PlanLecture 3:Lecture 3:

1. Fraisse Limits and Their Automaticity:1. Fraisse Limits and Their Automaticity:

a. Random Graphs.a. Random Graphs.

b. Universal Partial Order.b. Universal Partial Order.

2. The Isomorphism Problem for Automatic 2. The Isomorphism Problem for Automatic

Structures is Structures is ΣΣ1111-complete.-complete.

3. Conclusion: What is Next?3. Conclusion: What is Next?

Frasse LimitsLet K be a class of finite structures. Assume K

possesses the following properties:

1. Hereditary property (HP): If A is in K then any substructure of A is also in K.

2. Joint Embedding property (JEP): If A and B are in K then there is a C in K that contains both A and B.

Frasse Limits3. 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 kf=hg

Examples:

1. GRAPHS={finite graphs}

Frasse Limits2. LO={finite linear orders}

3. PO={finite partial orders}

4. BA={finite Boolean algebras}

5. LOU={finite linear orders with a unary predicate}

6. GRAPHSn={finite Kn-free graphs}

Frasse LimitsStructure 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.

Frasse Limits: Examples

1. F(GRAPHS) is the random graph.

2. F(LO) is the dense linear order.

3. F(PO) is the universal partial order.

4. F(BA) is the atomless Boolean algebra.

5. F(LOU) is the dense linear order with dense and co-dense unary predicate.

6. F(GRAPHSn) is the Kn-free random graph.

All these structures are -categorical and decidable.

We want to know which of these are automatic.

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.

Frasse Limits

Let A be an automatic structure. Consider the

sequence, called the standard approximation:

A0 A1 A2 …, where

An={v| v in A and |v|=n }.

Let Φ(x,y) be a fixed FO-formula. For every

n and every y in A define the function

cn,y: An {0,1}

Frasse Limits

cn,y(x)=1 if Φ(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| > n.

Frasse Limits

With y associate two objects:

1. Function Jy: AnQ, where Q is the state set of the automaton M recognizing Φ(x,y).

2. Subset Ky of Q defined by:

{s | M(s, y[n+1,…,|y|] is final }.

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 2|Q|.

2. The number of Jys is O(|An|).

These two claims prove the theorem.

Frasse Limits

Corollary: The following structures do not

have automatic presentations:

1. The random graph.

2. The universal partial order.

3. The random Kn-free graph.

Proof. We prove part 1, as an example.

Frasse Limits

Let Φ(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.

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.

The isomorphism problem

Theorem (Khoussainov, Nies, Rubin, Stephan)

The isomorphism problem for automatic

structures is Σ11-complete.

Proof. We code the isomorphism problem for

computable trees into the isomorphism

problem for automatic structures.

The isomorphism problem

Lemma (Goncharov, Knight) The isomorphism

problem for computable trees is Σ11-complete.

Lemma (Bennett). Any Turing machine is

equivalent to a reversible Turing machine.

We start with ({0,1}*1, prefix). This is an

automatic branching tree.

The isomorphism problem

Assumptions:

1. The domains of all Turing machines we consider are downward closed subsets of {0,1}*1.

2. Thus, we restrict ourselves to computable trees which are downward closed subsets of {0,1}*1.

3. All Turing machines are reversible.

4. Start configurations are words from {0,1}*1.

The isomorphism problem

Let T be a Turing machine. Constructions:

1. To each node w in {0,1}*1 attach branching tree. Denote the resulting structure by A1. A1 is automatic.

2. To each v in A1 not in {0,1}*1 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.

The isomorphism problem

3. To each v in {0,1}*1 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.

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).

What is Next?1. Study intrinsic state complexity of

structures (e.g. NFA presentations vs DFA presentations).

2. 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?)

3. Study the isomorphism problem for classes of automatic structures.

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.

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. Lohrey’s result)

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?