This is a sketch of the ideas used to create the Universal Turing Machine and prove the Halting Problem. The machine has n "operational" states plus a Halt state, where n is a positive integer, and one of the n states is distinguished as the starting state. Note that Post’s reformulation of the Turing machine is very much rooted in his Post 1936. Post’s Turing machine has only one kind of symbol and so does not rely on the Turing system of F and E-squares. Post’s Turing machine has a two-way infinite tape. The details of the encodings are found in the TM encoding details section below. The term actually predates the Turing Machine and refers to another notion of computation of functions. Why? De nition 2. • A Turing Machine that writes the maximum number of 1’s for its number of state ∑(n), Rado’s function, which is the maximum number of 1’s left on the tape. 4. This Demonstration shows two different enumerations for Turing machines with a halting state, following the formalism of the busy beaver. The n-state busy beaver game (or BB-n game), introduced in Tibor Radó's 1962 paper, involves a class of Turing machines, each member of which is required to meet the following design specifications: . The game. When it reaches either of these, the content of the tape is irrelevant and all we care about is which kind of halting state was reached. Turing Machines 4.1. A universal Turing machine is a Single tape Turing machine Two-tape Turing machine Reprogrammab le Truing machine None of them 3 U CO4 1 21. Conversely, the halting problem on finite state automata is easily decidable; all finite state automata halt. S(n) maximum number of moves that can be made by n -state halting Turing Machine. The reduced enumeration contains only those machines with the initial transition moving to the right to a state other than the halting and initial state Recall two de nitions from last class: De nition 1. Halting state of Turing machine are: Start and stop Accept and reject Start and reject Reject and allow 2 U CO4 1 20. Turing showed that the set of pairs ( M , w ) such that w is in H( M ) is recursively enumerable but not recursive. This class of languages is called therecursively enumerable languages. Key point. A language is Turing-recognizable if there exists a Turing machine which halts in an accepting state i its input is in the language. The halting problems asks that we determine whether or not a program, given an input, will halt (reach some final state). Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The complete enumeration contains all possible machines for the given number of states and a binary alphabet. 4.2. Post’s Turing machine halts when it reaches a state for which no actions are defined. Let H( M ) be the set of inputs w on which his machine halts. Turing machines can be encoded as strings, and other Turing machines can read those strings to peform \simulations". Turing proved that no algorithm exists that … A “Turing machine” is a device that manipulates symbols on a strip of tape according to a table of rules. The halting problem on usual computers is also decidable. The machine M in Alan Turing's paper accepted by just halting -- there is no final state. using nal state and halting are the same. The halting problem on Turing machines is undecidable. Most often, this question is formulated for ordinary deterministic Turing machines, and one asks whether these machines can reach a special “halting” head state. Thus it's important to specify the model. 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