Algorithms for hard problems
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Algorithms for hard problems Automata and tree automata. Juris Viksna, 2013. Finite deterministic automata. initial state. accepting state. transition. state. [Adapted from P.Drineas]. Finite deterministic automata. Finite Automaton (FA). : set of states. : input alphabet.

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Algorithms for hard problems Automata and tree automata

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Algorithms for hard problems

Automata and

tree automata

Juris Viksna, 2013


Finite deterministic automata

initial

state

accepting

state

transition

state

[Adapted from P.Drineas]


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]


Some basic results

Nondeterministic finite automaton M

[Adapted from R.Downey, M.Fellows]


Some basic results

Corresponding deterministic

finite automaton M

[Adapted from R.Downey, M.Fellows]


Some basic results

NDF with  transitions

[Adapted from R.Downey, M.Fellows]


Some basic results

Corresponding NDF without  transitions

[Adapted from R.Downey, M.Fellows]


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]


State minimization - example

[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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