The theory of successor extended by several predicates S everine - PowerPoint PPT Presentation

The theory of successor extended by several predicates S´ everine Fratani LaBRI , Universit´ e Bordeaux 1. LIAFA , Universit´ e Paris 7. URL:http://dept-info.labri.u-bordeaux.fr/ ∼ fratani T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 1

E XTENSION OF THE B¨ UCHI ’ S S EQUENTIAL C ALCULUS Question : For which monadic relations P 1 , . . . , P n is the MSO-theory of � N , +1 , P 1 , . . . , P n � decidable ? • [B¨ uchi 66] MSO-Th � N , +1 � is decidable. For a unique relation P : • [Elgot & Rabin 66], [Siefkes 70], [Carton & Thomas 00], [Fratani & S´ enizergues 06] . . . For several relations: • [Hosch 71] � N , +1 , P 1 , . . . , P m � with P i = { n 2 i } n ≥ 0 T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 2

S UMMARY • Higher Order Pushdown Automata • Integer sequences recognized by automata • Extension of the B¨ uchi’s Sequential Calculus • Perspectives T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 3

H IGHER O RDER P USHDOWN S TORES T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 4

P USHDOWN STORES (1- PDS ) b a c b a b .... top Three instructions are available: ω = bacbab ➜ reading of the top symbol : top( ω ) = b ➜ erasing of the top symbol : pop( ω ) = acbab ➜ adding of a symbol on the top: push a ( ω ) = abacbab T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 5

k - ITERATED PUSHDOWN STORES ( k - PDS ) • 1 - Pds ( A ) = A ∗ • ( k + 1) - Pds ( A ) = ( A · [ k - Pds ( A )]) ∗ Examples: • ω 2 = a [ ab ] b [ a ] a [ cab ] ∈ 2 - Pds • ω 3 = a [ a [ ab ] b [ a ] a [ cab ]] b [ a [ ab ] b [ a ] a [ cab ]] ∈ 3 - Pds T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 6

R EPRESENTATION WITH PLANAR TREES b a b a b b a b b b a b b a a b a b ω = a [ a [ bab ] b [ ab ] b [ b ]] b [ a [ ab ] a [ b ] b [ b ]] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 7

I NSTRUCTIONS Operation allowed on a k -pds: ➜ reading of the k top symbols: top ➜ erasing of the top i -pds, i ∈ [1 , k ] : pop i ➜ adding of the symbol a at level i , i ∈ [1 , k ] (with copy of the subtrees of level i − 1 ): push a,i T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 8

I NSTRUCTIONS Reading of the top symbols: b a b a b b a b b b a b b a a b a b top( ω ) = aab T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 9

I NSTRUCTIONS Erasing of the top i -pds: pop i b a b a b b a b b b a b b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 10

I NSTRUCTIONS Erasing the top 1 -pds: pop 1 a b a b b a b b b a b b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 11

I NSTRUCTIONS Erasing of the top 2 -pds: pop 2 a b b a b b b b b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 12

I NSTRUCTIONS Erasing of the top 3 -pds: pop 3 a b b b a a b b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 13

I NSTRUCTIONS Adding of a at level i : push a,i b a b b b a a b b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 14

I NSTRUCTIONS Adding of a at level 3 : push a, 3 b a b b b a b b b a a b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 15

I NSTRUCTIONS Adding of b at level 2 : push b, 2 : b a b a b b b a b b b b a a b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 16

I NSTRUCTIONS Adding of b at level 1 : push b, 1 b a b a b b b a b b b b a a b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 17

H IGHER O RDER P USHDOWN A UTOMATA [G REIBACH 70,M ASLOV 74] A = ( Q, Σ , A, ∆ , q 0 , F ) ∈ k - PA ➜ Q , Σ , A , ∆ finite sets ➜ q 0 ∈ Q , F ⊆ Q ➜ ( p, α, a k · · · a 1 , instr , q ) ∈ ∆ α α ( p, ω ) ( q, instr( ω )) with top( ω ) = a k · · · a 1 T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 18

H IGHER O RDER L ANGUAGES LANG k class of languages recognized by k -PA. LANG 0 � LANG 1 � LANG 2 � · · · ➜ LANG 0 = class of regular languages ➜ LANG 1 = class of algebraic languages ➜ LANG 2 = class of indexed languages [Aho68] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 19

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # ⊥ [ ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 20

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # 1 ⊥ [ a n ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 21

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # 1 a [ a n ⊥ ][ a n ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 22

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # 1 a [ ⊥ ] ⊥ [ a n ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 23

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # 1 ⊥ [ a n ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 24

E XAMPLE : A LANGUAGE OF LEVEL 2 L = { α n β n γ n | n ≥ 0 } n n n � �� � � �� � � �� � α · · · α β · · · β γ · · · γ # ⊥ [ ⊥ ] T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 25

I NTEGER S EQUENCES RECOGNIZED BY A UTOMATA T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 26

k - COMPUTABLE SEQUENCES Counter automata: k - CPA The level 1 of the store contains only the symbol a 1 . a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 2 b 2 b 2 a 2 a 2 b 2 a 3 b 3 T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 27

k - COMPUTABLE SEQUENCES A sequence s : N → N is k -computable if there exists a deterministic counter k -PA such that n z }| { a 1 a 1 a 2 α s ( n ) ε a k T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 28

E XAMPLE : LINEAR RECURRENCE s (0) = 2 and ∀ n ≥ 0 , s ( n + 1) = 2 s ( n ) + 1 . αα a 2 ε n+1 n n z }| { z }| { z }| { a 1 a 1 a 1 a 1 a 1 a 1 α a 2 a 2 a 2 T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 29

( k, N ) - COMPUTABLE SEQUENCES , N ⊆ N Sequences computables by a deterministic controlled k -PA. Test: is the top counter belong to N ? 3 ∈ N ? a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 a b b a a b a b T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 30

( k, N ) - COMPUTABLE SEQUENCES Example: s ( n ) = ⌊√ n ⌋ is (2 , N ) -computable, N = { n 2 | n ≥ 0 } n+1 n z }| { z }| { a 1 a 1 a 1 a 1 α if n + 1 ∈ N a 2 a 2 n+1 n z }| { z }| { a 1 a 1 a 1 a 1 ε if n + 1 / ∈ N a 2 a 2 T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 31

S OME ( k, N ) - COMPUTABLE SEQUENCES Proposition [FS06] ➜ Sequences solutions of a system of linear recurrence equations ( N -rational sequences) are 2-computable ➜ Sequences solutions of a system of polynomial recurrence equations with integer coefficients are 3-computable Proposition If u is an increasing sequence , then the sequence ⌊ u − 1 ⌋ is (2 , u ( N )) -computable. T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 32

C LOSURE PROPERTIES Theorem � � N N ➜ if s, t ∈ S k , k ≥ 2 , the sequence f + g ∈ S k . � � N N ➜ if s, t ∈ S k , k ≥ 3 , then s ⊙ t ∈ S k (the ordinary product), and if k +1 , then u t ∈ S � � N N k +1 . u ∈ S � � N N ➜ if s ∈ S k +1 and t ∈ S k , with k ≥ 2 , then s × t ∈ S k +1 (the � N convolution product) and s • t ∈ S k +1 (the formal power series substitution). ➜ if t ∈ S k , with k ≥ 2 , then the sequence s defined by: s (0) = 1 et s ( n + 1) = P n m =0 s ( m ) · t ( n − m ) (the convolution inverse of 1 − X × f ) belongs to S k +1 . T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 33

C LOSURE PROPERTIES Theorem (continuation) � N ➜ if s ∈ S k and t ∈ S ℓ , with k, l ≥ 2 , the the sequence s ◦ t (the � N sequence composition) belongs to S k + ℓ − 1 . ➜ For k ≥ 2 and every system of recurrent equations expressed by � N polynomial in S k +1 [ X 1 , . . . , X p ] , with initial conditions in N , every � N solution belongs to S k +1 . ➜ For k ≥ 2 , every system of recurrent equations expressed by � N polynomials with undetermined X 1 , . . . , X p , coefficients in S k +2 , � N exponants in S k +1 and initial conditions in N , every solution � N belongs to S k +2 . T HE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 34