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

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 1

EXTENSION OF THE B¨

UCHI’S SEQUENTIAL CALCULUS

Question : For which monadic relations P1, . . . , Pn is the MSO-theory of N, +1, P1, . . . , Pn decidable ?

• [B¨

uchi 66] MSO-ThN, +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, P1, . . . , Pm with Pi = {n2i}n≥0

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 2

SUMMARY

• Higher Order Pushdown Automata
• Integer sequences recognized by automata
• Extension of the B¨

uchi’s Sequential Calculus

• Perspectives

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 3

HIGHER ORDER PUSHDOWN STORES

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 4

PUSHDOWN 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: pusha(ω) = abacbab

THE 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:

• ω2 = a[ab] b[a] a[cab] ∈ 2-Pds
• ω3 = a[a[ab] b[a] a[cab]] b[a[ab] b[a] a[cab]] ∈ 3-Pds

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 6

REPRESENTATION WITH PLANAR TREES

b a a b a b b b b a b a b a b b b a

ω = a[a[bab] b[ab] b[b]] b[a[ab] a[b] b[b]]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 7

INSTRUCTIONS

Operation allowed on a k-pds:

➜ reading of the k top symbols: top ➜ erasing of the top i-pds, i ∈ [1, k]: popi ➜ adding of the symbol a at level i, i ∈ [1, k] (with copy of the subtrees of level i − 1): pusha,i

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 8

INSTRUCTIONS

Reading of the top symbols:

b a b a b b b b a b a b a b b b a a

top(ω) = aab

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 9

INSTRUCTIONS

Erasing of the top i-pds: popi

b a a b a b b b b a b a b a b b b a

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 10

INSTRUCTIONS

Erasing the top 1-pds: pop1

a a b a b b b b a b a b a b b b a

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 11

INSTRUCTIONS

Erasing of the top 2-pds: pop2

a b b b b a b a b a b b b a

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 12

INSTRUCTIONS

Erasing of the top 3-pds: pop3

a b a b a b b b

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 13

INSTRUCTIONS

Adding of a at level i: pusha,i

a b a b a b b b b

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 14

INSTRUCTIONS

Adding of a at level 3: pusha,3

a b a b a b b b b a a b b b a b a

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 15

INSTRUCTIONS

Adding of b at level 2: pushb,2:

a b a b a b b b a b a b b a a b b a b b

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 16

INSTRUCTIONS

Adding of b at level 1: pushb,1

a b a b a b b b a b a b b a a b b a b b

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 17

HIGHER ORDER PUSHDOWN AUTOMATA [GREIBACH 70,MASLOV 74]

A = (Q, Σ, A, ∆, q0, F) ∈ k-PA

➜ Q, Σ, A, ∆ finite sets ➜ q0 ∈ Q, F ⊆ Q ➜ (p, α, ak · · · a1, instr, q) ∈ ∆

α (p, ω) (q, instr(ω)) α

with top(ω) = ak · · · a1

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 18

HIGHER ORDER LANGUAGES

LANGk class of languages recognized by k-PA. LANG0 LANG1 LANG2 · · ·

➜ LANG0 = class of regular languages ➜ LANG1 = class of algebraic languages ➜ LANG2 = class of indexed languages [Aho68]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 19

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0} β · · · β γ · · · γ #

n

α · · · α

n n

⊥ [⊥]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 20

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0}

1

⊥ [an ⊥] β · · · β γ · · · γ #

n

α · · · α

n n THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 21

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0}

1

β · · · β γ · · · γ #

n

α · · · α

n n

a[an ⊥][an ⊥]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 22

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0}

1

β · · · β γ · · · γ #

n

α · · · α

n n

a[⊥] ⊥ [an ⊥]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 23

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0}

1

β · · · β γ · · · γ #

n

α · · · α

n n

⊥ [an ⊥]

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 24

EXAMPLE: A LANGUAGE OF LEVEL 2

L = {αnβnγn | n ≥ 0} ⊥ [⊥] β · · · β γ · · · γ #

n

α · · · α

n n THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 25

INTEGER SEQUENCES RECOGNIZED BY AUTOMATA

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 26

k-COMPUTABLE SEQUENCES

Counter automata: k-CPA The level 1 of the store contains

• nly the symbol a1.

a2 b2 b2 a2 a2 b3 b2 a3 a1 a1 a1 a1 a1 a1 a1 a1 a1 a1

THE 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

ak a1 a2 a1

z }| {

ε αs(n)

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 28

EXAMPLE: LINEAR RECURRENCE

s(0) = 2 and ∀n ≥ 0, s(n + 1) = 2s(n) + 1.

n+1 n n

a2 ε αα α a1 a2 a1

z }| {

a1 a2 a1

z }| {

a1 a2 a1

z }| {

THE 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?

a b b a a b b a a1 a1 a1 a1 a1 a1 a1 a1 a1 a1 3 ∈ N?

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 30

(k, N)-COMPUTABLE SEQUENCES

Example: s(n) = ⌊√n⌋ is (2, N)-computable, N = {n2 | n ≥ 0}

n+1 n+1 n n

if n + 1 ∈ N if n + 1 / ∈ N ε a1 a2 a1

z }| {

a1 a2 a1

z }| {

α a1 a2 a1

z }| {

a1 a2 a1

z }| {

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 31

SOME (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.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 32

CLOSURE PROPERTIES

Theorem

➜ if s, t ∈ S

• N

k , k ≥ 2, the sequence f + g ∈ S

• N

k .

➜ if s, t ∈ S

• N

k , k ≥ 3, then s ⊙ t ∈ S

• N

k (the ordinary product), and if

u ∈ S

• N

k+1, then ut ∈ S

• N

k+1.

➜ if s ∈ S

• N

k+1 and t ∈ Sk, with k ≥ 2, then s × t ∈ S

• N

k+1 (the

convolution product) and s • t ∈ S

• N

k+1 (the formal power series

substitution). ➜ if t ∈ Sk, with k ≥ 2, then the sequence s defined by: s(0) = 1 et s(n + 1) = Pn

m=0 s(m) · t(n − m) (the convolution inverse of

1 − X × f) belongs to Sk+1.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 33

CLOSURE PROPERTIES

Theorem (continuation)

➜ if s ∈ Sk and t ∈ S

• N

ℓ , with k, l ≥ 2, the the sequence s◦t (the

sequence composition) belongs to S

• N

k+ℓ−1.

➜ For k ≥ 2 and every system of recurrent equations expressed by polynomial in S

• N

k+1[X1, . . . , Xp], with initial conditions in N, every

solution belongs to S

• N

k+1.

➜ For k ≥ 2, every system of recurrent equations expressed by polynomials with undetermined X1, . . . , Xp, coefficients in S

• N

k+2,

exponants in S

• N

k+1 and initial conditions in N, every solution

belongs to S

• N

k+2.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 34

(k, N)-COMPUTABLE SEQUENCES

Proposition: If s is (k, N)-computable, then

➜ {αs(n) | n ≥ 0} is recognized by a k-CPAN ➜ Let Σs be the sequence defined for every n ≥ 0 by Σs(n) = Σm=n

m=0 s(m).

{αΣs(n) | n ≥ 0} is recognized by a deterministic k-CPAN. ➜ {αs(0)βαs(1)β · · · αs(n)β | n ≥ 0} is recognized by a deterministic k-CPAN.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 35

EXTENSIONS OF THE B¨

UCHI’S SEQUENTIAL CALCULUS

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 36

MSO LOGIC OVER COMPUTATION GRAPHS

Theorem[Car05,FS06,F05] The computation graph of a k-PA has a decidable MSO theory. Theorem[F05] If the MSO-theory of N, +1, N is decidable, then the computation graph of (k, N)-CPA has a decidable MSO-theory.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 37

APPLICATION TO THE SEQUENTIAL CALCULUS

Graph of A ∈ k-CPA

• N:

αs(0)

• −

→− →− → . . .

β

− →

αs(1)

• −

→− →− → . . .

β

− →

αs(2)

• −

→− →− → . . .

β

− → . . .

• Graph of N, +1, Σs(N):
• s(0)
• −

→− →− → . . .

s(1)

• −

→− →− → . . .

s(2)

• −

→− →− → . . . . . .

• ➜ The structure N, +1, Σs(N) is interpretable in the computation

graph of A ➜ If N, +1, N has a decidable MSO theory, then the graph of A has a decidable MSO theory.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 38

EXTENSIONS OF THE B¨

UCHI’S SEQUENTIAL CALCULUS

Theorem If N, +1, N has a decidable MSO theory, then for every (k, N)-computable sequence s, the MSO theory of N, +1, Σs(N) is decidable. Corollary

➜ N, +1, {⌊n√n⌋}n≥0 ➜ N, +1, {⌊n log n⌋}n≥0

have a decidable MSO theory.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 39

GENERALIZATION TO SEVERAL PREDICATES

Theorem If N, +1, N1, . . . , Nm has a decidable MSO theory, then for every (k, N)-computable sequence s (with

• N = (N1, . . . , Nm)), the structure

N, +1, Σs(N), Σs(N1), . . . , Σs(Nm) has a decidable MSO theory. Corollaire For every k1, . . . , km ≥ 0, the theory of N, +1, {nkm}n≥0, {nkmkm−1}n≥0, . . . , {nk1···km}n≥0 is decidable.

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 40

PERSPECTIVES

➜ Computable multiple sequences ➜ Decidability of N, +1 augmented with no-nested predicates Example: N, +1, {n2}n∈N, {n3 ∈ N} ➜ Equivalence problem for deterministic automata (if the computation graph is an infinite straight line)

THE THEORY OF SUCCESSOR EXTENDED BY SEVERAL PREDICATES 41