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Algorithms for hard problems Automata and tree automata

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Finite deterministic automata

Finite Automaton (FA)

: set of states

: input alphabet

: transition function d: Q×S Q

: initial state

: set of accepting states

L(M) = set of all words accepted by M

[Adapted from P.Drineas]

Finite non-deterministic automata

A word is accepted by NFA, if there exists an accepting path from

the initial state to a final state

[Adapted from P.Drineas]

Finite non-deterministic automata

Set of states, i.e.

Input aplhabet, i.e.

Transition function d: Q×(S) P(Q)

Initial state

Accepting states

L(M) = set of all words accepted by M

[Adapted from P.Drineas]

Some basic results

- the class of languages accepted by NFAs with -transitions is the same as the class of languages accepted by NFAs without -transitions
- the class of languages accepted by NFAs is the same as the class of languages accepted by DFAs

0,1

0,1

q3

1

q4

q1

q2

1

0,e

Nondeterministic finite automaton M

[Adapted from S.Yukita]

Some basic results

0

q010

q110

q000

q100

0

0

1

0

1

1

0

0

1

0

1

q011

q111

q001

q101

1

1

Deterministic finite automaton equivalent to M

1

[Adapted from S.Yukita]

Regular expressions

[Adapted from R.Downey, M.Fellows]

Regular languages

[Adapted from R.Downey, M.Fellows]

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]

Congruences

[Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Myhill’s congruence

[Adapted from R.Downey, M.Fellows]

Pumping Lemma

[Adapted from R.Downey, M.Fellows]

Myhill’s congruence

[Adapted from R.Downey, M.Fellows]

Construction of automata

[Adapted from R.Downey, M.Fellows]

Construction of automata

[Adapted from R.Downey, M.Fellows]

Construction of automata

[Adapted from R.Downey, M.Fellows]

State minimization

[Adapted from R.Downey, M.Fellows]

State minimization

[Adapted from R.Downey, M.Fellows]

Regular grammars

A right regular grammar is a formal grammar (N, Σ, P, S) such that all

the production rules in P are of one of the following forms:

A → a - where A is a non-terminal in N and a is a terminal in Σ

A → aB - where A and B are in N and a is in Σ

A → ε - where A is in N and ε denotes the empty string,

i.e. the string of length 0.

In a left regular grammar all rules obey the forms:

A → a - where A is a non-terminal in N and a is a terminal in Σ

A → Ba - where A and B are in N and a is in Σ

A → ε - where A is in N and ε is the empty string.

Both right and left grammars generate regular languages

Automata and parameterized algorithms

[Adapted from J.Flum,M.Grohe]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree automata

[Adapted from R.Downey, M.Fellows]

Tree grammars

[Adapted from R.Downey, M.Fellows]

Tree grammars

[Adapted from R.Downey, M.Fellows]

Tree grammars - example

[Adapted from R.Downey, M.Fellows]

Normalized tree grammars

[Adapted from R.Downey, M.Fellows]

Normalized tree grammars

[Adapted from R.Downey, M.Fellows]

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]

Regular tree expressions

[Adapted from R.Downey, M.Fellows]

Regular tree expressions

[Adapted from R.Downey, M.Fellows]

Regular tree expressions

[Adapted from R.Downey, M.Fellows]

Kleene’s theorem for trees (II)

[Adapted from R.Downey, M.Fellows]

Kleene’s theorem for trees (II)

[Adapted from R.Downey, M.Fellows]

Equivalence relation for tree languages

[Adapted from R.Downey, M.Fellows]

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]

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