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the +> state (via conditional branch instructions.) + +A Turing Complete (TC) system is a system that abides by, or can be reduced to, +the above description. + +TC also serves as a _definition of computability_ and provides a formal basis +for conceiving of computation at a theoretical level. + +All Turing Complete systems are functionally equivalent. This means they can +simulate each other given enough time and memory. Similarly a TC system can in +principle perform any computation that any other programmable computer can +perform. This is true for _other_ TC systems and also those that are not TC +however the inverse doesn't hold: a non-TC system cannot emulate a TS system. +For instance a calculator cannot do what a TC smart phone can do. But a smart +phone can act as a calculator. + +Completeness applies to the hardware of computers as well as their software. + +Turing Completeness is the theoretical basis of the practical concept of a +"general-purpose computer": a general-purpose computer is such because it is +TC - it can in theory compute anything that is computable. + +Most modern programming languages are Turing Complete in that they can, in +theory, be used to compute anything that is computable. + +What about Universal Turing Machines eh? diff --git a/zk/Turing_machines.md b/zk/Turing_machines.md index 4ed8e17..79c5452 100644 --- a/zk/Turing_machines.md +++ b/zk/Turing_machines.md @@ -1,19 +1,27 @@ --- tags: - theory-of-computation + - Turing --- ## What is a Turing Machine? -A Turing Machine consists of an infinitely long tape, that has been divided up -into cells. Each cell can contain either a 1, a 0 or an empty space. Above one -cell of the tape is a head, which can either move left or right, and can read -the symbols written in the cells. The head is also capable of erasing symbols -and writing new symbols into the cells. +Turing Machine is a machine that contains mutable state, executes sequences of +simple instructions that read and write that state, and can pick different +execution paths depending on the state (via conditional branch instructions.) -![Turing_machines_01.gif](Turing_machines_01.gif) The direction that the -head moves, which values it erases, and which values it writes in, are dependent -on a set of instructions provided to the machine.0 +This is a general expression, it is typically expanded as follows. + +The machine consists of an infinitely long tape, that has been divided up into +cells. Each cell can contain either a 1, a 0 or an empty space. Above one cell +of the tape is a head, which can either move left or right, and can read the +symbols written in the cells. The head is also capable of erasing symbols and +writing new symbols into the cells. + +![Turing_machines_01.gif](Turing_machines_01.gif) + +The direction that the head moves, which values it erases, and which values it +writes in, are dependent on a set of instructions provided to the machine. Different sets of instructions can be divided into **states.** States are like sub-routines and can themselves feature as part of instructions.