Results of the Golden 1960s Wolfgang Thomas Francqui Lecture, Mons, - - PowerPoint PPT Presentation

Results of the Golden 1960’s

Wolfgang Thomas Francqui Lecture, Mons, April 2013

Golden Times

Wolfgang Thomas

Overview

• 1. Background: MSO-Logic
• 2. B¨

uchi automata

• 3. Determinization
• 4. Tree automata
• 5. Rabin’s Tree Theorem
• 6. Regular trees

Wolfgang Thomas

Background: MSO-Logic

Wolfgang Thomas

Tarski’s Problem

G¨

• del’s and Turing’s results implied:

The first-order theory of (N, +, ·, 0, 1) is undecidable. Alfred Tarski asked: Is the monadic second-order theory of (N, +1, 0) decidable? Today we call this a model-checking problem: Is the model-checking problem

(N, +1, 0) | = ϕ?

w.r.t. MSO -logic decidable? Other names: S1S, SC, B¨ uchi’s arithmetic

Wolfgang Thomas

Alfred Tarski

Wolfgang Thomas

MSO Logic over (N, +1, 0)

We have first-order variables x, y, z, . . . ranging over natural numbers set variables X, Y, Z, . . . ranging over sets of natural numbers terms formed from first-order variables and 0 by application of “+1” atomic formulas s = t and X(t) for terms s, t and set variables X connectives ¬, ∨, ∧, →, ↔ and quantifiers ∃, ∀

Wolfgang Thomas

Example Formulas

Over graphs (V, E), we can express 3-colorability:

∃X1∃X2∃X3(Partition(X1, X2, X3) ∧∀x∀y(E(x, y) →

ij(Xi(x) ∧ Xj(y))))

Over (N, +1, 0) the induction axiom:

∀X(X(0) ∧ ∀y(X(y) → X(y + 1)) → ∀zX(z))

Over (N, +1, 0) the existence of automaton runs (e.g., for three states):

∃X1∃X2∃X3 (Partition(X1, X2, X3) ∧ transition and acceptance condition)

Wolfgang Thomas

Transitive Closure

The relation ≤ is the transitive closure of successor. We have x ≤ y iff for all sets X containing x and closed under successor, X(y) holds Notation: x < y, ∃ωy . . . ... for ∀x∃y(x < y ∧ . . ., etc. Taking closure under predecessor starting from y, a quantifier

• f finite sets suffices (weak MSO logic).

For any MSO-formula ϕ(z, z′), we write

ϕ∗(x, y) :=

∀X(X(x) ∧ ∀z, z′(X(z) ∧ϕ(z, z′) → X(z′)) → X(y))

Wolfgang Thomas

Example

“Each set with two successive elements contains an even number” First define “y is even”: Set ϕ2(z, z′)

:=

(z + 1) + 1 = z′

Even(y) := ϕ∗

2(0, y)

Then we take the following formula:

∀X(∃x(X(x) ∧ X(x + 1)) → ∃y(X(y) ∧ Even(y)))

Wolfgang Thomas

B¨ uchi Automata

Wolfgang Thomas

Wolfgang Thomas

• J. Richard B¨

uchi

Wolfgang Thomas

Sets versus Words

A set K ⊆ N can be identified with the infinite 0-1-word αK where αK(i) = 1 iff i ∈ K.

αP = 0 0 1 1 0 1 0 1 . . .

A tuple (K1, . . . Kn) corresponds to an ω-word over {0, 1}n

αEven,P = (1

0)(0 0)(1 1)(0 1)(1 0)(0 1) . . .

An MSO-formula ϕ(X1, . . . , Xn) defines an ω-language:

L(ϕ) = {α(K1,...,Kn) | (N, +1, 0) |

= ϕ[K1, . . . , Kn]}

L is MSO definable (over (N, +1, 0)) iff L = L(ϕ) for some

MSO-formula ϕ. Consider alphabets Σ = {0, 1}n for notational simplicity.

Wolfgang Thomas

B¨ uchi’s Version of “B¨ uchi Automaton”

Σω

1 : (∃r) · A[r(0)] ∧ ∀t B[i(t), r(t), r(t′)] ∧ (∃ωt) C[r(t)]

B¨ uchi showed closure properties of this formula class and derived that this is a normal form of formulas of S1S. Consequence: Each formula of S1S can be transformed into a B¨ uchi automaton. MTh(N, +1, 0) is decidable. This was new kind of “quantifier elimination”.

Wolfgang Thomas

B¨ uchi-Automata

Format: A = (Q, Σ, q0, ∆, F) with finite state-set Q, initial state q0, set F ⊆ Q of final states, transition relation ∆ ⊆ Q × Σ × Q

A accepts the input word α ∈ Σω if there is a run ̺ of A on α

such that ∃ωi ̺(i) ∈ F

L(A) := {α ∈ Σω | A accepts α}

is the ω-language recognized by A.

L is called B¨

uchi recognizable if L = L(A) for some B¨ uchi automaton A.

Wolfgang Thomas

Periodicity

Given A = (Q, Σ, q0, ∆, F) define

Wpq = {w ∈ Σ∗ | A : p

w

→ q}

Then

L(A) =

q∈F Wq0q · Wω q,q

An ω-language is B¨ uchi recognizable iff it is a finite union of

ω-languages U · Vω with regular U, V ⊆ Σ∗

B¨ uchi’s Theorem: An ω-language is MSO-definable iff it is B¨ uchi recognizable

Wolfgang Thomas

From Automata to MSO-Logic

q0 q1 q2 1 1 ϕA(X) :=

∃Y0∃Y1∃Y2(Partition(Y0, Y1, Y2) ∧ Y0(0) ∧∀x((Y0(x) ∧ ¬X(x) ∧ Y1(x + 1)) ∨(Y1(x) ∧ ¬X(x) ∧ Y0(x + 1)) ∨(Y1(x) ∧ X(x) ∧ Y2(x + 1)) ∨(Y2(x) ∧ X(x) ∧ Y2(x + 1))) ∧∀x∃y(x < y ∧ Y2(y)))

Wolfgang Thomas

From MSO-Logic to Automata

Proceed essentially by induction on formulas The difficult point is complementation. Given B¨ uchi’s Theorem, we have two immediate applications:

• 1. The MSO-theory of (N, +1, 0) is decidable.
• 2. MSO-formulas can be rewritten as EMSO-formulas.

Wolfgang Thomas

Complementation

B¨ uchi’s approach to complementation for his Σω

1 -formulas:

Represent the complement-ω-language as a finite union of sets U · Vω with regular U, V. As U, V he used equivalence classes of an equivalence relation:

u ∼A v :⇔ A : p

u

→ q ⇔ A : p

v

→ q

and A : p

u

→ q via F iff A : p

v

→ q via F ∼A is a finite congruence, and each ∼A-class is a regular.

For ∼A-classes U, V either UVω ⊆ L(A) or

UVω ∩ L(A) = O

Then: L(A) = {UVω|UVω ∩ L(A) = O}

Wolfgang Thomas

Ramsey’s Theorem

Given a coloring of all pairs (i, j) of natural numbers with

i < j, there is an infinite subset H ⊆ N and a fixed color c

such that each pair (i, j) with i, j ∈ H, i < j is colored with c. Given ∼A take as color for (i, j) the ∼A-class of α[i, j) The coloring is additive: the colors of (i, j) and (j, k) determine the color of (i, k). Consequence: Each ω-word belongs to a set UVω where U, V are ∼A-classes and moreover V · V ⊆ V.

Wolfgang Thomas

Determinization

Wolfgang Thomas

McNaughton’s Theorem

• R. McNaughton

Each B¨ uchi automaton an be transformed into a (deterministic) Muller automaton.

Wolfgang Thomas

Muller Automata

Format: A = (Q, Σ, q0, δ, F) with δ : Q × Σ → Q, F = {F1, . . . , Fk} where Fi ⊆ Q Acceptance: A accepts α iff for the unique run ̺ we have

• F∈F

(

• q∈F

∃ωi ̺(i) = q ∧

• q∈Q\F

¬∃ωi ̺(i) = q)

Write Aq for the det. B¨ uchi automaton (Q, Σ, q0, δ, {q}).

L(A) =

• F∈F

(

• q∈F

L(Aq) ∩

• q∈Q\F

L(Aq)) L is Muller recognizable iff L is a Boolean combination of

deterministic-B¨ uchi recognizable ω-languages.

Wolfgang Thomas

Deterministic B¨ uchi Automata in Logic

Given a finite automaton A. There is a monadic second-order formula ϕ(y) which expresses over an ω-word α: “the initial segment up to position y is accepted by A” In ϕ(y) one uses quantifiers “bounded by y”:

∃x(x ≤ y ∧ . . .), ∃X(∀z(X(z) → z ≤ y) ∧ . . .),

similarly for ∀.

L is deterministic-B¨

uchi recognizable iff it is definable in the form

∀x∃y(x < y ∧ϕ(y)) where ϕ(y) is bounded in y.

There are only two unbounded quantifiers (x and y), all other quantifiers are bounded to a finite domain.

Wolfgang Thomas

McNaughton’s Theorem Logically

Given a B¨ uchi recognizable ω-language of the form

U · Vω with ∼A-classes U, V where VV ⊆ V

The task is to express “α ∈ U · Vω” by a Boolean combination of formulas

∀x∃y(x < y ∧ϕ(y)) where ϕ(y) is bounded in y

This amounts to a drastic reduction of quantifier complexity.

Wolfgang Thomas

The Merge Relation

Given B¨ uchi automaton A and an ω-word α:

k ≃α k′(m) means: α[k, m) ∼A α[k′, m)

“k, k′ merge at m”

[For the following, more details are in: W. Th., Automata on Infinite Objects, Handbook of Theor. Comput Sci., Elsevier 1990]

Wolfgang Thomas

Down to Three Quantifiers

U, V stand for ∼A-classes, and V · V ⊆ V

Then: α ∈ U · Vω iff

∃k0(α[0, k0) ∈ U ∧ ∃ωk∃ℓ(α[k0, k) ∈ V ∧ k0 ∼A k(ℓ)))

Wolfgang Thomas

A Syntactic Detail

∃k0(α[0, k0) ∈ U ∧ ∃ωk∃ℓ(α[k0, k) ∈ V ∧ k0 ∼A k(ℓ)))

We want a formula

∃k0(α[0, k0) ∈ U ∧ ∃ωℓC(k0, ℓ) with C bounded in ℓ

Set C(k0, ℓ) :=

∃k(k0 < k < ℓ ∧ α[k0, k) ∈ V ∧ k0 ∼A k(ℓ) ∧ there is no m < ℓ with k0 ∼A k(m))

Wolfgang Thomas

Down to Two Quantifiers

Consider the condition

∃k0(α[0, k0) ∈ U ∧ ∃ωℓC(k0, ℓ))

We want to exchange ∃k0 and ∃ωℓ. Natural idea: Say ∃ωℓ∃k0 < ℓ C(k0, ℓ) and k0 is minimal. But the minimal k0 with α(0, k0) ∈ U may be incorrectly chosen; we can take one among those minimal k which finally do not merge.

Wolfgang Thomas

Two Unbounded Quantifiers Suffice

Let N be the numbers of ∼A-classes. The previous condition is equivalent to:

N

r=1 [there are infinitely may ℓ such that

among the smallest (k1, . . . , kr) ≤ ℓ with α[0, ki) ∈ U that do not merge at ℓ we have C(ki, ℓ) for some i

∧

there are only finitely many such ℓ where the maximum of those smallest (k1, . . . , kr) ≤ ℓ that do not merge at ℓ actually increases]

Wolfgang Thomas

A Combinatorial Formulation

Let C be an additive finite coloring of pairs over N. There are bounded formulas ϕc,ℓ(y) and ψℓ,c,d(y) such that A set {k0 < k1 . . . } exists with C(0, k0) = c and

C(ki, ki+1 = d

iff

(N, <, C) | = |C|

ℓ=1(∃x∀y > x ϕc,ℓ(y) ∧ ∀x∃y > x ψc,d,ℓ(y))

Wolfgang Thomas

A ”combinatorial result”?

Paul Erd¨

• s

Wolfgang Thomas

Parity Automata

accept with a special format of Boolean combination about infinitely many visits to states: Given a coloring c : Q → {0, . . . , m} a run ̺ is accepting if the maximal color occurring infinitely

• ften in it is even

We show later: Muller automata can be transformed into parity automata.

Wolfgang Thomas

Three Problems

Wolfgang Thomas

Tree Automata

Wolfgang Thomas

The Model T2

The structure of the infinite binary tree is

T2 = ({0, 1}∗, S0, S1, ε)

where Si is the i-th successor function:

S0(u) = u0, S1(u) = u1

The theory S2S is set of S2S-sentences which are true in T2 It is also called the monadic second-order theory (short: the monadic theory) of the infinite binary tree, denoted by

MTh2(T2)

A labelled binary tree can be presented as T : {0, 1}∗ → Σ.

Wolfgang Thomas

Example Formulas

Definition of x y (“node x is prefix of node y”):

ϕ∗

s(x, y) with ϕs(z, z′) := z0 = z′ ∨ z1 = z′

∀X

(X(y) ∧ ∀z(X(z0) → X(z)) ∧ ∀z(X(z1) → X(z))) → X(x)

• Chain(X)

(“X is linearly ordered by ”):

∀x∀y((X(x) ∧ X(y)) → (x y ∨ y x))

Path(X)

(“X is a path, i.e. a maximal chain”):

Chain(X) ∧ ¬∃Y(X ⊆ Y ∧ X Y ∧ Chain(Y)) X ⊆ Y: ∀z(X(z) → Y(z)) X = Y: ∀z(X(z) ↔ Y(z))

Wolfgang Thomas

Further Formulas

x < y

(“x is lexicographically before y”):

∃z(z0 x ∧ z1 y) ∨ ( x y ∧ x y)

Finite(X):

“each subset Y of X has a minimal and a maximal element w.r.t. <”

∀Y( (Y ⊆ X ∧ Y O) → (∃y “y is <-minimal in Y” ∧ ∃y “y is <-maximal in Y” ))

Wolfgang Thomas

Format of Tree Automata

A = (Q, Σ, q0, ∆, Acc) where

∆ ⊆ Q × Σ × Q × Q

A transition (q, a, q1, q2) allows the automaton in state q at an

a-labelled node u to proceed to states q1, q2 at the two

successor nodes u0, u1 A B¨ uchi / Muller / parity tree automaton

A = (Q, Σ, q0, ∆, F/F/c) accepts the tree t

if there exists a run ̺ of A on t such that on each path of ̺ the acceptance condition is satisfied.

Wolfgang Thomas

Example

T1 = {t ∈ Tω

{a,b} |

exists path through t with infinitely many b} recognized by a B¨ uchi tree automaton: Guess an appropriate path and on it check that infinitely often

b occurs by visiting in the next step a final state.

Wolfgang Thomas

Rabin’s Tree Theorem

Wolfgang Thomas

Michael O. Rabin

Wolfgang Thomas

Equivalence Logic vs. Automata

A tree language is definable in S2S iff it is recognizable by a parity tree automaton. Everything works as before, but complementation and emptiness test are now more difficult. We shall use theorems on infinite games, proved later (in the first lecture on games).

Wolfgang Thomas

Acceptance via Games

With any parity tree automaton A = (Q, Σ, q0, ∆, c) and any input tree t associate a game ΓA,t between two players “Automaton” and “Pathfinder” on the tree t

Wolfgang Thomas

ΓA,t

First Automaton picks a transition from ∆ which can serve to start a run at the root of the input tree. Then Pathfinder decides on a direction (left or right) to proceed to a son of the root. Then Automaton chooses again a transition for this node (compatible with the first transition and the input tree). Then Pathfinder reacts again by branching left or right from the momentary node, etc. Play gives a sequence of transitions (and hence a state sequence from Q), built up along a path chosen by Pathfinder. Automaton wins the play iff the constructed state sequence satisfies the parity condition. We speak of a “parity game”.

Wolfgang Thomas

Game Positions

Positions of Automaton are the triples

(tree node w, tree label t(w), state q at w)

By choice of a transition τ of the form (q, t(w), q′, q′′), a position of Pathfinder is reached. Positions of Pathfinder are the triples

(tree node w, tree label t(w), transition τ at w)

These positions with the moves define a “game graph”. Run Lemma: The tree automaton A accepts the input tree t iff in the parity game ΓA,t there is a positional winning strategy for player Automaton from the initial position (ε, t(ε), q0)

Wolfgang Thomas

Three Results on Parity Games

• 1. Parity games are positionally determined: From a given

start position one of the two players has a winning strategy, which moreover is positional.

• 2. The set of positions of a parity game graph from which a

given player wins is MSO-definable (in the MSO-language for game graphs).

• 3. For parity games over finite game graphs one can decide

for any position who wins from this position.

Wolfgang Thomas

Complementation Proof: Outline

Complementation of tree automata means to express the condition that a given automaton A does not accept t by acceptance of another automaton. Non-acceptance by A means non-existence of a winning strategy for Automaton in ΓA,t. Determinacy implies existence of a winning strategy for Pathfinder. We convert this strategy into an automaton strategy in a different game ΓB,t. This gives the desired complement automaton B.

Wolfgang Thomas

Applying Determinacy (Step 1)

Proof: Let A = (Q, Σ, q0, ∆, c) be a parity tree automaton. We find a parity tree automaton B accepting precisely the trees t ∈ Tω

Σ which are not accepted by A

Start with the following equivalences: For any tree t,

A does not accept t

iff (by Run Lemma) Automaton has no winning strategy from the initial position

(ε, t(ε), q0) in the parity game ΓA,t

iff (by Determinacy Theorem) (*) in ΓA,t, Pathfinder has a positional winning strategy from

(ε, t(ε), q0)

Wolfgang Thomas

Step 2

Reformulate (*) in the form “B accepts t” for some tree automaton B Pathfinder’s strategy is a function f from the set

{0, 1}∗ × Σ × ∆ of his vertices into the set {0, 1} of directions.

Decompose this function into a family

(fw : Σ × ∆ → {0, 1})

• f “local instructions”, parameterised by w ∈ {0, 1}∗

The set I of possible local instructions i : Σ × ∆ → {0, 1} is finite, Thus Pathfinder’s winning strategy can be coded by the

I-labelled tree s with s(w) = fw

Wolfgang Thomas

Step 3

Let s∧t be the corresponding (I × Σ)-labelled tree with

s∧t(w) = (s(w), t(w)) for w ∈ {0, 1}∗

Now (*) is equivalent to the following: There is an I-labelled tree s such that for all sequences τ0τ1 . . . of transitions chosen by Automaton and for all (in fact for the unique)

π ∈ {0, 1}ω determined by τ0τ1 . . . via the strategy

coded by s, the generated state sequence violates the parity condition.

Wolfgang Thomas

More Detail

A reformulation of this yields: (1) There is an I-labelled tree s such that s∧t satisfies: (2) for all π ∈ {0, 1}ω (3) for all τ0τ1 . . . ∈ ∆ω (4) if the sequence s|π of local instructions ap- plied to the sequence of tree labels t|π and to the transition sequence τ0τ1 . . . indeed produces the path π, then the state sequence determined by τ0τ1 . . . violates the parity condition. Condition (4) describes a property of ω-words over

I × Σ × ∆ × {0, 1}

which obviously can be checked by a sequential parity automaton M4, independently of t.

Wolfgang Thomas

The Input-free Case

An input-free parity tree automaton A = (Q, q0, ∆, c) with

∆ ⊆ Q × Q × Q defines a simpler game ΓA:

Automaton has positions in in Q and chooses transitions from

Q × Q × Q

Pathfinder has positions in ∆ and chooses directions. The corresponding game graph is finite! Run Lemma (input-free case): A admits at least one successful run iff Automaton has a winning strategy in ΓA from position q0. The first condition is checked effectively by the 3rd result on parity games.

Wolfgang Thomas

Wolfgang Thomas

Rabin’s Tree Theorem

Rabin’s Tree Theorem The theory S2S is decidable. Proof Consider an S2S-sentence ϕ It can be transformed into an input-free parity tree automaton

A such that

the unlabelled infinite binary tree T2 satisfies ϕ iff A has some successful run. The second condition can be checked effectively.

Wolfgang Thomas

Regular Trees

Wolfgang Thomas

Rabin’s Basis Theorem

Recall: A nonempty regular ω-language contains an ultimately periodic ω-word. A corresponding result holds for nonempty tree languages which are recognized by parity tree automata. Rabin’s Basis Theorem A nonempty tree language recognized by a parity tree automaton contains a regular tree. A tree t ∈ Tω

Σ is called regular if it is “finitely generated” in the

following sense: There is a deterministic finite automaton equipped with output which tells for any given input w ∈ {0, 1}∗ which label is at node w (i.e. the value t(w)).

Wolfgang Thomas

Examples

Wolfgang Thomas

Rabin’s Basis Theorem: Proof

Assume A = (Q, Σ, q0, ∆, c) is a parity tree automaton. Proceed to an “input-guessing” (and input-free) tree automaton A′ with states in Q × Σ:

A′ guesses an input tree and works on it as A does. A′ may have several initial states.

Then: The input-free automaton A′ admits a successful run iff

T(A O, and a tree in T(A) is extracted from the second

components of the run. Thus a regular tree is generated.

Wolfgang Thomas

Wolfgang Thomas

Looking Back

B¨ uchi automata, Muller automata, and parity tree automata provide different versions of quantifier elimination: to Σ1

1, to Bool(Π0 2).

Tree automata provide a less radical way of quantifier elimination than B¨ uchi automata: An S2S-formula ϕ(X1, . . . , Xn) can be transformed into a formula with two second-order quantifiers: “There is a run on the tree given by X1, . . . , Xn such that on each path the acceptance condition is satisfied.” In logical terminology this is a Σ1

2-condition.

Wolfgang Thomas