41 lines
1.8 KiB
Markdown
41 lines
1.8 KiB
Markdown
---
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tags: [theory-of-computation, Turing]
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created: Friday, September 13, 2024
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---
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# Turing Completeness
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We know that a [Turing machine](Turing_machines.md) is a theoretical construct
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of a computer that:
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> contains mutable state, executes sequences of simple instructions that read
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> and write that state, and can pick different execution paths depending on the
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> state (via conditional branch instructions.)
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A Turing Complete (TC) system is a system that abides by, or can be reduced to,
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the above description.
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TC also serves as a _definition of computability_ and provides a formal basis
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for conceiving of computation at a theoretical level.
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All Turing Complete systems are functionally equivalent. This means they can
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simulate each other given enough time and memory. Similarly a TC system can in
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principle perform any computation that any other programmable computer can
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perform. This is true for _other_ TC systems and also those that are not TC
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however the inverse doesn't hold: a non-TC system cannot emulate a TS system.
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For instance a calculator cannot do what a TC smart phone can do. But a smart
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phone can act as a calculator.
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This concept of completeness is also expressed in terms of a Universal Turing
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Machine - a TM capable of simulating any other Turing machine. Given that the
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ability of a TM to simulate another TM is the condition for completeness, a TC
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system is also a UTM.
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Completeness applies to the hardware of computers as well as their software.
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Turing Completeness is the theoretical basis of the practical concept of a
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"general-purpose computer": a general-purpose computer is such because it is
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TC - it can in theory compute anything that is computable.
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Most modern programming languages are Turing Complete in that they can, in
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theory, be used to compute anything that is computable.
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