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Automata and natural language theory are topics lying at the heart of computer science. Both are linked to computational complexity and together, these.
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However some of the exercises require deeper understanding of the subject and some sophistication. Most of the questions are about regular languages and finite automata, and context-free languages and pushdown automata. A small collection of problems concerning various interesting properties of strings is also Included in the section 'miscellaneous'.

There are no problems related to decidability or the complexity of algorithms. The collection can be useful also because there are presently no exercise-books in the theory of automata and formal languages. Request changes or add full text files to a record.

Journal of Automata, Languages and Combinatorics

Email us: wrap warwick. Skip to content Skip to navigation. The Library. Login Admin. Unpublished Preview. Downloads per month over past year. Pumping lemma is used to check whether a grammar is context free or not. Let us take an example and show how it is checked. At first, choose a number n of the pumping lemma. Then vx has only 0s and 1s.


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Then uwy , which would have to be in L , has n 2s, but fewer than n 0s or 1s. A pushdown automaton is a way to implement a context-free grammar in a similar way we design DFA for a regular grammar. A PDA may or may not read an input symbol, but it has to read the top of the stack in every transition. The "turnstile" notation is used for connecting pairs of ID's that represent one or many moves of a PDA.

Regular Languages & Finite Automata (Solved Problem 4)

In final state acceptability, a PDA accepts a string when, after reading the entire string, the PDA is in a final state. From the starting state, we can make moves that end up in a final state with any stack values. The stack values are irrelevant as long as we end up in a final state. Then at state q 2 , if we encounter input 0 and top is Null, we push 0 into stack.

This may iterate. And if we encounter input 1 and top is 0, we pop this 0. Then at state q 3 , if we encounter input 1 and top is 0, we pop this 0.

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This may also iterate. And if we encounter input 1 and top is 0, we pop the top element. At state q 2 , the w is being read. In state q 3 , each 0 or 1 is popped when it matches the input. If any other input is given, the PDA will go to a dead state. If a grammar G is context-free, we can build an equivalent nondeterministic PDA which accepts the language that is produced by the context-free grammar G.

A parser can be built for the grammar G. Also, if P is a pushdown automaton, an equivalent context-free grammar G can be constructed where. Parsing is used to derive a string using the production rules of a grammar. It is used to check the acceptability of a string. Compiler is used to check whether or not a string is syntactically correct. A parser takes the inputs and builds a parse tree. Pop the non-terminal on the left hand side of the production at the top of the stack and push its right-hand side string. A Turing Machine is an accepting device which accepts the languages recursively enumerable set generated by type 0 grammars.

It was invented in by Alan Turing. A Turing Machine TM is a mathematical model which consists of an infinite length tape divided into cells on which input is given. It consists of a head which reads the input tape.

A state register stores the state of the Turing machine. After reading an input symbol, it is replaced with another symbol, its internal state is changed, and it moves from one cell to the right or left. If the TM reaches the final state, the input string is accepted, otherwise rejected. The following table shows a comparison of how a Turing machine differs from Finite Automaton and Pushdown Automaton.

Here the transition 1Rq 1 implies that the write symbol is 1, the tape moves right, and the next state is q 1.

CS 301 Languages and Automata

Similarly, the transition 1Lq 2 implies that the write symbol is 1, the tape moves left, and the next state is q 2. For a Turing machine, the time complexity refers to the measure of the number of times the tape moves when the machine is initialized for some input symbols and the space complexity is the number of cells of the tape written.

A TM accepts a language if it enters into a final state for any input string w. A language is recursively enumerable generated by Type-0 grammar if it is accepted by a Turing machine. A TM decides a language if it accepts it and enters into a rejecting state for any input not in the language. A language is recursive if it is decided by a Turing machine. There may be some cases where a TM does not stop. Such TM accepts the language, but it does not decide it. The basic guidelines of designing a Turing machine have been explained below with the help of a couple of examples.

Hence q 2 is the only accepting state. Let us assume that the input string is terminated by a blank symbol, B, at each end of the string. If M is in q 0 , on reading 0, it moves right, enters the state q 1 and erases 0.


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On reading 1, it enters the state q 2 and moves right. If M is in q 1 , on reading 0, it moves right and erases 0, i. On reaching the leftmost 1, it enters q 2 and moves right. If it reaches B, i. If M is in q 2 , on reading either 0 or 1, it moves right.

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On reaching B, it moves left and enters the state q 4. If M is in q 3 , it replaces B by 0, moves left and reaches the final state q f. If M is in q 4 , on reading either 0 or 1, it moves left. On reaching the beginning of the string, i. Multi-tape Turing Machines have multiple tapes where each tape is accessed with a separate head. Each head can move independently of the other heads. Initially the input is on tape 1 and others are blank. At first, the first tape is occupied by the input and the other tapes are kept blank. Next, the machine reads consecutive symbols under its heads and the TM prints a symbol on each tape and moves its heads.

Multi-track Turing machines, a specific type of Multi-tape Turing machine, contain multiple tracks but just one tape head reads and writes on all tracks. Here, a single tape head reads n symbols from n tracks at one step. It accepts recursively enumerable languages like a normal single-track single-tape Turing Machine accepts. So, here the transitions are not deterministic. The computation of a non-deterministic Turing Machine is a tree of configurations that can be reached from the start configuration.

An input is accepted if there is at least one node of the tree which is an accept configuration, otherwise it is not accepted. If all branches of the computational tree halt on all inputs, the non-deterministic Turing Machine is called a Decider and if for some input, all branches are rejected, the input is also rejected. A Turing Machine with a semi-infinite tape has a left end but no right end.

The left end is limited with an end marker. In each step, it reads the symbol on the tape under its head. It writes a new symbol on that tape cell and then it moves the head either into left or right one tape cell. A transition function determines the actions to be taken. It has two special states called accept state and reject state. If at any point of time it enters into the accepted state, the input is accepted and if it enters into the reject state, the input is rejected by the TM. In some cases, it continues to run infinitely without being accepted or rejected for some certain input symbols.

A linear bounded automaton is a multi-track non-deterministic Turing machine with a tape of some bounded finite length.


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The computation is restricted to the constant bounded area. The input alphabet contains two special symbols which serve as left end markers and right end markers which mean the transitions neither move to the left of the left end marker nor to the right of the right end marker of the tape.