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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 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.
2.
3.
4.
5.
6.
7.
8.
{0w1 | w is a word}.
{u101v | u,v are words}.
{u0a1…an | each ai is 0 or 1, u is a word}.
{w101 | w does not contain 101}.
{w | the length of w is a multiple of 3}.
Keene’s theorem.
The star height hierarchy.
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.
Regular Relations
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):
1
1
0
0
1





1
0
1
0
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.
Structures
A 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.
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(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: s1s2….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.
2.
3.
A is automatic over binary alphabet.
A is definable in
({0,1}*; L, R, prefix, EqL).
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),…)