Automatic Structures

Bakhadyr Khoussainov

Computer Science Department

The University of Auckland,

New Zealand

PlanLecture 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.

PlanLecture 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).

PlanLecture 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?

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 =a0 ….an 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. {u0a1…an | each ai 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 L1 and L2 be FA recognizable. Then the

following languages are FA recognizable:

1. The union of L1 and L2.

2. The intersection of L1 and L2.

3. The complement of L1.

Building Automata

Projection Operation:

Let Σ= Σ1x Σ2 be an alphabet. Let L be a

language over Σ.

Pr1(L)={w | u ((w,u) belongs to L) }

If L is regular then so is Pr1(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=1010100110.

Write them one below the other:

11001

1010100110 and form the word c(u,v):

Regular Relations

1

1

1

0

0

1

0

0

1

1

0

0

1

1

0

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.

StructuresA structure is a tuple

(A; P0, P1,…,Pn, F0, F1…,Fm),

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, P0, P1,…,Pn) is

automatic if

1. The domain A is a FA recognizable language, and

2. each predicate Pi 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.

Examples7. 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(x1,…,xn), builds an automaton that recognizes the set

{(a1,…,an) | A satisfies Φ(a1,…,an)}.

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(x1,…,xn) in FO+ω , builds an automaton

for the set:

{(a1,…,an) | A satisfies Φ(x1,…,xn)}.

Proof. Extend A to (A, <llex ). Now, any formula

ω x Φ(x,z) is equivalent to

y x (y<llexx & 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: e1(n)=2n, et(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 et(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 Bi

3. Linear Orders: Σ(η+2n)

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 s1,…., sm such that the word si simulates state i.

b. The word si is a binary sequence such that the jth component is 1 iff the jth component of the run on x is i.

c. The run should be accepting.

Definability Theorem I (Proof):

More formally, Φ(x) says: s1s2….sm:

1. The first digit of s1 is 1.

2. For any position p only one of words si has 1.

3. If pth digit of si is 1 and the pth digit of x is σ then (p+1)th digit of sj is 1, where

(i, σ)=j.

4. If the (|x|+1)th digit of sk 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),…)