Mechanical Computing: The Computational Complexity of Physical Devices

Computing Machine Models To gage the computational power of a family of mechanical computers, we will use a widely known abstract computational model known as the Turing machine, defined in this section. The Turing machine. The Turing machine model formulated by Alan Turing (1937) was the first complete mathematical model of an abstract computing machine that possessed universal computing power. The machine model has (i) a finite-state transition control for logical control of the machine processing, (ii) a tape with a sequence of storage cells containing symbolic values, and (iii) a tape scanner for reading and writing values to and from the tape cells, which could be made to move (left and right) along the tape cells. A machine model is abstract if the description of the machine transition mechanism or memory mechanism does not provide specification of the mechanical apparatus used to implement them in practice. Since Turing’s description did not include any specification of the mechanical mechanism for executing the finite-state transitions, it cannot be viewed as a concrete mechanical computing machine but instead as an abstract machine. Still it is a valuable computational model, due to its simplicity and very widespread use in computational theory. A universal Turing machine simulates any other Turing machine; it takes its input a pair consisting of a string providing a symbolic description of a Turing machine M and the input string x and simulates M on input x. Because of its simplicity and elegance, the Turing machine has come to be the standard computing model used for most theoretical works in computer science. Informally, the Church-Turing hypothesis states that a Turing machine model can simulate a computation by any “reasonable” computational model (we will discuss some other reasonable computational models below).computing machine that possessed universal computing power. The machine model has (i) a finite-state transition control for logical control of the machine processing, (ii) a tape with a sequence of storage cells containing symbolic values, and (iii) a tape scanner for reading and writing values to and from the tape cells, which could be made to move (left and right) along the tape cells. A machine model is abstract if the description of the machine transition mechanism or memory mechanism does not provide specification of the mechanical apparatus used to implement them in practice. Since Turing’s description did not include any specification of the mechanical mechanism for executing the finite-state transitions, it cannot be viewed as a concrete mechanical computing machine but instead as an abstract machine. Still it is a valuable computational model, due to its simplicity and very widespread use in computational theory. A universal Turing machine simulates any other Turing machine; it takes its input a pair consisting of a string providing a symbolic description of a Turing machine M and the input string x and simulates M on input x. Because of its simplicity and elegance, the Turing machine has come to be the standard computing model used for most theoretical works in computer science. Informally, the Church-Turing hypothesis states that a Turing machine model can simulate a computation by any “reasonable” computational model (we will discuss some other reasonable computational models below). Computational problems. A computational problem is given an input string specified by a string over a finite alphabet; determine the Boolean answer: 1 if the answer is yes and otherwise 0. For simplicity, we generally will restrict the input alphabet to be the binary alphabet {0,1}. The input size of a computational problem is the number of input symbols, which is the number of bits of the binary specification of the input. (Note: It is more common to make these definitions in terms of language acceptance. A language is a set of strings over a given finite alphabet of symbols. A computational problem can be identified with the language consisting of all strings over the input alphabet where the answer is 1. For simplicity, we defined each complexity class as the corresponding class of problems.) Recursively computable problems and undecidable problems. There is a large class of problems, known as recursively computable problems, that Turing machines compute in finite computations, that is, always halting in finite time with the answer. There are certain problems that are not recursively computable; these are called undecidable problems. The halting problem is given a Turing machine description and an input, output 1 if the Turing machine ever halts and else output 0. Turing 2 Mechanical Computing: The Computational Complexity of Physical Devices