ∧ ↓1 XF(a ∧ ↑1 ∼))
∃x (a(x) ∧ ∃y (x < y ∧ a(y ) ∧ x ∼ y ))
F(a ∧ ↓1 XF(b ∧ XF(a ∧ ↑1 ∼)))
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
FO over data words
Register automata
From LTL↓ to register automata
Theorem
LTL↓n (X, X−1 , U, U−1 )
LogSpace
−→
2ARAn .
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
[Kaminski & Francez, TCS ’94]
‘. . . it is very likely that
[decidability of language containment of 1NRA in 1NRA 1 ]
can be extended to infinite data words.’
[French, TIME ’03], [Demri, Lazić & Nowak, TIME ’05],
[Lisitsa & Potapov, TIME ’05]:
SAT<ω
Registers
LTL↓ (X, F)
LTL↓ (X, U)
↓
LTL (X, F, F−1 )
1
SATω
2
Σ01 -comp.
Stéphane Demri, Ranko Lazić
1
2
Σ11 -comp.
Σ11 -comp.
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
LTL↓1 (X, U) SAT
LogSpace
1ARA1 ¬EMP
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Incrementing Counter Automata
inc.c
ifzero.c
a
ε
b
ifzero.c
dec.c
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
LTL↓1 (X, U) SAT
LogSpace
1ARA1 ¬EMP
PSpace
ICA ¬EMP
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Proof.
1. Quotient runs by ∼.
2. Represent levels and steps using counters.
3. For infinite data words, use [Miyano & Hayashi, TCS ’84]:
weak parity alternating 7→ Büchi nondeterministic.
4. Incrementing errors cannot cause a false acceptance.
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
end∨
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
q0
3
b ∧ ↑1 ∼
>
a ⇒ ↑1 6∼
q
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
end∨
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
q0
3
b ∧ ↑1 ∼
>
a ⇒ ↑1 6∼
q0, 0
q
- q
a
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
q0, 0
q
a
q0
3
b ∧ ↑1 ∼
>
a ⇒ ↑1 6∼
end∨
- q
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
q0, 0
q0, 1
- q
a
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
q
- q
a
q0, 0
q0, 1
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
q
- q
a
q0, 0
q0, 1
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
- q
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
q
- q
a
q0, 0
q0, 1
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
q0, 4
- q
b
- q
a
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
q
- q
a
q0, 0
q0, 1
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
q0, 4
- q
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
0
- q
a
q0, 0
q0, 1
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
q0, 4
- q
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 0
q0, 1
0
- 1
a
- q
a
- q0, 1
- q
b
Stéphane Demri, Ranko Lazić
q0, 4
- q
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
- q0, 1
0
- 1
a
- 2
a
- q
b
Stéphane Demri, Ranko Lazić
q0, 4
- q
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 4
0
- 1
a
- 2
a
- 1
b
Stéphane Demri, Ranko Lazić
- q
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
q0, 4
0
- 1
a
- 2
a
- 1
b
Stéphane Demri, Ranko Lazić
- 0
b
- q
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
0
- 1
a
- 2
a
- 1
b
Stéphane Demri, Ranko Lazić
- 0
b
- 1
a
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
q
4
a ⇒ ↓1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ∧ ↑1 ∼
q0
3
>
a ⇒ ↑1 6∼
end∨
0
- 1
a
- 2
a
- 1
b
Stéphane Demri, Ranko Lazić
- 0
b
- 1
a
- 0
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
inc.c
ifzero.c
a
ε
b
ifzero.c
dec.c
0
- 1
a
- 2
a
- 1
b
Stéphane Demri, Ranko Lazić
- 0
b
- 1
a
- 0
b
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Infinite words
G (a ⇒ ↓1 X((a ⇒ ¬↑1 ∼) U (b ∧ ↑1 ∼)))
We should accept:
a
a
b
a
b
a
b
a
b
···
a
a
b
a
b
a
a
a
a
···
but reject:
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
inc.c2 inc.c1 a
ifzero.c1
ε
dec.c2 dec.c1
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
b ε
Stéphane Demri, Ranko Lazić
ifzero.c2
ε
a
inc.c1 inc.c2
b dec.c1 dec.c2
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
LTL↓1 (X, F) SAT
LTL↓1 (X, U) SAT
LogSpace
1ARA1 ¬EMP
1NRA1 ¬UNI
LogSpace
PSpace
LogSpace
ICA ¬EMP
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Proof.
Encode computations of ICA as data words:
inc.c inc.c dec.c inc.c dec.c dec.c iszero.c
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Proof.
Encode computations of ICA as data words:
inc.c inc.c dec.c inc.c dec.c dec.c iszero.c
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Proof.
Encode computations of ICA as data words:
inc.c inc.c dec.c inc.c dec.c dec.c iszero.c
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
LTL↓1 (X, F) SAT
LTL↓1 (X, U) SAT
LogSpace
1ARA1 ¬EMP
1NRA1 ¬UNI
LogSpace
PSpace
LogSpace
ICA ¬EMP
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Theorem
¬EMP<ω
¬EMPω
Minsky CA
Σ01 -complete
Σ11 -complete
Incrementing CA
R (a), not PR (b)
Π01 -complete (c)
Proof.
(a) Reverse computations, and obtain Reset Petri net Coverability
[Dufourd, Finkel & Schnoebelen, ICALP ’98].
(b) Reverse computations, and adapt [Schnoebelen, IPL ’02]:
Reachability is not PR for Lossy Channel Systems.
(c) Adapt [Ouaknine & Worrell, FoSSaCS ’06]:
Recur. State for Insertion Chan. Mach. with Empt. Test.
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
[Kaminski & Francez, TCS ’94]
‘. . . it is very likely that
[decidability of language containment of 1NRA in 1NRA 1 ]
can be extended to infinite data words.’
[French, TIME ’03], [Demri, Lazić & Nowak, TIME ’05],
[Lisitsa & Potapov, TIME ’05]:
SAT<ω
Registers
LTL↓ (X, F)
LTL↓ (X, U)
↓
LTL (X, F, F−1 )
1
SATω
2
Σ01 -comp.
Stéphane Demri, Ranko Lazić
1
2
Σ11 -comp.
Σ11 -comp.
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
[Demri & Lazić, LICS ’06]
1NRA1 ¬UNIω is Π01 -complete.
[French, TIME ’03], [Demri, Lazić & Nowak, TIME ’05],
[Lisitsa & Potapov, TIME ’05]:
SAT<ω
Registers
LTL↓ (X, F)
LTL↓ (X, U)
↓
LTL (X, F, F−1 )
1
SATω
2
Σ01 -comp.
Stéphane Demri, Ranko Lazić
1
2
Σ11 -comp.
Σ11 -comp.
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
[Demri & Lazić, LICS ’06]
1NRA1 ¬UNIω is Π01 -complete.
[French, TIME ’03], [Demri, Lazić & Nowak, TIME ’05],
[Lisitsa & Potapov, TIME ’05], [Demri & Lazić, LICS ’06]:
SAT<ω
Registers
LTL↓ (X, F)
LTL↓ (X, U)
↓
LTL (X, F, F−1 )
1
R \ PR
R \ PR
Σ01 -comp.
SATω
2
Σ01 -comp.
Σ01 -comp.
Σ01 -comp.
Stéphane Demri, Ranko Lazić
1
Π01 -comp.
Π01 -comp.
Σ11 -comp.
2
Σ11 -comp.
Σ11 -comp.
Σ11 -comp.
LTL with the Freeze Quantifier and Register Automata
LTL with freeze
Register automata
Expressiveness
Complexity of satisfiability
From RA to ICA
From ICA to LTL with freeze and RA
Complexity results
Characterisation of languages
Characterisation of languages
Corollary
{L(C) : C an Incrementing CA}
= {f (str(L(φ))) : f a homomorphism, φ in LTL ↓1 (X, F)}
= {f (str(L(φ))) : f a homomorphism, φ in LTL ↓1 (X, U)}
Stéphane Demri, Ranko Lazić
LTL with the Freeze Quantifier and Register Automata