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

Algorithms for hard problems

Automata and

tree automata

Juris Viksna, 2013


Finite deterministic automata

Finite deterministic automata

initial

state

accepting

state

transition

state

[Adapted from P.Drineas]


Finite deterministic automata1

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

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 automata1

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

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 results1

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 results2

Some basic results

Nondeterministic finite automaton M

[Adapted from R.Downey, M.Fellows]


Some basic results3

Some basic results

Corresponding deterministic

finite automaton M

[Adapted from R.Downey, M.Fellows]


Some basic results4

Some basic results

NDF with  transitions

[Adapted from R.Downey, M.Fellows]


Some basic results5

Some basic results

Corresponding NDF without  transitions

[Adapted from R.Downey, M.Fellows]


Regular expressions

Regular expressions

[Adapted from R.Downey, M.Fellows]


Regular languages

Regular languages

[Adapted from R.Downey, M.Fellows]


Regular languages languages accepted by dfa nfa

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]


Regular languages languages accepted by dfa nfa1

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]


Regular languages languages accepted by dfa nfa2

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]


Regular languages languages accepted by dfa nfa3

Regular languages = languages accepted by DFA/NFA

[Adapted from R.Downey, M.Fellows]


Congruences

Congruences

[Adapted from R.Downey, M.Fellows]


Myhill nerode theorem

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Myhill nerode theorem1

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Myhill nerode theorem2

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Myhill nerode theorem3

Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Myhill s congruence

Myhill’s congruence

[Adapted from R.Downey, M.Fellows]


Pumping lemma

Pumping Lemma

[Adapted from R.Downey, M.Fellows]


Myhill s congruence1

Myhill’s congruence

[Adapted from R.Downey, M.Fellows]


Construction of automata

Construction of automata

[Adapted from R.Downey, M.Fellows]


Construction of automata1

Construction of automata

[Adapted from R.Downey, M.Fellows]


Construction of automata2

Construction of automata

[Adapted from R.Downey, M.Fellows]


State minimization

State minimization

[Adapted from R.Downey, M.Fellows]


State minimization1

State minimization

[Adapted from R.Downey, M.Fellows]


State minimization example

State minimization - example

[Adapted from R.Downey,

M.Fellows]


Regular grammars

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

Automata and parameterized algorithms

[Adapted from J.Flum,M.Grohe]


Tree automata

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree automata1

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree automata2

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree automata3

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree automata4

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree automata5

Tree automata

[Adapted from R.Downey, M.Fellows]


Tree grammars

Tree grammars

[Adapted from R.Downey, M.Fellows]


Tree grammars1

Tree grammars

[Adapted from R.Downey, M.Fellows]


Tree grammars example

Tree grammars - example

[Adapted from R.Downey, M.Fellows]


Normalized tree grammars

Normalized tree grammars

[Adapted from R.Downey, M.Fellows]


Normalized tree grammars1

Normalized tree grammars

[Adapted from R.Downey, M.Fellows]


Kleene s theorem for trees

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]


Kleene s theorem for trees1

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]


Kleene s theorem for trees2

Kleene’s theorem for trees

[Adapted from R.Downey, M.Fellows]


Regular tree expressions

Regular tree expressions

[Adapted from R.Downey, M.Fellows]


Regular tree expressions1

Regular tree expressions

[Adapted from R.Downey, M.Fellows]


Regular tree expressions2

Regular tree expressions

[Adapted from R.Downey, M.Fellows]


Kleene s theorem for trees ii

Kleene’s theorem for trees (II)

[Adapted from R.Downey, M.Fellows]


Kleene s theorem for trees ii1

Kleene’s theorem for trees (II)

[Adapted from R.Downey, M.Fellows]


Equivalence relation for tree languages

Equivalence relation for tree languages

[Adapted from R.Downey, M.Fellows]


Trees myhill nerode theorem

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Trees myhill nerode theorem1

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


Trees myhill nerode theorem2

Trees - Myhill-Nerode theorem

[Adapted from R.Downey, M.Fellows]


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