Instantaneous description - recognizable language, Theory of Computation

De?nition (Instantaneous Description) (for both DFAs and NFAs)

An instantaneous description of A = (Q,Σ, δ, q0, F), either a DFA or an NFA, is a pair hq,wi ∈ Q×Σ*, where q the current state and w is the portion of the input under and to the right of the read head.

The directly computes relation is also the same.

De?nition (Directly Computes Relation) (for both DFAs and NFAs without ε-transitions).

210_recognizable language.png

 

Note that while this is de?ned identically for both DFAs and NFAs, in the case of NFAs it is no longer even partial functional; an ID may well have many successors. Moreover, it is no longer true that the only IDs without successors are those in which w = ε. The effect of this is that the transition function (δ(q, σ)) as we de?ned it earlier is no longer total or even functional: there may be some q and σ for which δ(q, σ) is not de?ned (in notation: δ(q, σ)↑) and there may be some q and σ for which there are multiple states which could be the value of δ(q, σ). We can accommodate both of these possibilities by taking the transition function for NFAs to be a function returning a set of states.

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