Automatic Structures Bakhadyr Khoussainov Computer Science Department The University of Auckland, New Zealand

Plan Lecture 1: 1. Motivation. 2. Finite Automata. Examples. 3. Building Automata. 4. Automatic Structures. Definition. 5. Examples. 6. Decidability Theorems I and II. 7. Definability Theorems.

Plan Lecture 2: 1. Automatic Boolean Algebras. 2. Automatic Linear Orders and Ranks. 3. Automatic Trees and Ranks. 4. Automatic Versions of Konig’s Lemma. 5. Definability and Intrinsic Regularity: a) Decidability Theorem III. b) Example: Intrinsic Regularity in ( , S).

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?

Motivation Refinement of the theory of computable structures A part of feasible mathematics Generalization of the theory of finite models A natural generalization of automata theory Automatic groups Infinite state systems. Roots go back to the late 50s and the 60s to early developments of automata theory by Buchi, Elgot, Eilenberg, Kleene, Rabin, Sheperdson.

Finite Automata Fix an alphabet Σ. An automaton consists of: 1.A finite set S of states. 2.A subset I of S. States in I are initial states. 3.A transition diagram Δ: SxΣ → P(S) 4.A subset F of S. States in F are called final states. Automata can be represented as directed labeled graphs.

Finite Automata Let w =a 0 ….a n be a word. The word is accepted by the automaton if there exists an accepting run of the automaton on the word. L(A)={w | w is accepted by A} Language L is FA recognizable if L=L(A) for some automaton A.

Examples and Some Results 1.{0w1 | w is a word}. 2.{u101v | u,v are words}. 3.{u0a 1 …a n | each a i is 0 or 1, u is a word}. 4.{w101 | w does not contain 101}. 5.{w | the length of w is a multiple of 3}. 6.Keene’s theorem. 7.The star height hierarchy. 8.NFA and DFA are equivalent (a few words).

Building Automata Let L 1 and L 2 be FA recognizable. Then the following languages are FA recognizable: 1.The union of L 1 and L 2. 2.The intersection of L 1 and L 2. 3.The complement of L 1.

Building Automata Projection Operation: Let Σ= Σ 1 x Σ 2 be an alphabet. Let L be a language over Σ. Pr 1 (L)={w |  u ((w,u) belongs to L) } If L is regular then so is Pr 1 (L).

Regular Relations Consider a binary relation R on the set Σ *. Thus, R  Σ * x Σ *. We want to define what it means that R is FA recognizable. There are several ways to define FA recognizable relations. There are research schools that study questions of this type. We follow Buchi ’ s original definition published in1960.

We define the convolution of R. Take words u and v; Say, u=11001,v= Write them one below the other: and form the word c(u,v): Regular Relations

c(u,v) is called the convolution of (u,v). Consider c(R)={c(u,v) | (u,v) belongs to R}. Note, c(R) is a language over new finite alphabet. Definition (Buchi and Elgot, 1960,1961). The relation R is FA recognizable (equivalently, regular) if its convolution c(R) is FA recognizable.

Structures A structure is a tuple (A; P 0, P 1,…,P n, F 0, F 1 …,F m ), where 1.each P is a predicate symbol, and 2.each F is a functional symbol. Assumptions: a) A is a countable set. b) Consider relational structures in which each function F is replaced by its graph.

Structures Examples: a) Graphs (V; E). b) Partial orders (P;  ). c) Linear orders (L;  ). d) Trees (T;  ). e) Groups (G; +). f) Boolean algebras (B; , ∩, /, 0,1). g) Rings (R; +, x, 0,1).

Definition: Automatic Structure (Hodgson 1976, Khoussainov and Nerode 1994) A structure A=(A, P 0, P 1,…,P n ) is automatic if 1.The domain A is a FA recognizable language, and 2.each predicate P i is a FA recognizable language.

Definition: Automatic Structure To describe an automatic structure one needs to explicitly specify: The alphabet. A finite automaton that recognizes the domain of the structure. Finite automata recognizing all the predicates of the structure.

Examples: 1.The successor structure ({1}*; S), where S(w)=w1 2.The 2 successors structure ({0,1}*; L, R), where L(w)=w0 and R(w)=w1. 3.The linear order ({1}*; <), where w<u iff the length of w is less than that of u. 4.The binary tree ({0,1}*;  prefix ), where x  prefix y iff x is a prefix of y.

Examples 5. The word structure ({0,1}*; L, R, < pref, EqL), where EqL(x,y) iff |x|=|y|. 6. The structure (N; +), where numbers are represented as binary words with least significant digits written from left to right and rightmost digit not being 0.

Examples 7. The Presburger arithmetic (N; S, +,  ), where numbers are represented in binary. 8. Arithmetic with weak division (N; S, +, , | 2 ), where x | 2 y iff x is a power of two and y is a multiple of x.

Examples 9. Let T be a Turing machine. Consider the graph (Conf(T), E), where Conf(T) is the space of all configurations of T, and E(x,y) if there is a one-step transition from configuration x into y via T. 10. The structure ({0,1}*1;  lex ). This is a dense linearly ordered set.

Decidability Theorem I (Hodgson 1976, Khoussainov and Nerode, 1994) Let A be an automatic structure. There exists an algorithm that, given a FO formula Φ(x 1, …,x n ), builds an automaton that recognizes the set {(a 1, …,a n ) | A satisfies Φ(a 1, …,a n )}. Proof. By induction on the length of the formula Φ. The disjunction corresponds to the union, negation to the complementation, and  to projection operations.

Corollaries 1.The first order theory, that is, the set of all first order sentences true in any given automatic structure is decidable. 2. The first order theory of Presburger arithmetic (N; S, 0, <, +) is decidable.

Decidability Theorem II (Gradel and Blumensath, in LICS 2000) Let A be an automatic structure. There exists an algorithm that, given a formula Φ(x 1, …,x n ) in FO+  ω, builds an automaton for the set: {(a 1, …,a n ) | A satisfies Φ(x 1, …,x n )}. Proof. Extend A to (A, < llex ). Now, any formula  ω x Φ(x,z) is equivalent to  y  x (y< llex x & Φ(x,z) ).

Corollaries: 4. Let (T; <) be an automatic finitely branching infinite tree. Then it has a regular infinite path. Proof. Consider (T;<, < llex ). Here is a FO+  ω definition of an infinite path. Good(x) if any y below or equal to x is the < llex -first immediate successor of its parent such that there are infinitely many z above y.

Comment: Consider: e 1 (n)=2 n, e t (n)=the tower of 2s of length t to the power of n. The  quantifier brings non-determinism. The negation which follows  brings exponential blow up in the number of states. So, the t blocks of the negation symbol followed by  in a formula yields an automaton with e t (n) number of states.

Comment: If A is automatic then the time complexity of the algorithm deciding the theory of A is non-elementary. Theorem (Blumensath, Gradel, LICS 2000). The time complexity of the first order theory of (N; S, +, <, | 2 ) is non-elementary. M. Lohrey (2003): The theory of any automatic finitely branching graph is double exponential. F. Fleadtke (2003): The known lower bound for Presburger arithmetic is matched via automata.

Definition: Automatic Presentations (Khoussainov and Nerode 1994) Let A be a structure. 1.An automatic presentation of A, or equivalently, automatic copy of A, is any automatic structure isomorphic to A. 2.If A has an automatic presentation then A is called FA presentable.

Automata Presentable Structures: Examples 1. The group (Z; +). More generally, finitely generated Abelian groups. 2. Boolean Algebras B i  3. Linear Orders: Σ(η+2 n ) 4. Graphs. 5. Equivalence Structures.

Definability Theorem I (Buchi 1960, Elgot 1961, Eilenberg, Elgot and Sheperdson 1969, Bruere et al. 1994, Blumensath and Gradel 1999) A structure A has an automatic presentation iff A is isomorphic to a structure definable in ({0,1}*; L, R,  prefix, EqL). Proof. One direction is clear. The other direction: Let A be an automatic. Fact: We can assume that the alphabet is {0,1}.

Definability Theorem I (Proof): It suffices to show that any regular relation R over {0,1} is definable. Say, for simplicity, R is unary. Assume M accepts R: 1. {1,….,m} are the states of M; 1 is the initial state 2.  is the transition table. 3. F is the set of all accepting states.

Definability Theorem I (Proof) Want to build Φ(x) such that for all w in {0,1} * the word w is in R iff Φ(w) is true. The formula needs to say the following: a.There exist words s 1, …., s m such that the word s i simulates state i. b.The word s i is a binary sequence such that the j th component is 1 iff the j th component of the run on x is i. c.The run should be accepting.

Definability Theorem I (Proof): More formally, Φ(x) says:  s 1  s 2 ….  s m : 1. The first digit of s 1 is For any position p only one of words s i has If p th digit of s i is 1 and the p th digit of x is σ then (p+1) th digit of s j is 1, where  (i, σ)=j. 4. If the (|x|+1) th digit of s k is 1 then k is in F. All these can be expressed in the FO logic.

Definability Theorem II ( Gradel & Blumeansath, 2000 ) The following are equivalent: 1. A is automatic over binary alphabet. 2. A is definable in ({0,1}*; L, R,  prefix, EqL). 3. A is definable in (N; S, +, , | 2 ).

Definability Theorem III ( Nabebin 1976, Blumensath 1999) A structure A has an automatic presentation over a unary alphabet if and only if it is isomorphic to a structure definable in (  ; , mod(2), mod(3), mod(4),…)