Algorithms for hard problems Automata and tree automata

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# Algorithms for hard problems Automata and tree automata - PowerPoint PPT Presentation

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

Juris Viksna, 2013

Finite deterministic automata

initial

state

accepting

state

transition

state

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

Finite non-deterministic automata

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

the initial state to a final state

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

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

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

Some basic results

Nondeterministic finite automaton M

Some basic results

Corresponding deterministic

finite automaton M

Some basic results

NDF with  transitions

Some basic results

Corresponding NDF without  transitions

Regular expressions

Regular languages

Regular languages = languages accepted by DFA/NFA

Regular languages = languages accepted by DFA/NFA

Regular languages = languages accepted by DFA/NFA

Regular languages = languages accepted by DFA/NFA

Congruences

Myhill-Nerode theorem

Myhill-Nerode theorem

Myhill-Nerode theorem

Myhill-Nerode theorem

Myhill’s congruence

Pumping Lemma

Myhill’s congruence

Construction of automata

Construction of automata

Construction of automata

State minimization

State minimization

State minimization - example

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

Tree automata

Tree automata

Tree automata

Tree automata

Tree automata

Tree automata

Tree grammars

Tree grammars

Tree grammars - example

Normalized tree grammars

Normalized tree grammars

Kleene’s theorem for trees

Kleene’s theorem for trees

Kleene’s theorem for trees

Regular tree expressions

Regular tree expressions

Regular tree expressions

Kleene’s theorem for trees (II)

Kleene’s theorem for trees (II)

Equivalence relation for tree languages