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Extra resources for Algorithms of informatics, vol. 1

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19. 2-1 Give a DFA which accepts natural numbers divisible by 9. 2-2 Give a DFA which accepts the language containing all words formed by a. an even number of 0’s and an even number of 1’s, b. an even number of 0’s and an odd number of 1’s, c. an odd number of 0’s and an even number of 1’s, d. an odd number of 0’s and an odd number of 1’s. 2-3 Give a DFA to accept respectively the following languages: L1 = {an bm | n ≥ 1, m ≥ 0}, L2 = {an bm | n ≥ 1, m ≥ 1}, n m L3 = {a b | n ≥ 0, m ≥ 0}, L4 = {an bm | n ≥ 0, m ≥ 1}.

Wk , pk )} (without move on the input tape). A pushdown automaton is deterministic, if for any q ∈ Q and z ∈ W we have • |δ(q, a, z)| ≤ 1, ∀a ∈ Σ ∪ {ε} and • if δ(q, ε, z) = ∅, then δ(q, a, z) = ∅, ∀a ∈ Σ. We can associate to any pushdown automaton a transition table, exactly as in the case of finite automata. The rows of this table are indexed by elements of Q, the columns by elements from Σ ∪ {ε} and W (to each a ∈ Σ ∪ {ε} and z ∈ W will correspond a column). At intersection of row corresponding to state q ∈ Q and column corresponding to a ∈ Σ ∪ {ε} and z ∈ W we will have pairs (w1 , p1 ), .

2. 10 Relations between regular grammars and finite automata. To any regular grammar one may construct an NFA which accepts the language generated by that grammar. Any NFA can be transformed in an equivalent DFA. To any DFA one may construct a regular grammar which generates the language accepted by that DFA. 12 results that the class of regular languages coincides with the class of languages accepted by NFA’s and also with class of languages accepted by DFA’s. The result of these three theorems is illustrated in Fig.

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