Advice Automatic Structures and Uniformly Automatic Classes Faried - - PowerPoint PPT Presentation

Advice Automatic Structures and Uniformly Automatic Classes

Faried Abu Zaid 1, Erich Grädel 2, Frederic Reinhardt 2

1TU Ilmenau 2RWTH Aachen University 1 / 17

Automatic Structures

Idea: present structure by automata d = (AD, A≈, (AR)R∈τ)

◮ Domain is a regular set ◮ All relations are recognised by synchronous automata

◮ L(AR) = {x1 ⊗ · · · ⊗ xr | (x1, . . . , xr) ∈ R} ◮ Convolution:

a b b a a b a b b a ⊗ (a, b) (b, a) (b, ) (a, ) (a, b) (, b) (, a) =

2 / 17

Example

(N, +) is automatic:

◮ Encode n ∈ N by reverse binary expansion ◮ Correctness of addition checked by “carry procedure”

1 n1 1 1

• n2

1 1 1 1

• 1

1 1 1

3 / 17

Why Should we Care?

Theorem

Every automatic structure has a decidable first-order theory.

◮ Can be extended by

◮ modulo-quantifiers ◮ cardinality-quantifiers ◮ a weak form of second order quantification (words) 4 / 17

Advice Automatic Structures

In many domains being automatic is rather restrictive:

◮ A finitely generated group is automatic iff it is

Abelian-by-finite.

◮ (Q, +) is not automatic. ◮ An infinite Boolean algebra is automatic iff it is isomorphic

to some Bωn.

◮ An integral domain is ω-automatic iff it is finite.

Extension: automata with access to some fixed advice ⇒ automatic structures with advice α. some useful fixed advice . . . some encoding.. .

5 / 17

Motivation: Addition over the Rational Numbers

Idea (for simplicity Q/Z):

◮ q ∈ [0, 1) ∩ Q ⇒ q = ∑n i=2 ai i!

with ai < i

◮ Encode q = ∑n i=2 ai/i! by bin2(a2)# · · · # binn(an) ◮ bini(x) = bin(x) padded to length ⌈log(i)⌉ ◮ Corresponding language is automatic with advice

α := bin(2)# bin(3)# bin(4)# · · ·

◮ (i+a) i!

=

1 (i−1)! + a i!

1 α # 1 1 # 1 # 1 1 # · · · q1 # 1 # 1 · · · q2 1 # 1 # 1 · · · # 1 # 1 1 · · ·

6 / 17

Uniformly Automatic Classes

Definition

A class C of τ-structures is uniformly automatic if it can be presented by a single presentation c and a set of advices P.

◮ If P has a decidable MSO-theory we say C is strongly

automatic.

◮ If P is regular we say C is regularly automatic. ◮ Remark: In this setting consider also automata over finite

words/trees.

Example

The class of all . . .

◮ countable linear orders is regularly ω-tree-automatic ◮ finite graphs of treewidth/cliquewidth at most c is

regularly tree-automatic

◮ finite graphs of pathwidth/linear cliquewidth at most c is

regularly automatic

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Corollary

• 1. The FO-theory of a uniformly ω-automatic class C with advice

set P is decidable, if the MSO-theory of P is decidable.

• 2. The FO-theory of a structure A that is ω-automatic with advice

α is decidable, if the MSO-theory of α is decidable.

8 / 17

Torsion free Abelian Groups

Theorem

The class of all torsion free Abelian groups of rank 1 is regularly ω-automatic.

Theorem (Baer)

The t.f.a.g.o.r. 1 are up to isomorphism the subgroups of (Q, +).

◮ c ∈ (N ∪ {∞})ω: Qc =

• z

p

α1 1 ·...·p αk k | z ∈ Z, αi < ci

• Lemma

For all (ni)i≥1 ∈ Nω let c = (ci)i≥1 with ci := ∑∞

j=1 max{k ∈ N | pk i divides nj} then every q ∈ Qc can be

written uniquely as sgn(q)

• z +

k

∑

i=1

ai n1 · · · · · ni

• 9 / 17

Limitations: Bounded rank

Theorem

No class of countable cancellative commutative semigroups with unbounded rank is uniformly ω-automatic.

Corollary

Every countable cancellative commutative semigroup that is ω-automatic with advice has finite rank.

10 / 17

Closure under direct products

◮ C∗ := {A1 × A2 × . . . × Ak | Ai ∈ C} ◮ Cω := {∏ω i=0 Ai | {Ai | i ∈ N} ⊆ C} ◮ C(ω) := {∏<ω i=0 Ai | {Ai | i ∈ N} ⊆ C} (weak direct product)

Lemma

Let C be a uniformly ω-tree-automatic class of structures. Then C∗, Cω and C(ω) are uniformly ω-tree-automatic. If moreover C has a presentation with a regular advice set, then so do C∗, Cω and C(ω). α1

,

α2

,

α3

,...

· · ·

α1 α2 α3 11 / 17

Abelian Groups

Corollary

There is a regularly ω-tree-automatic presentation of the class of all Abelian groups up to elementary equivalence.

Proof.

◮ Every Abelian group is elementarily equivalent to a group

• f the form ⊕∞

i=1Gκ1 i with κi ≤ ω and Gi ≤ Q or Gi ≤ Q/Z. ◮ {G | G ≤ Q} and {G | G ≤ Q/Z} are regularly

ω-automatic.

◮ ({G | G ≤ Q}∗ ∪ {G | G ≤ Q/Z}∗)(ω) is regularly

ω-tree-automatic.

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Abelian Divisible Groups

Corollary

The class of countable divisible Abelian groups is strongly ω-tree-automatic.

Proof.

◮ Every countable divisible Abelian group is isomorphic to a

group of the form ∞

i=1 Gκi i with κi ≤ ω and

Gi = Z(n∞) ∼ = Z[n]/Z or Gi = Q.

◮ CDiv = {Z[n]/Z | n ≥ 2} ∪ {Q} is uniformly ω-automatic

with parameter set P := {(bin(n)#)ω | n ≥ 2} ∪ {bin(2)#bin(3)# . . .} which has a decidable MSO-theory.

13 / 17

Limitations: Countable Boolean Algebras

Theorem

A countably infinite Boolean algebra is ω-automatic with advice iff it is automatic.

Theorem

A class of countably infinite Boolean algebras is uniformly ω-automatic iff it is a finite class of automatic infinite Boolean algebras.

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Limitations: Forbidden Substructures

Theorem

No countable advice automatic structure contains.. .

◮ a pairing function, ◮ the free semigroup over two generators or ◮ (N, ·)

as substructure.

Corollary

(Q, ·) and free group over at least two generators are not advice automatic.

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Limitations: Integral Domains

Theorem

No infinite integral domain has an injective ω-automatic presentation with advice.

Corollary

No countably infinite integral domain is ω-automatic with advice.

Theorem

The field of reals (R, +, ·) is not ω-automatic with advice.

16 / 17

Open Problems

◮ Which torsion free abelian groups of rank 2 are

ω-automatic with advice?

◮ Are there other domains where automatic presentations

significantly benefit from advice?

◮ Is the field of reals in automatic in any sense?

◮ Rabin asked this already in 1968! 17 / 17