Turing machines are automata without a separate output tape. They can not only read the input tape but also write on it.
Furthermore the input tape is infinite at both ends and initially the input word is written on the tape from left to right consecutively and the rest of the cells contain a special symbol, the BLANK symbol.
A Turing machine does not always halt. Due to its infinite tape the Turing machine never steps off it. The Turing machine can only halt due to partiality, so its delta function is always partial.
In this case the partiality of the delta function is not a mistake or a problem to be solved but a good property.
b Definícióc [Turing Machine] A formal septuple is called a Turing automaton where
• K: is the finite set of states,
• V: is the input alphabet, namely the alphabet of the language to be recognized,
• W: is the output alphabet, namely the alphabet of symbols that can be written on the tape, where ,
• : is the state transition function, ,
• : is the initial state of the automaton, ,
• B: is the blank symbol, ,
• F: is the finite set of accepting states .
The functioning of the automaton can be divided up to the following steps. Initially the automaton is in initial state and the input tape contains symbols of an input word from left to right consecutively and the write-read head is over the leftmost cell of the tape.
When functioning the read head reads a symbol from the input tape, then considering the symbol and the actual state it transits state according to the delta function. Then it writes a symbol back on the tape and moves the head left or right.
The automaton halts if the function is not interpreted with the actual input symbol and state.
Although the Turing machine reads a symbol from the input tape in each step, it can not step off it , since it is infinite in both directions. Thus the automaton does not surely halt after n steps.
1.1. Configuration of Turing Machines, Computational Process, Recognition
b Definícióc [Configuration of Turing Machines] The configuration of a Turing machine is a formal triplet where is the word left from the write-read head on the input tape , q is the actual state and is the word under and right from the write-read head.
The head is over the first character of word . Words , do not contain the blank symbol, so before and after the tape only contains BLANK symbols.
Turing automatons
Based on the configuration of the automaton we can know every important information regarding the state of the automaton and based on that, after reloading the configuration the automaton can continue the process.
The initial configuration of a Turing machine is the triplet and is the input word. The configuration when functioning is some triplet. The terminal configuration is some triplet.
The delta function is interpreted on a couplet . It consists of form lines, where it assigns a new state to the couplet and writes a symbol on the tape , and defines if the head should move left or right.
b Definícióc [Computational Process of Turing Machines] A configuration of a Turing machine is a halting configuration if the function is not interpreted on the couplet.
A sequence of configurations, whose first element is the initial configuration and every other element is the result of the previous one so that the function is implemented on it, is called computational process.
b Definícióc [Accepting State] A computational process is a complete computational process if the last configuration of the sequence is a halting configuration, and the Turing machine recognizes (accepts) a word, and if the initial configuration can be transited into one halting configuration through a complete computational process, and .
Anyway, the definition above can be well implemented with both deterministic and nondeterministic cases.
There is no point in differentiating partial and non-partial cases because a Turing machine can not be complete.
1.2. Analyzing the Functioning of a Turing Machine
A Turing machine does not always stop. It can run into an infinite loop moving alternately between two cells of the input tape. Moreover, it can move right (or left) infinitely after reading a symbol.
If the delta function is not partial, the Turing machine will never halt.
The word written on the infinite tape by the Turing machine is not necessarily consecutive. There can be BLANK cells in between the written word or word parts.
Let us examine some possible situations when the automaton is functioning and we give it a correct input word.
• , it halts after some (even less than n) steps and it is in an accepting state.
• The automaton halts after some steps but not in an accepting state (it is nondeterministic and chose a wrong path).
• The automaton does not halt (it is nondeterministic with a wrong path).
Let us also consider some cases when the input word is wrong.
• The automaton halts after some steps but not in an accepting state (both in nondeterministic and deterministic cases).
• The automaton does not halt.
Now, let us analyze a concrete example which reveals how a word is analyzed by a Turing machine that is constructed from the following elements: The input alphabet of the automaton , the set of states , and the set of symbols that can be written on the tape , and the set of accepting states that contains one element .
The delta function of the automaton in a table format is the following:
Turing automatons
The delta function does not execute the recognition of words but reflects them in the middle. (the input word is 01011.)
An other definition of the automaton above can be seen in the following image:
b Definícióc [Recursively Enumerable Languages] A language is recursively enumerable if there is a Turing machine that recognizes it and a language is recursive if there is a Turing machine that recognizes it.
The automaton halts even with words that are not in the language.
The class of phrase like languages is the same as the class of recursively enumerable languages so Turing machines are recognizing automata of type-0 languages. According to this a Turing machine can be constructed to every type-0 language and vica versa. The recognizing automata of type-1 languages are special Turing machines.
Turing automatons