Algorithms for hard problems Automata and tree automata Juris Viksna, 2015

Finite deterministic automata initial state accepting state transition [Adapted from P.Drineas]

Finite deterministic automata [Adapted from P.Drineas] Finite Automaton (FA) : set of states : input alphabet : transition function  : Q×   Q : initial state : set of accepting states L(M) = set of all words accepted by M

Finite non-deterministic automata [Adapted from P.Drineas] A word is accepted by NFA, if there exists an accepting path from the initial state to a final state

Finite non-deterministic automata [Adapted from P.Drineas] Set of states, i.e. Input aplhabet, i.e. Transition function  : Q×(   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 q3q3 q1q1 q2q2 0,1 1 0,  q4q4 0,1 1 [Adapted from S.Yukita] Nondeterministic finite automaton M

Some basic results [Adapted from S.Yukita] q 010 q 000 q q 110 q 011 q 001 q 101  q Deterministic finite automaton equivalent to M

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-Nerode theorem [Adapted from R.Downey, M.Fellows]

Myhill’s congruence [Adapted from R.Downey, M.Fellows]

Myhill’s congruence Proof.  L has f.i.   L has f.i. Obvious (from definitions)  L has f.i.   L has f.i. ??? Assume the contrary. Then there are infinite number of  L equivalence classes x 1,x 2,x 3,... Consider all pairs u,v that can be used to show nonequivalence of x i and x j for some i  j by considering ux i v and ux j v. If there is finite number of choices of u in such pairs then there will be u for which we will have an infinite number of  L equivalences classes in form ux i. Otherwise u 1,u 2,u 3,... will form an infinite set of  L equivalences classes.

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]

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]

Regular tree expressions [Adapted from R.Downey, M.Fellows]

Regular tree expressions [Adapted from R.Downey, M.Fellows] X  Y is still forgotten

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) Unfortunately the definition of M(V,j,k) is never clearly explained... The current best guess is that M( ,j,k) refers to RE for reaching state j when processing trees with leaves labelled from . For V  Q the labels are correspondingly from the set V . The final RE is union of all M( ,j,n+1) for j  F. Initial M( ,j,0) is union of RE of both type a  and type a 1  a...  a a s. M(V,j,0) additionally allows for q i instead of a i. [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]