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Plan. Lecture 3: 1. Fraisse Limits and Their Automaticity: a. Random Graphs. b. Universal Partial Order. 2. The Isomorphism Problem for Automatic Structures is Σ 1 1 -complete. 3. Conclusion: What is Next?. Frasse Limits. Let K be a class of finite structures. Assume K
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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: • 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.
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 • 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.
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). • 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: • The random graph. • The universal partial order. • 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: • The domains of all Turing machines we consider are downward closed subsets of {0,1}*1. • Thus, we restrict ourselves to computable trees which are downward closed subsets of {0,1}*1. • All Turing machines are reversible. • Start configurations are words from {0,1}*1.
The isomorphism problem Let T be a Turing machine. Constructions: • To each node w in {0,1}*1 attach branching tree. Denote the resulting structure by A1. A1 is automatic. • 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 A3adjoin 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? • 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.
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?
