Presburger Arithmetic Reversal-Bounded Counter Machines St ephane - PowerPoint PPT Presentation

Presburger Arithmetic Reversal-Bounded Counter Machines St´ ephane Demri (demri@lsv.fr) October 7th, 2016

Slides and lecture notes http://www.lsv.fr/˜demri/notes-de-cours.html https://wikimpri.dptinfo.ens-cachan.fr/doku. php?id=cours:c-2-9-1

Plan of the lecture ◮ Previous lecture : ◮ Introduction to Presburger arithmetic. ◮ Decidability and quantifier elimination. ◮ Automata-based approach. ◮ Presburger sets are the semilinear sets. ◮ Application: Parikh image of regular languages. ◮ Introduction to reversal-bounded counter machines.

The previous lecture in 2 slides (1/2) ◮ First-order theory FO ( N ) on � N , ≤ , + � : ϕ ::= ⊤ | ⊥ | t ≤ t ′ | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ∃ x ϕ | ∀ x ϕ ◮ Presburger sets = {� v ( x 1 ) , . . . , v ( x n ) � ∈ N n : v | def � ϕ ( x 1 , . . . , x n ) � = ϕ } ◮ Quantifier-free fragment ⊤ | ⊥ | t ≤ t ′ | t ≡ k t ′ | t = t ′ | t < t ′ | t ≥ t ′ | t > t ′ (plus Boolean connectives) ◮ The satisfiability problem for the quantifier-free fragment is NP-complete.

Previous lecture in 2 slides (2/2) ◮ For every ϕ , there is a quantifier-free formula ϕ ′ such that 1. free ( ϕ ′ ) ⊆ free ( ϕ ) . 2. ϕ ′ is logically equivalent to ϕ . 3. ϕ ′ can be effectively built from ϕ . ◮ Presburger arithmetic is decidable. ◮ Alternative proof with the automata-based approach: “Presburger sets as regular languages of finite words”

Semilinear Sets

Formulae with one free variable def ϕ ( x ) = ( x � = 1 ∧ x � = 2 ) ∧ ( x = 0 ∨ ( x ≥ 3 ∧ ∃ y ( x = 3 + 2y ))) � ϕ ( x ) � = { 0 } ∪ { 3 + 2 n : n ≥ 0 } ◮ After the value 3, every two value belongs to � ϕ ( x ) � . • ◦ ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • . . . ◮ This can be generalized. X ⊆ N is ultimately periodic def ⇔ there exist N ≥ 0 and P ≥ 1 such that for all n ≥ N , we have n ∈ X iff n + P ∈ X . prefix period of length P � �� � � �� � • ◦ ◦ • ◦ • ◦ • • ◦ • • • • ◦ • • • • ◦ • • • • ◦ • • . . .

Examples of ultimately periodic sets ◮ The set of even numbers is ultimately periodic (with N = 0 and P = 2). ◮ The set of odd numbers is ultimately periodic (with N = 0 and P = 2). ◮ � x ≡ k k ′ � is ultimately periodic (with N = 0 and P = k ). ◮ Ultimately periodic sets are closed under union, intersection and complementation.

Proof for complementation ◮ Suppose X is ultimately periodic and X = N � X . ◮ The statements below are equivalent for n ≥ N : ◮ n ∈ X , ◮ n �∈ X (by definition of X ), ◮ n + P �∈ X ( X is ultimately periodic with parameters N and P ), ◮ n + P ∈ X (by definition of X ). ◮ X is ultimately periodic too and the same parameters N and P can be used. • ◦ ◦ • ◦ • ◦ • • ◦ • • • • ◦ • • • • ◦ • • • • ◦ • • . . . ◦ • • ◦ • ◦ • ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ . . .

Ultimately periodic sets X are Presburger sets � � ( x � = k ) ∧ [( x = k ) ∨ k ∈ [ 0 , N − 1 ] � X k ∈ [ 0 , N − 1 ] ∩ X � (( x ≥ N ) ∧ ( ∃ y ( x = k + P y )))] k ∈ [ N , N + P − 1 ] ∩ X It remains to show the converse result.

Semilinear sets of dimension 1 For every formula ϕ ( x ) with a unique free variable x, � ϕ � is an ultimately periodic set. ◮ Formula ϕ ( x ) with a unique free variable x. ◮ ϕ ′ : equivalent quantifier-free formula. ◮ ϕ ′ is a Boolean combination of atomic formulae of one of the forms below: ⊤ , ⊥ , x ≤ k , x ≡ k k ′ . ◮ Each atomic formula defines an ultimately periodic set and ultimately periodic sets are closed under union, intersection and complementation. ◮ So � ϕ ′ � = � ϕ � is ultimately periodic.

Semilinear sets ◮ A linear set X is defined by a basis b ∈ N d and a finite set of periods P = { p 1 , . . . , p m } ⊆ N d : m � X = { b + λ i p i : λ 1 , . . . , λ m ∈ N } i = 1 ◮ A linear set: �� 3 � 2 � 4 � � � � + i × + j × : i , j ∈ N 4 5 7 ◮ A semilinear set is a finite union of linear sets. ◮ Each semilinear set can be represented by a finite set of pairs of the form � b , P � .

Ultimately periodic sets are semilinear sets ◮ Ultimately periodic set X with parameters N and P . � � X = ( { n } ) ∪ ( { n + λ P : λ ∈ N } ) n ∈ [ 0 , N − 1 ] ∩ X n ∈ [ N , N + P − 1 ] ∩ X ◮ { n } is a linear set with no period. ◮ { n + λ P : λ ∈ N } is a linear set with basis n and unique period P .

The fundamental characterisation [Ginsburg & Spanier, PJM 66] ◮ For every Presburger formula ϕ with d ≥ 1 free variables, � ϕ � is a semilinear subset of N d . ◮ For every semilinear set X ⊆ N d , there is ϕ such that X = � ϕ � . ◮ The class of semilinear sets are effectively closed under union, intersection, complementation and projection. ◮ For instance, ( X 1 = � ϕ 1 � and X 2 = � ϕ 2 � ) imply X 1 ∩ X 2 = � ϕ 1 ∧ ϕ 2 � ◮ Presburger formula for �� 3 � 2 � 4 � � � � + i × + j × : i , j ∈ N 4 5 7 ∃ y , y ′ ( x 1 = 3 + 2y + 4y ′ ∧ x 2 = 4 + 5y + 7y ′ )

X = { 2 n : n ∈ N } is not a Presburger set ◮ Ad absurdum , suppose that X is semilinear. ◮ Since X is infinite, there are b ≥ 0 and p 1 , . . . , p m > 0 ( m ≥ 1) such that m � def Y = { b + λ i p i : λ 1 , . . . , λ m ∈ N } ⊆ X i = 1 ◮ There exists 2 α ∈ Y such that p 1 < 2 α . ◮ By definition of Y , we have 2 α + p 1 ∈ Y . ◮ But, 2 α < 2 α + p 1 < 2 α + 1 , contradiction.

X = { n 2 : n ∈ N } is not a Presburger set ◮ Ad absurdum , suppose that X is semilinear. ◮ Since X is infinite, there are b ≥ 0 and p 1 , . . . , p m > 0 ( m ≥ 1) such that m � def Z = { b + λ i p i : λ 1 , . . . , λ m ∈ N } ⊆ X i = 1 ◮ Let N ∈ N be such that N 2 ∈ Z and ( 2 N + 1 ) > p 1 . ◮ Since Z is a linear set, we also have ( N 2 + p 1 ) ∈ Z . ◮ However ( N + 1 ) 2 − N 2 = ( 2 N + 1 ) > p 1 . ◮ Hence N 2 < N 2 + p 1 < ( N + 1 ) 2 , contradiction.

A VASS weakly computing multiplication     1 0 0 1         1 0     ,     0 1         0 0     0 0 q 0   0   0 0     0   0       − 1   0       0   0         0 0 0   0 0 0   0       0 0     q 1 q 2     − 1 1           0 1 − 1     0 0   1   0     0     0   0

Weak multiplication     0 a �� a � c 0 b � �     �     0 c ∈ N 3 | ∃ � ∗ ∈ N 3 , � q 0 ,     → � q 1 , − � = b d     0 d     f e  0   e  0 f �� n � � ∈ N 3 : p ≤ n × m m . p

Weak multiplication in a VASS ◮ Suppose there is ϕ ( x 1 , . . . , x 6 ) such that       a 0 a b 0 b             c 0 c � ∗       � ϕ ( x 1 , . . . , x 6 ) � = { | � q 0 , → � q 1 , − �}       d 0 d        e   0   e  f 0 f ◮ Formula ψ ( x ) below verifies � ψ ( x ) � = { n 2 | n ∈ N } ∃ x 1 , . . . , x 5 ϕ ( x 1 , . . . , x 5 , x ) ∧ x 1 = x 2 ∧ ∀ x ′ ( x ′ > x ) ⇒ ¬∃ x 3 , x 4 , x 5 ϕ ( x 1 , . . . , x 5 , x ′ ) Contradiction!

Parikh Image of Regular Languages

Parikh image ◮ Σ = { a 1 , . . . , a k } with ordering a 1 < · · · < a k .   n 1 n 2    ∈ N k where each n j is the ◮ Parikh image of u ∈ Σ ∗ :   .   . .  n k number of occurrences of a j in u . � 3 � ◮ Parikh image of u = a b a a b , written Π( u ) , is . 2 ◮ Definition for Parikh image extends to languages. ◮ The Parikh image of any context-free language is semilinear. [Parikh, JACM 66] ◮ Effective computation from pushdown automata.

Bounded languages ◮ Language L ⊆ Σ ∗ bounded def ⇔ L ⊆ u ∗ 1 · · · u ∗ n for some words u 1 , . . . , u n in Σ ∗ . ◮ L ⊆ Σ ∗ is bounded and regular iff it is a finite union of languages of the form u 0 v ∗ 1 u 1 · · · v ∗ k u k ◮ The Parikh images of bounded and regular languages are semilinear (i.e. Presburger sets).

Counting letters in bounded and regular languages ◮ Parikh image of u 0 v ∗ 1 u 1 · · · v ∗ k u k is equal to { b + λ 1 p 1 + · · · λ k p k : λ 1 , . . . , λ k ∈ N } with ◮ b = Π( u 0 ) + · · · + Π( u k ) , ◮ p i = Π( v i ) for every i ∈ [ 1 , k ] . ◮ Finite union of such languages handled by finite unions of linear sets. ◮ Then, constructing a Presburger formula for the Parikh image easily follows.