Transition graphs, Theory of Computation

We represented SLk automata as Myhill graphs, directed graphs in which the nodes were labeled with (k-1)-factors of alphabet symbols (along with a node labeled ‘?' and one labeled ‘?') and the edges were labeled with individual alphabet symbols. The k-factors of the automaton could be recovered by appending the symbol on an edge to the factor of the node it is incident from. The key value of the graphs is the way that they capture the set of all computations of the automaton in a concise form: every computation of the automaton corresponds to a path through the automaton from ‘?' to ‘?' and vice versa. The su?x substitution closure property is, in essence, a consequence of this fact. All that is signi?cant about the initial portion of a computation is the node it ends on. All strings that lead to the same node are equivalent in the sense that any continuation that extends one of them to form a string that is accepted will extend any of them to form a string that is accepted, and any continuation that leads one of them to be rejected will lead any of them to be rejected.

In adapting this idea for LTk automata, we have to confront the fact that the last k - 1 symbols of the input are no longer enough to characterize the initial portion of a string. We now will also need the record of all k-factors which occurred in that initial portion. To accommodate this, we will extend the labeling of our nodes to include sets of k-factors. The node set will be pairs in which the ?rst component is a k - 1 factor (the last k - 1 symbols of the input) and the second component is a set of k-factors. At the initial node, not having scanned any of the input yet, we have seen no k-factors, that is, the initial set of k-factors is empty (∅). The label of the initial node, then is (?, ∅).

Posted Date: 3/21/2013 3:21:04 AM | Location : United States







Related Discussions:- Transition graphs, Assignment Help, Ask Question on Transition graphs, Get Answer, Expert's Help, Transition graphs Discussions

Write discussion on Transition graphs
Your posts are moderated
Related Questions
These assumptions hold for addition, for instance. Every instance of addition has a unique solution. Each instance is a pair of numbers and the possible solutions include any third

Applying the pumping lemma is not fundamentally di?erent than applying (general) su?x substitution closure or the non-counting property. The pumping lemma is a little more complica

When an FSA is deterministic the set of triples encoding its edges represents a relation that is functional in its ?rst and third components: for every q and σ there is exactly one

The Myhill-Nerode Theorem provided us with an algorithm for minimizing DFAs. Moreover, the DFA the algorithm produces is unique up to isomorphism: every minimal DFA that recognizes

In Exercise 9 you showed that the recognition problem and universal recognition problem for SL2 are decidable. We can use the structure of Myhill graphs to show that other problems

Computer has a single unbounded precision counter which you can only increment, decrement and test for zero. (You may assume that it is initially zero or you may include an explici




The path function δ : Q × Σ* → P(Q) is the extension of δ to strings: This just says that the path labeled ε from any given state q goes only to q itself (or rather never l