Plan

1
Plan

• Lecture 3
• 1. Fraisse Limits and Their Automaticity
• a. Random Graphs.
• b. Universal Partial Order.
• 2. The Isomorphism Problem for Automatic
• Structures is S11-complete.
• 3. Conclusion What is Next?

2
Frasse Limits

• Let K be a class of finite structures. Assume K
• possesses the following properties
• Hereditary property (HP) If A is in K then any
  substructure of A is also in K.
• Joint Embedding property (JEP) If A and B are in
  K then there is a C in K that contains both A and
  B.

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Frasse Limits

• 3. Amalgamation property (AP) Let A, B, C be in
  K. Let f C ? A and g C ? B be embeddings.
  There is a D and embeddings
• k A ? D and h B ? D such that kfhg
• Examples
• 1. GRAPHSfinite graphs

4
Frasse Limits

• 2. LOfinite linear orders
• 3. POfinite partial orders
• 4. BAfinite Boolean algebras
• 5. LOUfinite linear orders with a unary
  predicate
• 6. GRAPHSnfinite Kn-free graphs

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Frasse Limits

• Structure A is ultra-homogeneous if any partial
• finite automorphism of A can be extended to an
• automorphism.
• The age of structure A is the class of all
  finite
• substructures of A.
• Theorem. If class K has HP, JEP and AP then there
  
• is a unique ultra-homogeneous structure F(K),
• called Fraisse limit of K, whose age is K.

6
Frasse Limits Examples

• F(GRAPHS) is the random graph.
• F(LO) is the dense linear order.
• F(PO) is the universal partial order.
• F(BA) is the atomless Boolean algebra.
• F(LOU) is the dense linear order with dense and
  co-dense unary predicate.
• F(GRAPHSn) is the Kn-free random graph.
• All these structures are ?-categorical and
  decidable.
• We want to know which of these are automatic.

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Frasse Limits

• We know that the following have automatic
• copies
• F(LO) is the dense linear order.
• F(LOU) is the dense linear order with dense and
  co-dense unary predicate.
• We know that F(BA), the atomless Boolean
• algebra, does not have automatic copy.

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Frasse Limits

• Let A be an automatic structure. Consider the
• sequence, called the standard approximation
• A0 ? A1 ? A2 ? , where
• Anv v in A and vn .
• Let F(x,y) be a fixed FO-formula. For every
• n and every y in A define the function
• cn,y An ? 0,1

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Frasse Limits

• cn,y(x)1 if F(x,y) is true and
• cn,y(x)0 otherwise.
• Theorem (Khoussainov, Rubin, Stephan)
• If A is automatic then the number of functions
• of type cn,y is bounded by C An for some
• constant C.
• Proof. We can assume y gt n.

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Frasse Limits

• With y associate two objects
• 1. Function Jy An?Q, where Q is the state set
  of the automaton M recognizing F(x,y).
• Subset Ky of Q defined by
• s M(s, yn1,,y is final .

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Frasse Limits

• Claim 1 If cn,y ? cn,v then (Jy,Ky) ?
  (Jv,Kv).
• Hence, cn,y ? (Jy,Ky) .
• Claim 2.
• 1. The number of Kys is at most 2Q.
• 2. The number of Jys is O(An).
• These two claims prove the theorem.

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Frasse Limits

• Corollary The following structures do not
• have automatic presentations
• The random graph.
• The universal partial order.
• The random Kn-free graph.
• Proof. We prove part 1, as an example.

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Frasse Limits

• Let F(x,y) be E(x,y) (the edge relation). Let
• A0 ? A1 ? A2 ? . be the standard
• approximation. For An, if X, Y is a partition
• of An then there exists y such that E(x,y) is
• true for all x in X, and E(x,y) is false for all
  x
• in Y. Hence, the number of functions of type
• cn,y is 2n. This is a contradiction.

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The Isomorphism Problem

• Consider the following set
• (A,B) A and B are automatic A ? B.
• This set is called the isomorphism problem
• for automatic structures.
• Goal Find the complexity of the isomorphism
• problem for automatic structures.

15
The isomorphism problem

• Theorem (Khoussainov, Nies, Rubin, Stephan)
• The isomorphism problem for automatic
• structures is S11-complete.
• Proof. We code the isomorphism problem for
• computable trees into the isomorphism
• problem for automatic structures.

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The isomorphism problem

• Lemma (Goncharov, Knight) The isomorphism
• problem for computable trees is S11-complete.
• Lemma (Bennett). Any Turing machine is
• equivalent to a reversible Turing machine.
• We start with (0,11, ?prefix). This is an
• automatic ? branching tree.

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The isomorphism problem

• Assumptions
• The domains of all Turing machines we consider
  are downward closed subsets of 0,11.
• Thus, we restrict ourselves to computable trees
  which are downward closed subsets of 0,11.
• All Turing machines are reversible.
• Start configurations are words from 0,11.

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The isomorphism problem

• Let T be a Turing machine. Constructions
• To each node w in 0,11 attach ? branching
  tree. Denote the resulting structure by A1. A1
  is automatic.
• To each v in A1 not in 0,11 attach ? many
  chains of length n for every natural number n,
  and one ? chain. Denote the resulting structure
  by A2. The structure A2 is automatic.

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The isomorphism problem

• 3. To each v in 0,11 attach ? many chains of
  length n for every natural number n. Denote the
  resulting structure by A3. The structure A3
  is automatic.
• 4. To structure A3 adjoin the configuration
  space Conf(T). Adjoin ? many chains of length n
  (n??) for each n. Denote the resulting structure
  by A(T). A(T) is an automatic structure.

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The isomorphism problem

• Claim 1. T halts on w iff every chain attached to
  w is finite.
• Claim 2. The set w T halts on w is definable
  in the language L(?1,?).
• Claim 3. A(T1) ?A(T2) iff
• domain(T1)?domain(T2).

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What is Next?

• Study intrinsic state complexity of structures
  (e.g. NFA presentations vs DFA presentations).
• Prove structural theorems for classes of
  automatic structures, e.g. characterize the
  isomorphism types of linear orders, trees,
  groups,(Does (Q,) have an automatic copy?)
• Study the isomorphism problem for classes of
  automatic structures.

22
What is next?

• 4. Characterize intrinsic regularity of
  relations, e.g. is ? intrinsically regular in
  (Z,)?
• 5. Develop the model theory of automatic
  structures, e.g. construct automatic models for
  given theories.
• 6. Study derivative structures, e.g. automatic
  automorphism groups, of automatic structures.

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What is Next?

• 7. Develop the theory of tree or ?-automatic
  structures.
• 8. Time complexity of model checking in automatic
  structures when does an automatic structure have
  a feasible time complexity? (e.g. Lohreys
  result)

24
The Key Point

• Informal Definition (with Moshe Vardi)
• A structure is automatic if its theory in a
• given logic can be proved to be decidable via
• automata theoretic methods.
• Question If the theory of A is decidable, is
• then A automatic?