Formal languages

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The formation rules are usually recursive. Some rules postulate that such and such expressions belong to the language in question. Some other rules establish how to build well-formed expressions from other expressions belonging to the language. It is assumed that nothing else is a well-formed expression. For example, the language of propositional calculus could be defined as follows.

CS Formal Languages

The formation rules are then that every English letter and every letter with an index is a formula, and if and are formulas, then so are , , , , and. Formation rules are sufficient for defining simple languages. More syntactically complex languages are defined by means of grammars or regular expressions. The formation rules of propositional calculus and most other formation rules can be straightforwardly transformed into grammar productions. For example, the formation rule becomes the production , where is the start symbol. This entry contributed by Alex Sakharov author's link. When the input is exhausted, the DFA halts.

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Characterizing the language. Each DFA recognizes a formal languageā€”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.


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Java implementation. Nondeterministic finite-state automata. The behavior of a DFA is deterministic : for each input symbol and each state, there is exactly one possible state transition. 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. Kleene's theorem. There is a striking connection among regular expressions, DFAs, and NFAs that has dramatic practical and theoretical consequences. RE recognition. Kleene's theorem serves as the basis for a solution to the recognition problem for REs.

Theoretische Informatik

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. Creative Exercises Harvester. Write a Pattern and Matcher client Harvester. Web crawling.

Develop a program WebCrawler. Search and replace. Write a filter SearchAndReplace.

Formal English and informal English

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.

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Write a regular expression for binary strings with at least two 0s but not consecutive 0s. 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. Answer : subzero. Find all English words that start with g, contain the trigraph pev and end with e. Answer : grapevine. 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. Answer : proprietorier. Find all words that contain the four letters a, s, d, and f, not necessarily in that order.

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Solution : cat words. 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.

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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. Yes: 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.

siterecu.ga Write a regular expression to check whether a sequence contains two or more repeats of the the GATA tetranucleotide. Modify Validate. Hint: use the? Write a Java regular expression to match various spellings of Libyan dictator Moammar Gadhafi's last name using the folling template: i starts with K, G, Q, ii optionally followed by H, iii followed by AD, iv optionally followed by D, v optionally followed by H, vi optionally followed by AF, vii optionally followed by F, vii ends with I.

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.


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Creative Exercises Harvester.