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Chapter 1: Problem 72

Let \(M_{1}\) and \(M_{2}\) be DFAs that have \(k_{1}\) and \(k_{2}\) states, respectively, and then let \(U=L\left(M_{1}\right) \cup L\left(M_{2}\right)\). a. Show that if \(U \neq \emptyset\), then \(U\) contains some string \(s\), where \(|s|<\max \left(k_{1}, k_{2}\right)\). b. Show that if \(U \neq \Sigma^{*}\), then \(U\) excludes some string \(s\), where \(|s|

Short Answer

Expert verified
In summary, for a non-empty union \(U=L(M_{1}) \cup L(M_{2})\), there exists a string \(s\) with \(|s|<\max \left(k_{1}, k_{2}\right)\), and if \(U \neq \Sigma^{*}\), there exists a string \(s\) with \(|s|

Step by step solution

01

Understanding the problem

First, make sure you understand each symbol and concept. DFAs are like machines that process input sequences and produce outputs. They have a finite number of states \(k\). Languages of DFAs are set of strings that DFA accepts. Union of two languages is the set of all strings that are accepted by at least one of the two DFAs.
02

Proving existence of a string in U smaller than max(k1, k2)

If \(U \neq \emptyset\), this means there are some strings accepted by \(M_{1}\) or \(M_{2}\). According to the pigeonhole principle, if a DFA has \(k\) states, it must accept a string \(s\) where \(|s|
03

Proving the exclusion of a string in U smaller than k1*k2

Now suppose \(U \neq \Sigma^{*}\) which means there is at least one string not in the language accepted by either \(M_{1}\) or \(M_{2}\). If every string with length less than \(k_{1}*k_{2}\) was included, then a super DFA \(M_{3}\) that includes all state transitions from both \(M_{1}\) and \(M_{2}\) can be created, having at most \(k_{1}*k_{2}\) states because each state in \(M_{3}\) represents a different pair of states from \(M_{1}\) and \(M_{2}\). Since \(U \neq \Sigma^{*}\) there must exist a string that \(M_{3}\) doesn't accept, which, by the pigeonhole principle, must be of a length less than or equal to the number of states in \(M_{3}\), in other words \(|s|

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Deterministic Finite Automata (DFA)
Deterministic Finite Automata (DFA) are foundational in the study of computational theory. They are abstract machines used to recognize patterns within input strings. A DFA consists of a finite set of states, and by reading a string of symbols, it transitions from one state to another according to a set of rules called a transition function.

When processing an input, a DFA starts in a specific state, called the start state, and processes the string symbol by symbol. If the string ends and the DFA is in an accepting state, the input is recognized; otherwise, it is rejected. This accept or reject mechanism is used to determine whether the string belongs to the language defined by that DFA. With its deterministic nature, for every state and input symbol, there is exactly one state the DFA can move to, ensuring predictability and consistency in computation.
Pigeonhole Principle
The pigeonhole principle is a simple yet profound mathematical concept used to reason about partitions and the distribution of objects. It states that if you have more objects than containers to put them in, at least one container must hold more than one object.

In the context of computational theory, we can think of the states of a DFA as containers and the input symbols of a string as objects. If a DFA has a finite number of states and processes an input string longer than the number of states, then the DFA must revisit at least one state - creating what we can refer to as a 'loop' or cycle. This is crucial when proving properties about languages recognized by automata because it guarantees the existence of certain types of strings within or outside a language, as demonstrated in the textbook exercise.
Union of Languages
The union of languages is a fundamental operation in formal language theory, akin to the logical 'OR'. For two languages, say Language A and Language B, their union Language C consists of all the strings that are either in Language A or in Language B, or in both.

The union operation is crucial when we work with automata like DFAs because it allows building a language that encompasses the flexibility of multiple input descriptions. Therefore, when combining two DFAs to perform a union operation, we create a new set of rules that capture the essential behavior of both underlying automata. This new DFA will accept a string if at least one of the original DFAs would have accepted it, illustrating the expansive nature of the union operation.
State Transition Graph
A state transition graph visually represents the behavior of a DFA. It consists of nodes representing states, and directed edges labeled with input symbols that depict the transitions from one state to another. The graph starts at the initial state and has specially marked nodes for accepting (final) states.

The graph is a powerful tool to intuitively understand how a DFA processes input strings. For example, by following a path along the edges, one can determine whether a particular string will be accepted or not. Additionally, cycles in the graph are indicators of repetitive patterns and can be crucial in proving the lengths of strings that can or cannot be accepted by the DFA. Visualizing a DFA's operation through its state transition graph can make understanding the automaton's language conceptually easier.

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Most popular questions from this chapter

Convert the following regular expressions to NFAs using the procedure given in Theorem 1.54. In all parts, \(\Sigma=\\{\mathrm{a}, \mathrm{b}\\}\). a. \(\mathrm{a}(\mathrm{abb})^{*} \cup \mathrm{b}\) b. \(a^{+} \cup(a b)^{+}\) c. \(\left(\mathrm{a} \cup \mathrm{b}^{+}\right) \mathrm{a}^{+} \mathrm{b}^{+}\)

a. Show that if \(M\) is a DFA that recognizes language \(B\), swapping the accept and nonaccept states in \(M\) yields a new DFA recognizing the complement of \(B\). Conclude that the class of regular languages is closed under complement. b. Show by giving an example that if \(M\) is an NFA that recognizes language \(C\), swapping the accept and nonaccept states in \(M\) doesn't necessarily yield a new NFA that recognizes the complement of \(C\). Is the class of languages recognized by NFAs closed under complement? Explain your answer.

Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts, \(\Sigma=\\{\mathrm{a}, \mathrm{b}\\} .\) a. \(\\{w \mid w\) has at least three a's and at least two b's \({ }^{\text {A }} \mathbf{b} .\\{w \mid w\) has exactly two a's and at least two b's \(\\}\) c. \(\\{w \mid w\) has an even number of a's and one or two b's \(\\}\) A. \(\\{w \mid w\) has an even number of a's and each a is followed by at least one b \(\\}\) e. \(\\{w \mid w\) starts with an a and has at most one b \(\\}\) f. \(\\{w \mid w\) has an odd number of a's and ends with a b \(\\}\) g. \(\\{w \mid w\) has even length and an odd number of a's \(\\}\)

Let \(N\) be an NFA with \(k\) states that recognizes some language \(A\). a. Show that if \(A\) is nonempty, \(A\) contains some string of length at most \(k\). b. Show, by giving an example, that part (a) is not necessarily true if you replace both \(A\) 's by \(\bar{A}\). c. Show that if \(\bar{A}\) is nonempty, \(\bar{A}\) contains some string of length at most \(2^{k}\). d. Show that the bound given in part (c) is nearly tight; that is, for each \(k\), demonstrate an NFA recognizing a language \(A_{k}\) where \(\overline{A_{k}}\) is nonempty and where \(\overline{A_{k}}\) 's shortest member strings are of length exponential in \(k\). Come as close to the bound in (c) as you can.

a. Let \(A\) be an infinite regular language. Prove that \(A\) can be split into two infinite disjoint regular subsets. b. Let \(B\) and \(D\) be two languages. Write \(B \Subset D\) if \(B \subseteq D\) and \(D\) contains infinitely many strings that are not in \(B\). Show that if \(B\) and \(D\) are two regular languages where \(B \Subset D\), then we can find a regular language \(C\) where \(B \Subset C \Subset D\).
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