Transcript Document

Plan
Lecture 3:
1. Fraisse Limits and Their Automaticity:
a. Random Graphs.
b. Universal Partial Order.
2. The Isomorphism Problem for Automatic
Structures is Σ11-complete.
3. Conclusion: What is Next?
Frasse Limits
Let 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 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 kf=hg
Examples:
1. GRAPHS={finite graphs}
Frasse Limits
2. 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 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.
Frasse Limits: Examples
1.
2.
3.
4.
5.
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.
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: AnQ, 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?