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¨ odel’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: ∃ X 1 ∃ X 2 ∃ X 3 ( Partition ( X 1 , X 2 , X 3 ) ∧∀ x ∀ y ( E ( x , y ) → � i � j ( X i ( x ) ∧ X j ( 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): ∃ X 1 ∃ X 2 ∃ X 3 ( Partition ( X 1 , X 2 , X 3 ) ∧ 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 , ∃ ω y . . . ... for ∀ x ∃ y ( x < y ∧ . . . , etc. Taking closure under predecessor starting from y , a quantifier of 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 ϕ 2 ( 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 α K where α K ( i ) = 1 iff i ∈ K . α P = 0 0 1 1 0 1 0 1 . . . A tuple ( K 1 , . . . K n ) corresponds to an ω -word over { 0, 1 } n α Even, P = ( 1 0 )( 0 0 )( 1 1 )( 0 1 )( 1 0 )( 0 1 ) . . . An MSO-formula ϕ ( X 1 , . . . , X n ) defines an ω -language: L ( ϕ ) = { α ( K 1 ,..., K n ) | ( N , + 1, 0 ) | = ϕ [ K 1 , . . . , K n ] } 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” 1 : ( ∃ r ) · A [ r ( 0 )] ∧ ∀ t B [ i ( t ) , r ( t ) , r ( t ′ )] ∧ ( ∃ ω t ) C [ r ( t )] Σ ω 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 uchi automaton. MTh ( N , + 1, 0 ) is decidable. B¨ This was new kind of “quantifier elimination”. Wolfgang Thomas

B¨ uchi-Automata Format: A = ( Q , Σ , q 0 , ∆ , F ) with finite state-set Q , initial state q 0 , set F ⊆ Q of final states, transition relation ∆ ⊆ Q × Σ × Q A accepts the input word α ∈ Σ ω if there is a run ̺ of A on α such that ∃ ω i ̺ ( i ) ∈ F L ( A ) : = { α ∈ Σ ω | A accepts α } is the ω -language recognized by A . uchi recognizable if L = L ( A ) for some B¨ L is called B¨ uchi automaton A . Wolfgang Thomas

Periodicity Given A = ( Q , Σ , q 0 , ∆ , F ) define W pq = { w ∈ Σ ∗ | A : p w → q } Then q ∈ F W q 0 q · W ω L ( A ) = � q , q An ω -language is B¨ uchi recognizable iff it is a finite union of ω -languages U · V ω 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 1 0 1 q 0 q 1 q 2 0 ϕ A ( X ) : = ∃ Y 0 ∃ Y 1 ∃ Y 2 ( Partition ( Y 0 , Y 1 , Y 2 ) ∧ Y 0 ( 0 ) ∧∀ x (( Y 0 ( x ) ∧ ¬ X ( x ) ∧ Y 1 ( x + 1 )) ∨ ( Y 1 ( x ) ∧ ¬ X ( x ) ∧ Y 0 ( x + 1 )) ∨ ( Y 1 ( x ) ∧ X ( x ) ∧ Y 2 ( x + 1 )) ∨ ( Y 2 ( x ) ∧ X ( x ) ∧ Y 2 ( x + 1 ))) ∧∀ x ∃ y ( x < y ∧ Y 2 ( 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 uchi’s approach to complementation for his Σ ω B¨ 1 -formulas: Represent the complement- ω -language as a finite union of sets U · V ω with regular U , V . As U , V he used equivalence classes of an equivalence relation: u v u ∼ A v : ⇔ A : p → q ⇔ A : p → q u v and A : p → q via F iff A : p → q via F ∼ A is a finite congruence, and each ∼ A -class is a regular. For ∼ A -classes U , V either UV ω ⊆ L ( A ) or UV ω ∩ L ( A ) = � O Then: L ( A ) = � { UV ω | UV ω ∩ 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 UV ω 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 , q 0 , δ , F ) with δ : Q × Σ → Q , F = { F 1 , . . . , F k } where F i ⊆ Q Acceptance: A accepts α iff for the unique run ̺ we have � � ∃ ω i ̺ ( i ) = q ∧ � ¬∃ ω i ̺ ( i ) = q ) ( F ∈F q ∈ F q ∈ Q \ F Write A q for the det. B¨ uchi automaton ( Q , Σ , q 0 , δ , { q } ) . � � � L ( A ) = ( L ( A q ) ∩ L ( A q )) F ∈F q ∈ F q ∈ Q \ F 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 · V ω with ∼ A -classes U , V where VV ⊆ V The task is to express “ α ∈ U · V ω ” 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 · V ω iff ∃ k 0 ( α [ 0, k 0 ) ∈ U ∧ ∃ ω k ∃ ℓ ( α [ k 0 , k ) ∈ V ∧ k 0 ∼ A k ( ℓ ))) Wolfgang Thomas

A Syntactic Detail ∃ k 0 ( α [ 0, k 0 ) ∈ U ∧ ∃ ω k ∃ ℓ ( α [ k 0 , k ) ∈ V ∧ k 0 ∼ A k ( ℓ ))) We want a formula ∃ k 0 ( α [ 0, k 0 ) ∈ U ∧ ∃ ω ℓ C ( k 0 , ℓ ) with C bounded in ℓ Set C ( k 0 , ℓ ) : = ∃ k ( k 0 < k < ℓ ∧ α [ k 0 , k ) ∈ V ∧ k 0 ∼ A k ( ℓ ) ∧ there is no m < ℓ with k 0 ∼ A k ( m )) Wolfgang Thomas

Down to Two Quantifiers Consider the condition ∃ k 0 ( α [ 0, k 0 ) ∈ U ∧ ∃ ω ℓ C ( k 0 , ℓ )) We want to exchange ∃ k 0 and ∃ ω ℓ . Natural idea: Say ∃ ω ℓ ∃ k 0 < ℓ C ( k 0 , ℓ ) and k 0 is minimal. But the minimal k 0 with α ( 0, k 0 ) ∈ U may be incorrectly chosen; we can take one among those minimal k which finally do not merge. Wolfgang Thomas