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We begin with some important definitions. A symbol is our basic building block, typically a character or a digit. An alphabet is a finite set of symbols. A string is a finite sequence of alphabet symbols.
A formal language is a set of strings possibly infinite , all over the same alphabet. Now, we consider some examples. We begin with examples of formal languages over the binary alphabet. The simplest way to specify a formal language is to enumerate its strings. One complication is that languages can be large or infinite sets, so we often use informal descriptions. We use whichever alphabet is suitable to the task at hand when working with formal languages: Here are some example languages over different alphabets: How do we completely and precisely define formal languages?
This task is known as the specification problem for formal languages. Our informal English-language descriptions do the job in some cases but are rather inadequate in others. Given a language L and a string x , the recognition problem is to answer the following question: We now consider an important class of formal languages known as the regular languages , for which we can solve the specification and recognition problems.
We use the union , concatenation , and closure operations on sets, along with parentheses , to specify a regular language. The concatenation RS of two formal languages R and S is the set of all strings that can be created by appending a string from R to a string from S.
Note that each time we take a string from R , we are free to use any string in the set. We use parentheses or rely on a defined operator precedence order to specify in which order should the operators be applied. For REs, closure is performed before concatenation, and concatenation is performed before union. A regular expression RE is a string of symbols that specifies a formal language. Every regular expression is either an alphabet symbol, specifying the singleton set containing that symbol, or composed from the following operations where R and S are REs: R S , specifying the union of the sets R and S , Concatenation: RS , specifying the concatenation of the sets R and S , Closure: R , specifying the same set as R.
A formal language is regular if and only if it can be specified by an RE. Here are some examples: Our definition of REs is a minimal one that includes the four basic operations that characterize regular languages concatenation, union, closure, and parentheses.
In practice, it is useful to make various additions to this set. Generalized REs support a number of shorthand notations, such as the following: A list or range of symbols enclosed in square brackets  matches any symbol in the list or range.
Several escape sequences consisting of a backslash followed by an alphabet symbol match a defined set of symbols. Extensions to the closure operation. Accordingly, Java REs have the following option for specifying restrictions on the number of repetitions of the closure operation: Java includes many more shorthands and extensions that we will not explore.
Regular expressions in Java. If s is any Java String and re is any regular expression, then s. A DFA is an abstract machine that consists of A finite number of states , each of which is designated as either an accept state or a reject state. A set of transitions that specify how the machine changes state.
Each state has one transition for each symbol in the alphabet. A tape reader initially positioned at the first symbol of an input string and capable only of reading a symbol and moving to the next symbol. We represent each DFA as a directed graph where accept states are vertices labeled Yes , reject states are vertices labeled No , and each transition is a directed edge that is labeled with a symbol from the alphabet.
All DFAs start at state 0 with an input string on the tape and the tape head on the leftmost symbol in the input string. The machine operates by reading a symbol, moving the tape head right one position, and then changing state as specified by the transition labeled with the input symbol just read.
When the input is exhausted, the DFA halts. Each DFA recognizes a formal language—the set of all strings it accepts. For example, the above DFA recognizes all binary strings for which the number of b s is a multiple of 3. The behavior of a DFA is deterministic: A nondeterministic finite automaton NFA is the same as a DFA, but with restrictions on the transitions leaving each state removed, so that Multiple transitions labeled with the same symbol are allowed.
Unlabeled state transitions null transitions are allowed. Following a null transition does not consume an input symbol. Not all symbols need be included among the transitions leaving each state. An NFA accepts a string if there is any sequence of transitions that can take the machine from the start state to an accept state. In the NFA at left, when in state 0 and reading an a , it can choose to stay in state 0 or make the transition to state 1.
It recognizes binary strings whose second-to-last symbol is a. In the NFA at right, when it state 0, it can follow the null transition to state 1, without consuming an input symbol. It recognizes binary strings that do not contain the substring bba. There is a striking connection among regular expressions, DFAs, and NFAs that has dramatic practical and theoretical consequences.
Kleene's theorem serves as the basis for a solution to the recognition problem for REs. Simulate the operation of the NFA on the given input string. That is the approach taken by Java to implement its matches method that we considered earlier. Limits on the power of DFAs. For example, the language containing all binary strings with an equal number of a and b symbols is not regular. Machines with more power. One simple way to define a machine that can recognize more languages is to add a pushdown stack to the DFA, yielding a machine known as a pushdown automaton PDA.
It is not difficult to develop a PDA that can recognize binary strings with an equal number of a and b symbols. In the next section, we will consider Turing machines , which is the abstract machine that lies at the heart of computer science. Exercises Give an RE that specifies each of the following languages over the binary alphabet.
All strings except empty string Contains at least three consecutive b s Starts with a and has odd length, or starts with b and has even length No consecutive b s Any string except bb or bbb Starts and ends with the same symbol Contains at least two a s and at most one b Solutions: Write a Pattern and Matcher client Harvester. Develop a program WebCrawler. Write a filter SearchAndReplace.
Write a regular expression for each of the following sets of binary strings. Use only the basic operations. The last one is by far the trickiest.
Write a regular expression for binary strings with at least two 0s but not consecutive 0s. For each of the following, indicate how many bit strings of length exactly are matched by the regular expression: Write a Java regular expression to match phone numbers, with or without area codes.
The area codes should be of the form or Find all English words that end with nym. Final all English words that contain the trigraph bze. Find all English words that start with g, contain the trigraph pev and end with e. Find all English words that contain the trigraph spb and have at least two r's. Find the longest English word that can be written with the top row of a standard keyboard. Find all words that contain the four letters a, s, d, and f, not necessarily in that order.
Write a Java regular expression, for use with Validate. Modify the previous exercise to make the - optional, so that is considered a legal input. Write a Java regular expression to match all strings that contain exactly five vowels and the vowels are in alphabetical order. Write a Java regular expression to match valid OS X file names.
Such a file name consists of any sequence of characters other than a colon. Additionally, it cannot begin with a period. Given a string s that represents the name of an IP address in dotted quad notation, break it up into its constituent pieces, e. Make sure that the four fields are numeric. Write a Java regular expression to describe valid IP addresses of the form a.
Write a Java regular expression to match license plates that start with 4 digits and end with two uppercase letters. Write a regular expression to extract the coding sequence from a DNA string. Write a regular expression to check whether a sequence contains two or more repeats of the the GATA tetranucleotide.
Write a Java regular expression to match various spellings of Libyan dictator Moammar Gadhafi's last name using the folling template: