Advice Automatic Structures and Uniformly Automatic Classes Faried Abu Zaid 1 , Erich Grädel 2 , Frederic Reinhardt 2 1 TU Ilmenau 2 RWTH Aachen University 1 / 17
Automatic Structures Idea: present structure by automata d = ( A D , A ≈ , ( A R ) R ∈ τ ) ◮ Domain is a regular set ◮ All relations are recognised by synchronous automata ◮ L ( A R ) = { x 1 ⊗ · · · ⊗ x r | ( x 1 , . . . , x r ) ∈ R } ◮ Convolution: ( a , b ) a b = a ⊗ a ( b , a ) ( a , b ) b b ( b , � ) ( a , � ) ( � , b ) ( � , a ) b a b a 2 / 17
Example ( N , +) is automatic: ◮ Encode n ∈ N by reverse binary expansion ◮ Correctness of addition checked by “carry procedure” n 1 1 0 1 0 0 1 � n 2 0 1 1 1 0 1 � 1 1 0 0 1 0 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 B ω n . ◮ 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 ): a i ◮ q ∈ [ 0, 1 ) ∩ Q ⇒ q = ∑ n with a i < i i = 2 i ! ◮ Encode q = ∑ n i = 2 a i / i ! by bin 2 ( a 2 ) # · · · # bin n ( a n ) ◮ bin i ( 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 ) ! + a 1 = ◮ i ! i ! · · · 1 0 # 1 1 # 1 0 0 # 1 0 1 # α q 1 · · · 0 0 # 1 0 # 0 1 0 � � � � � q 2 · · · 0 1 # 1 0 # 0 0 1 � � � � � · · · 0 0 # 0 1 # 0 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 7 / 17
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 , +) . � � z ◮ c ∈ ( N ∪ { ∞ } ) ω : Q c = k | z ∈ Z , α i < c i α k α 1 1 · ... · p p Lemma For all ( n i ) i ≥ 1 ∈ N ω let c = ( c i ) i ≥ 1 with c i : = ∑ ∞ j = 1 max { k ∈ N | p k i divides n j } then every q ∈ Q c can be written uniquely as � � k a i ∑ sgn ( q ) z + n 1 · · · · · n i i = 1 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 ∗ : = { A 1 × A 2 × . . . × A k | A i ∈ C} ◮ C ω : = { ∏ ω i = 0 A i | { A i | i ∈ N } ⊆ C} ◮ C ( ω ) : = { ∏ < ω i = 0 A i | { A i | i ∈ N } ⊆ C} (weak direct product) Lemma Let C be a uniformly ω -tree-automatic class of structures. Then C ∗ , C ω and C ( ω ) are uniformly ω -tree-automatic. If moreover C has a presentation with a regular advice set, then so do C ∗ , C ω and C ( ω ) . α 1 α 2 α 3 α 1 , , ,... α 2 · · · α 3 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 of the form ⊕ ∞ i = 1 G κ 1 i with κ i ≤ ω and G i ≤ Q or G i ≤ 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. 12 / 17
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 G κ i i with κ i ≤ ω and G i = Z ( n ∞ ) ∼ = Z [ n ] / Z or G i = Q . ◮ C Div = { 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. 14 / 17
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. 15 / 17
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
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