[LAC] Equivalence of DFA and NFA

It is a surprising fact that every language that can be described by some NFA can also be described by some DFA. Moreover, the DFA has more states than NFA. The smallest DFA can have 2^n states while the smallest NFA for the same language has only n states.

We can thus convert an NFA into a DFA by considering each possible set of NFA states as a single DFA state!

subset construction

The subset construction starts from an NFA N = (Qn, Σ, δn, q0, Fn). Its goal is the description of a DFA D = (Qd, Σ, δd, {q0}, Fd} such that L(D) = L(N).
where
δD(A)(P,x) = U{δ(q,x) | q ∈ P}
FD(A) = {P∈P(Q)|P∩F≠∅}

It might be confused if we just say its definition, i’ll show how the NFA converted to DFA.

Example

The language is same as before–accpets all strings that end in “01”
We have figured out its NFA:
enter image description here

So first, we want to write a complete subset construction of it.
So how many possible states are there in the DFA?
2^3 = 8 states, we use a transition table and write down all the states.
(notice the final or accepting states now are not just one, may be a lot)
It is a ongoing version of transition table.
enter image description here
Then we fill in the table about the result after input a symbol.
enter image description here

ok, by now, what we write is a compelete subset construction including all states

If you write these sets(notice, states are sets now) into a alphabet, we can make a new and more clear transition table:
enter image description here
Of the eight states in the above image, starting in the start state B, we can only reach state B, E and F. The other five states are inaccessible from the start state and may as well not be there. We often can avoid the exponential-time step of construction transition table entries for every subset of states if we perform “lazy evaluation” on the subsets, as follows.

  • Basis: We know for certain that the singleton set consisting only of N’s start state is accessible.
  • Induction: Suppose we have determined that set S of states is accessible; then fro each input symbol a, compute the set of states δd(S, a); we know that these sets of states will also be accessible.

In this way, we can converge the subset construction. We can draw a transition diagram by this converged transition table:
enter image description here

Example question

enter image description here

We can write out the following transition table:
enter image description here

If we simplify it:
enter image description here

评论

发表评论

此博客中的热门博文

[MLE] W2 Multivariate linear regression

图片
Multivariate linear regression What we have talked about in week 1 is linear regression with single feature. For example, in the house pricing predicting problem, we only use the size of house (x) to predict the price (y) . But what if there are other features like age, bedroom numbers and so on? Linear regression with multiple variables is also known as “multivariate linear regression” According to the graph above, we now have new representation symbols: n = number of features m = number of training examples \(x^{(i)}\) = \(i^{th}\) training example \(x^{(i)}_j\) = feature j in \(i^{th}\) training example 1416 is \(x_{1}^{(2)}\) in this case, m is 47 for example, n is 4. Hypothesis function The multivariable form of the hypothesis function accommodating these multiple features is as follows: if we make \(x_0\) to 1, the \(h_{\theta}(x)\) will be \(\theta^Tx\), which is \(\theta\) transpose matrix multiply by x, where \(\theta\) is a vector of 1 by(n+1) dimension an
阅读全文

[MLE] W1 Introduction

图片
Before get started The blog is based on the machine learning course on Coursera, taught by Andrew Ng . From today, i will begin my study trip on Machine learning, which will be labelled as MLE. This course will last 11 weeks and today is week 1. The application of machine learning is everywhere: Database mining Applications can’t program by hand Self-customizing program Understanding human learning(brain, real AI) Introduction What is Machine Learning Two definitions of machine learning are offered: Arthur Samuel described it as: “the field of study that gives computers the ability to learn without being explicitly programmed.” This is an older, informal definition. Tom Mitchell provides a more modern definition: “A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E.” If we raise a example of what is machine learning: If
阅读全文

[AIM] MetaHeuristics

图片
Day2 Outline Metaheuristics -definition Single point based search method(simple stochastic local search) Iterated local search Scheduling Tabu Search Metaheuristic what is a metaheuristic? A metaheuristic is a high-level problem independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms. what is the difference between heuristic and metaheuristic? Heuristic : Heuristics are problem-dependent techniques. As such, they usually are adapted to the problem at hand and they try to take full advantage of the particularities of this problem. However, because they are often too greedy, they usually get trapped in a local optimum and thus fail, in general, to obtain the global optimum solution. Metaheuristic : Meta-heuristics, on the other hand, are problem-independent techniques . As such, they do not take advantage of any specificity of the problem and, therefore, can be used as black boxes . In general, t
阅读全文