Backward Deterministic Finite-State Automata on Infinite Words
Thomas Wilke
6 March 2017
A General View of Automata
Automata run on objects.
Runs are defined by
a transition condition (local requirement),
an initial condition (front),
a final condition (end).
A General View of Determinism
Crucial requirements:
Locally, there is exactly one way to proceed.
For a given object, there is exactly one way to start.
Thus: There is exactly one initial run.
Acceptance is determined by the final condition (for the initial run).
If the final condition is closed under negation, then closure under complementation is
immediate.
Ordinary Finite-State ω-Automata
Run: sequence of states and symbols, s0 a0 s1 a1 . . .
Initial condition: set of states I , s0 ∈ I .
Transition condition: ∆ ⊆ S × A × S, ⟨si , ai , si+1 ⟩ ∈ ∆ for every i.
Final condition (Büchi): B ⊆ S × A × S, inf(s0 a0 s1 a1 . . . ) ∩ B ≠ ∅.
Final condition (generalized Büchi): B ⊆ 2S×A×S , inf(s0 a0 s1 a1 . . . ) ∩ B ≠ ∅ for every
B ∈ B.
Final condition (Muller): M ⊆ 2S×A×S , inf(s0 a0 s1 a1 . . . ) ∈ M.
Determinism: I is a singleton set and ∆ a function S × A → S.
(Generalized) Büchi: not closed under negation; Muller: closed under negation.
Backward Deterministic ω-Automata
Things are reversed:
condition
initial condition
final condition
transition condition
ordinary
I
B or B or M
∆
backward
B or B or M
I
∆
Determinism:
The transition relation ∆ is a function S ← A × S.
There is exactly one initial run.
Immediate closure under complementation.
Formal Definition (Carton & Michel)
Ingredients:
S a finite set of states
ρ∶ A × S → S a transition function
B ⊆ 2S×A×S a generalized Büchi condition (B ⊆ 2A×S is enough)
F ⊆ S a final set
Transition condition: s0 a0 s1 a1 . . . with ρ(ai , si+1 ) = si for every i.
Initial condition: B is satisfied.
Crucial (semantic) requirement: There is exactly one initial run.
Final condition: s0 ∈ F .
Immediate closure under complementation!
An Example
“Infinitely many a.”
Decidability
The semantic property “being a backward deterministic ω-automaton” is decidable in
polynomial time.
Backward deterministic iff:
For every u ∈ A+ , there is exactly one state q such that u is a loop at q.
Emptiness and equivalence are decidable in polynomial time (even in non-deterministic
logarithmic space).
Closure Properties
Complementation: immediate.
Intersection: ordinary product construction.
Union: via de Morgan with intersection and complementation.
Projection (and other operators): not immediate.
From Generalized Büchi to Büchi (Carton & Michel)
For every backward deterministic automaton with n states and m transition Büchi sets
there exists an equivalent backward deterministic Büchi automaton with n2m states.
There is an exponential lower bound.
Completeness (Carton & Michel)
Every regular ω-language is recognized by a backward deterministic Büchi automaton
with (3n)n states.
Two different proofs in the journal version of the original paper.
Involved construction with two essential ingredients:
run DAG’s and ranks
lifting construction
(Complete) Run DAG of an Ordinary Büchi Automaton
Classification of Vertices
Peeling
Kupferman & Vardi, Carton & Michel
Peeling:
1. Remove all finitary vertices.
2. Remove all B-free vertices.
Vertices on final runs stay.
If peeling does not change anything, all vertices are on some final run (B-recurring
vertices).
If a non-finitary vertex is removed, then on almost every level at least one vertex is
removed.
The B-recurring vertices remain after n peeling steps.
General Picture—Ranks
If a backward deterministic ω-automaton could “compute” ranks on complete run
DAGs with respect to any fixed Büchi automaton, then . . .
The Lifting Construction—Computing Finitary Vertices
Determining all finitary vertices
From Formulas to Automata, Inductively
Given ϕ; construct automaton for ϕ.
The automaton guesses which subformulas are true and which aren’t.
A state: set of subformulas of ϕ.
The backward deterministic transition function:
p ∈ ρ(a, s) iff p ∈ a
′
. . . ¬p . . .
′
ψ ∨ ψ ∈ ρ(a, s) iff ψ ∈ ρ(a, s) ∨ ψ ∈ ρ(a, s)
... ∧ ...
Xψ ∈ ρ(a, s) iff ψ ∈ s
Fψ ∈ ρ(a, s) iff ψ ∈ ρ(a, s) or Fψ
′
...G...
′
′
ψUψ ∈ ρ(a, s) iff ψ ∈ ρ(a, s) or (ψ ∈ ρ(a, s) and ψUψ ∈ s)
Büchi sets:
Fψ ∶ {Ψ ∣ ψ ∈ Ψ or Fψ ∉ Ψ}
′
′
′
ψUψ ∶ {Ψ ∣ ψ ∈ Ψ or ψUψ ∉ Ψ}
...G...
...R...
...R...
Counter-Free Automata
From Automata to Formulas, Inductively
Given a counter-free automaton, construct a formula.
First case, every symbol induces a permutation: every symbol induces the
identity.—Easy!
Second case, there is some symbol c such that q0 ∉ ρ(c, S).
The Separation Theorem
←
Ð←
Ð
F(q ∧ X G p)
←
Ð
G p ∧ p ∧ p Ůq
For every future-past temporal formula, there exists an equivalent separated formula
(boolean combination of pure past, pure future, and pure present formulas).
ω-Bimachines
Temporal formulas define functions ω → {0, 1}, example: p ∧ Xp
ω-bimachine: a finite-state automaton + a backward deterministic finite-state ω + . . .
Automata-Theoretic Proof of the Separation Theorem
Given a future-past temporal formula, construct an equivalent separated formula.
1. Turn the given formula into a counter-free ω-bimachine.
The lifting construction is important!
2. Turn the counter-free ω-bimachine into a separated formula.
The above tranlation from counter-free backward deterministic ω-automata to
temporal logic is important!
Typical Questions
Can a property be expressed
without X?
with F only?
without nesting U?
by nesting U at most k times?
Initial equivalence for backward deterministic automata
q ≡A q ′
iff
ρ(u, q) ∈ F ↔ ρ(u, q ′ ) ∈ F for each u ∈ A∗
Remark
≡A is a left congruence, that is, whenever q ≡A q ′ , then ρ(u, q) ≡A ρ(u, q ′ ) for all u.
Lemma
Whether or not q ≡A q ′ holds can be determined non-deterministically in logarithmic
space.
Characterization of F-expressibility
Theorem
Let L be a propositional ω-language recognized by a backward deterministic automaton
A. TFAE:
(A) L is F-expressible.
(B) 1) A/≡A doesn’t have one of the following patterns.
2) For all u and v , if u(0) = v (0) and occ(u) = occ(v ), then uÁ ≡A v Á.
The Example From Before
GFa ∧ GFb
1. There are two ≡A -classes (clouds).
2. The mixed words have their anchors in the orange class, the others in the yellow
class.
Conclusion
Backward deterministic ω-automata and temporal logic are
a good match!