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Automaton (NFA) (with ε-transitions) is a 5-tuple: (Q,Σ, δ, q0, Fi where Q, Σ, q0 and F are as in a DFA and T ⊆ Q × Q × (Σ ∪ {ε}).
We must also modify the de?nitions of the directly computes relation and the path function to allow for the possibility that ε-transitions may occur anywhere in a computation or path. The ε-transition from state 1 to state 3 in the example, for instance, allows the automaton on input ‘a' to go from state 0 not only to state 1 but also to immediately go on to state 3. Similarly, it allows the automaton, when in state 1 with input ‘b', to move ?rst to state 3 and then take the ‘b' edge to state 0 or, when in state 0 with input ‘a', to move ?rst to state 2 and then take the ‘a' edge to state 3. Thus, on a given input ‘σ', the automaton can take any sequence of ε-transitions followed by exactly one σ-transition and then any sequence of ε-transitions. To capture this in the de?nition of δ we start by de?ning the function ε-Closure which, given a state, returns the set of all states reachable from it by any sequence of ε-transitions.
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When we study computability we are studying problems in an abstract sense. For example, addition is the problem of, having been given two numbers, returning a third number that is
Kleene called this the Synthesis theorem because his (and your) proof gives an effective procedure for synthesizing an automaton that recognizes the language denoted by any given r
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Application of the general suffix substitution closure theorem is slightly more complicated than application of the specific k-local versions. In the specific versions, all we had
The fact that regular languages are closed under Boolean operations simpli?es the process of establishing regularity of languages; in essence we can augment the regular operations
The Emptiness Problem is the problem of deciding if a given regular language is empty (= ∅). Theorem 4 (Emptiness) The Emptiness Problem for Regular Languages is decidable. P
To see this, note that if there are any cycles in the Myhill graph of A then L(A) will be infinite, since any such cycle can be repeated arbitrarily many times. Conversely, if the
Theorem The class of ?nite languages is a proper subclass of SL. Note that the class of ?nite languages is closed under union and concatenation but SL is not closed under either. N
A problem is said to be unsolvable if no algorithm can solve it. The problem is said to be undecidable if it is a decision problem and no algorithm can decide it. It should be note
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