Automatic Structures

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Automatic Structures

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

2
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.

3
Plan

• Lecture 2
• 1. Automatic Boolean Algebras.
• 2. Automatic Linear Orders and Ranks.
• 3. Automatic Trees and Ranks.
• 4. Automatic Versions of Konigs Lemma.
• 5. Definability and Intrinsic Regularity
• a) Decidability Theorem III.
• b) Example Intrinsic Regularity in (?, S).

4
Plan

• Lecture 3
• 1. Fraisse Limits and Their Automaticity
• a. Random Graphs.
• b. Universal Partial Order.
• 2. The Isomorphism Problem for Automatic
• Structures is S11-complete.
• 3. Conclusion What is Next?

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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.

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Finite Automata

• Fix an alphabet S. An automaton consists of
• A finite set S of states.
• A subset I of S. States in I are initial states.
• A transition diagram ? SxS ? P(S)
• A subset F of S. States in F are called final
  states.
• Automata can be represented as directed
• labeled graphs.

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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 LL(A) for
• some automaton A.

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Examples and Some Results

• 0w1 w is a word.
• u101v u,v are words.
• u0a1an 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.
• Keenes theorem.
• The star height hierarchy.
• NFA and DFA are equivalent (a few words).

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Building Automata

• Let L1 and L2 be FA recognizable. Then the
• following languages are FA recognizable
• The union of L1 and L2.
• The intersection of L1 and L2.
• The complement of L1.

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Building Automata

• Projection Operation
• Let S S1x S2 be an alphabet. Let L be a
• language over S.
• Pr1(L)w ? u ((w,u) belongs to L)
• If L is regular then so is Pr1(L).

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Regular Relations

• Consider a binary relation R on the set S.
• Thus, R? S x S. 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 Buchis original definition
• published in1960.

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Regular Relations

• We define the convolution of R. Take words
• u and v Say, u11001,v1010100110.
• Write them one below the other
• 11001
• 1010100110 and form the word c(u,v)

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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.

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Structures

• A structure is a tuple
• (A P0, P1,,Pn, F0, F1,Fm),
• where
• each P is a predicate symbol, and
• 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.

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Structures

• Examples
• a) Graphs (V E).
• b) Partial orders (P ?).
• c) Linear orders (L ?).
• d) Trees (T ?).
• e) Groups (G ).
• f) Boolean algebras (B ?, n, /, 0,1).
• g) Rings (R , x, 0,1).

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Definition Automatic Structure (Hodgson 1976,
Khoussainov and Nerode 1994)

• A structure A(A, P0, P1,,Pn) is
• automatic if
• The domain A is a FA recognizable language, and
• each predicate Pi is a FA recognizable language.

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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.

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Examples

• The successor structure (1 S), where S(w)w1
• The 2 successors structure (0,1 L, R), where
  L(w)w0 and R(w)w1.
• The linear order (1 lt), where wltu iff the
  length of w is less than that of u.
• The binary tree (0,1 ?prefix), where
• x ?prefix y iff x is a prefix of y.

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Examples

• 5. The word structure
• (0,1 L, R, ltpref, EqL),
• where EqL(x,y) iff xy.
• 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.

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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.

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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
• (0,11 ?lex ).
• This is a dense linearly ordered set.

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Decidability Theorem I (Hodgson 1976,
Khoussainov and Nerode, 1994)

• Let A be an automatic structure. There exists
• an algorithm that, given a FO formula F(x1,,xn),
  builds an automaton that recognizes the set
• (a1,,an) A satisfies F(a1,,an).
• Proof. By induction on the length of the
• formula F. The disjunction corresponds to the
• union, negation to the complementation, and
• ? to projection operations.

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Corollaries

• 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, lt, ) is decidable.

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Decidability Theorem II (Gradel and Blumensath,
in LICS 2000)

• Let A be an automatic structure. There
• exists an algorithm that, given a formula
• F(x1,,xn) in FO?? , builds an automaton
• for the set
• (a1,,an) A satisfies F(x1,,xn).
• Proof. Extend A to (A, ltllex ). Now, any formula
  
• ?? x F(x,z) is equivalent to
• ?y ?x (yltllexx F(x,z) ).

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Corollaries

• 4. Let (T lt) be an automatic finitely
• branching infinite tree. Then it has a regular
• infinite path.
• Proof. Consider (Tlt, ltllex ). Here is a FO ??
• definition of an infinite path. Good(x) if any
  
• y below or equal to x is the ltllex-first
• immediate successor of its parent such that
• there are infinitely many z above y.

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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.

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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, , lt, 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.

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Definition Automatic Presentations (Khoussainov
and Nerode 1994)

• Let A be a structure.
• An automatic presentation of A, or equivalently,
  automatic copy of A, is any automatic structure
  isomorphic to A.
• If A has an automatic presentation then A is
  called FA presentable.

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Automata Presentable Structures Examples

• 1. The group (Z ). More generally, finitely
• generated Abelian groups.
• 2. Boolean Algebras Bi?
• 3. Linear Orders S(?2n)
• 4. Graphs.
• 5. Equivalence Structures.

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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.

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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.

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Definability Theorem I (Proof)

• Want to build F(x) such that for all w in 0,1
• the word w is in R iff F(w) is true. The
• formula needs to say the following
• There exist words s1,., sm such that the word si
  simulates state i.
• 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.
• The run should be accepting.

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Definability Theorem I (Proof)

• More formally, F(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 s then (p1)th digit of sj is 1, where
• ?(i, s)j.
• 4. If the (x1)th digit of sk is 1 then k is in
  F.
• All these can be expressed in the FO logic.

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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 ).

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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),)