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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

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]